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class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Matrices"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Matrices</span> </div> </a> <ul id="toc-Matrices-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Linear_systems" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Linear_systems"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Linear systems</span> </div> </a> <ul id="toc-Linear_systems-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Endomorphisms_and_square_matrices" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Endomorphisms_and_square_matrices"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Endomorphisms and square matrices</span> </div> </a> <button aria-controls="toc-Endomorphisms_and_square_matrices-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Endomorphisms and square matrices subsection</span> </button> <ul id="toc-Endomorphisms_and_square_matrices-sublist" class="vector-toc-list"> <li id="toc-Determinant" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Determinant"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.1</span> <span>Determinant</span> </div> </a> <ul id="toc-Determinant-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Eigenvalues_and_eigenvectors" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Eigenvalues_and_eigenvectors"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.2</span> <span>Eigenvalues and eigenvectors</span> </div> </a> <ul id="toc-Eigenvalues_and_eigenvectors-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Duality" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Duality"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>Duality</span> </div> </a> <button aria-controls="toc-Duality-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Duality subsection</span> </button> <ul id="toc-Duality-sublist" class="vector-toc-list"> <li id="toc-Dual_map" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Dual_map"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.1</span> <span>Dual map</span> </div> </a> <ul id="toc-Dual_map-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Inner-product_spaces" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Inner-product_spaces"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.2</span> <span>Inner-product spaces</span> </div> </a> <ul id="toc-Inner-product_spaces-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Relationship_with_geometry" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Relationship_with_geometry"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>Relationship with geometry</span> </div> </a> <ul id="toc-Relationship_with_geometry-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Usage_and_applications" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Usage_and_applications"> <div class="vector-toc-text"> <span class="vector-toc-numb">8</span> <span>Usage and applications</span> </div> </a> <button aria-controls="toc-Usage_and_applications-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Usage and applications subsection</span> </button> <ul id="toc-Usage_and_applications-sublist" class="vector-toc-list"> <li id="toc-Functional_analysis" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Functional_analysis"> <div class="vector-toc-text"> <span class="vector-toc-numb">8.1</span> <span>Functional analysis</span> </div> </a> <ul id="toc-Functional_analysis-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Scientific_computation" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Scientific_computation"> <div class="vector-toc-text"> <span class="vector-toc-numb">8.2</span> <span>Scientific computation</span> </div> </a> <ul id="toc-Scientific_computation-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Geometry_of_ambient_space" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Geometry_of_ambient_space"> <div class="vector-toc-text"> <span class="vector-toc-numb">8.3</span> <span>Geometry of ambient space</span> </div> </a> <ul id="toc-Geometry_of_ambient_space-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Study_of_complex_systems" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Study_of_complex_systems"> <div class="vector-toc-text"> <span class="vector-toc-numb">8.4</span> <span>Study of complex systems</span> </div> </a> <ul id="toc-Study_of_complex_systems-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Fluid_mechanics,_fluid_dynamics,_and_thermal_energy_systems" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Fluid_mechanics,_fluid_dynamics,_and_thermal_energy_systems"> <div class="vector-toc-text"> <span class="vector-toc-numb">8.5</span> <span>Fluid mechanics, fluid dynamics, and thermal energy systems</span> </div> </a> <ul id="toc-Fluid_mechanics,_fluid_dynamics,_and_thermal_energy_systems-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Extensions_and_generalizations" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Extensions_and_generalizations"> <div class="vector-toc-text"> <span class="vector-toc-numb">9</span> <span>Extensions and generalizations</span> </div> </a> <button aria-controls="toc-Extensions_and_generalizations-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Extensions and generalizations subsection</span> </button> <ul id="toc-Extensions_and_generalizations-sublist" class="vector-toc-list"> <li id="toc-Module_theory" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Module_theory"> <div class="vector-toc-text"> <span class="vector-toc-numb">9.1</span> <span>Module theory</span> </div> </a> <ul id="toc-Module_theory-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Multilinear_algebra_and_tensors" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Multilinear_algebra_and_tensors"> <div class="vector-toc-text"> <span class="vector-toc-numb">9.2</span> <span>Multilinear algebra and tensors</span> </div> </a> <ul id="toc-Multilinear_algebra_and_tensors-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Topological_vector_spaces" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Topological_vector_spaces"> <div class="vector-toc-text"> <span class="vector-toc-numb">9.3</span> <span>Topological vector spaces</span> </div> </a> <ul id="toc-Topological_vector_spaces-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-See_also" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#See_also"> <div class="vector-toc-text"> <span class="vector-toc-numb">10</span> <span>See also</span> </div> </a> <ul id="toc-See_also-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Explanatory_notes" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Explanatory_notes"> <div class="vector-toc-text"> <span class="vector-toc-numb">11</span> <span>Explanatory notes</span> </div> </a> <ul id="toc-Explanatory_notes-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Citations" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Citations"> <div class="vector-toc-text"> <span class="vector-toc-numb">12</span> <span>Citations</span> </div> </a> <ul id="toc-Citations-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-General_and_cited_sources" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#General_and_cited_sources"> <div class="vector-toc-text"> <span class="vector-toc-numb">13</span> <span>General and cited sources</span> </div> </a> <ul id="toc-General_and_cited_sources-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Further_reading" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Further_reading"> <div class="vector-toc-text"> <span class="vector-toc-numb">14</span> <span>Further reading</span> </div> </a> <button aria-controls="toc-Further_reading-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Further reading subsection</span> </button> <ul id="toc-Further_reading-sublist" class="vector-toc-list"> <li id="toc-History_2" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#History_2"> <div class="vector-toc-text"> <span class="vector-toc-numb">14.1</span> <span>History</span> </div> </a> <ul id="toc-History_2-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Introductory_textbooks" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Introductory_textbooks"> <div class="vector-toc-text"> <span class="vector-toc-numb">14.2</span> <span>Introductory textbooks</span> </div> </a> <ul id="toc-Introductory_textbooks-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Advanced_textbooks" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Advanced_textbooks"> <div class="vector-toc-text"> <span class="vector-toc-numb">14.3</span> <span>Advanced textbooks</span> </div> </a> <ul id="toc-Advanced_textbooks-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Study_guides_and_outlines" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Study_guides_and_outlines"> <div class="vector-toc-text"> <span class="vector-toc-numb">14.4</span> <span>Study guides and outlines</span> </div> </a> <ul id="toc-Study_guides_and_outlines-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-External_links" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#External_links"> <div class="vector-toc-text"> <span class="vector-toc-numb">15</span> <span>External links</span> </div> </a> <button aria-controls="toc-External_links-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle External links subsection</span> </button> <ul id="toc-External_links-sublist" class="vector-toc-list"> <li id="toc-Online_Resources" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Online_Resources"> <div class="vector-toc-text"> <span class="vector-toc-numb">15.1</span> <span>Online Resources</span> </div> </a> <ul id="toc-Online_Resources-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Online_books" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Online_books"> <div class="vector-toc-text"> <span class="vector-toc-numb">15.2</span> <span>Online books</span> </div> </a> <ul id="toc-Online_books-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> </ul> </div> </div> </nav> </div> </div> <div class="mw-content-container"> <main id="content" class="mw-body"> <header class="mw-body-header vector-page-titlebar"> <nav aria-label="Contents" class="vector-toc-landmark"> <div id="vector-page-titlebar-toc" class="vector-dropdown vector-page-titlebar-toc vector-button-flush-left" title="Table of Contents" > <input type="checkbox" id="vector-page-titlebar-toc-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-titlebar-toc" class="vector-dropdown-checkbox " aria-label="Toggle the table of contents" > <label id="vector-page-titlebar-toc-label" for="vector-page-titlebar-toc-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" ><span class="vector-icon mw-ui-icon-listBullet mw-ui-icon-wikimedia-listBullet"></span> <span class="vector-dropdown-label-text">Toggle the table of contents</span> </label> <div class="vector-dropdown-content"> <div id="vector-page-titlebar-toc-unpinned-container" class="vector-unpinned-container"> </div> </div> </div> </nav> <h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">Linear algebra</span></h1> <div id="p-lang-btn" class="vector-dropdown mw-portlet mw-portlet-lang" > <input type="checkbox" id="p-lang-btn-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-p-lang-btn" class="vector-dropdown-checkbox mw-interlanguage-selector" aria-label="Go to an article in another language. Available in 94 languages" > <label id="p-lang-btn-label" for="p-lang-btn-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--action-progressive mw-portlet-lang-heading-94" aria-hidden="true" ><span class="vector-icon mw-ui-icon-language-progressive mw-ui-icon-wikimedia-language-progressive"></span> <span class="vector-dropdown-label-text">94 languages</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-af mw-list-item"><a href="https://af.wikipedia.org/wiki/Line%C3%AAre_algebra" title="Lineêre algebra – Afrikaans" lang="af" hreflang="af" data-title="Lineêre algebra" data-language-autonym="Afrikaans" data-language-local-name="Afrikaans" class="interlanguage-link-target"><span>Afrikaans</span></a></li><li class="interlanguage-link interwiki-als mw-list-item"><a href="https://als.wikipedia.org/wiki/Lineare_Algebra" title="Lineare Algebra – Alemannic" lang="gsw" hreflang="gsw" data-title="Lineare Algebra" data-language-autonym="Alemannisch" data-language-local-name="Alemannic" class="interlanguage-link-target"><span>Alemannisch</span></a></li><li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D8%AC%D8%A8%D8%B1_%D8%AE%D8%B7%D9%8A" title="جبر خطي – Arabic" lang="ar" hreflang="ar" data-title="جبر خطي" data-language-autonym="العربية" data-language-local-name="Arabic" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-an mw-list-item"><a href="https://an.wikipedia.org/wiki/Alchebra_lineal" title="Alchebra lineal – Aragonese" lang="an" hreflang="an" data-title="Alchebra lineal" data-language-autonym="Aragonés" data-language-local-name="Aragonese" class="interlanguage-link-target"><span>Aragonés</span></a></li><li class="interlanguage-link interwiki-ast mw-list-item"><a href="https://ast.wikipedia.org/wiki/%C3%81lxebra_llinial" title="Álxebra llinial – Asturian" lang="ast" hreflang="ast" data-title="Álxebra llinial" data-language-autonym="Asturianu" data-language-local-name="Asturian" class="interlanguage-link-target"><span>Asturianu</span></a></li><li class="interlanguage-link interwiki-az mw-list-item"><a href="https://az.wikipedia.org/wiki/X%C9%99tti_c%C9%99br" title="Xətti cəbr – Azerbaijani" lang="az" hreflang="az" data-title="Xətti cəbr" data-language-autonym="Azərbaycanca" data-language-local-name="Azerbaijani" class="interlanguage-link-target"><span>Azərbaycanca</span></a></li><li class="interlanguage-link interwiki-bn mw-list-item"><a href="https://bn.wikipedia.org/wiki/%E0%A6%B0%E0%A7%88%E0%A6%96%E0%A6%BF%E0%A6%95_%E0%A6%AC%E0%A7%80%E0%A6%9C%E0%A6%97%E0%A6%A3%E0%A6%BF%E0%A6%A4" title="রৈখিক বীজগণিত – Bangla" lang="bn" hreflang="bn" data-title="রৈখিক বীজগণিত" data-language-autonym="বাংলা" data-language-local-name="Bangla" class="interlanguage-link-target"><span>বাংলা</span></a></li><li class="interlanguage-link interwiki-zh-min-nan mw-list-item"><a href="https://zh-min-nan.wikipedia.org/wiki/S%C3%B2a%E2%81%BF-s%C3%A8ng_t%C4%81i-s%C3%B2%CD%98" title="Sòaⁿ-sèng tāi-sò͘ – Minnan" lang="nan" hreflang="nan" data-title="Sòaⁿ-sèng tāi-sò͘" data-language-autonym="閩南語 / Bân-lâm-gú" data-language-local-name="Minnan" class="interlanguage-link-target"><span>閩南語 / Bân-lâm-gú</span></a></li><li class="interlanguage-link interwiki-ba mw-list-item"><a href="https://ba.wikipedia.org/wiki/%D2%BA%D1%8B%D2%99%D1%8B%D2%A1%D0%BB%D1%8B_%D0%B0%D0%BB%D0%B3%D0%B5%D0%B1%D1%80%D0%B0" title="Һыҙыҡлы алгебра – Bashkir" lang="ba" hreflang="ba" data-title="Һыҙыҡлы алгебра" data-language-autonym="Башҡортса" data-language-local-name="Bashkir" class="interlanguage-link-target"><span>Башҡортса</span></a></li><li class="interlanguage-link interwiki-be mw-list-item"><a href="https://be.wikipedia.org/wiki/%D0%9B%D1%96%D0%BD%D0%B5%D0%B9%D0%BD%D0%B0%D1%8F_%D0%B0%D0%BB%D0%B3%D0%B5%D0%B1%D1%80%D0%B0" title="Лінейная алгебра – Belarusian" lang="be" hreflang="be" data-title="Лінейная алгебра" data-language-autonym="Беларуская" data-language-local-name="Belarusian" class="interlanguage-link-target"><span>Беларуская</span></a></li><li class="interlanguage-link interwiki-be-x-old mw-list-item"><a href="https://be-tarask.wikipedia.org/wiki/%D0%9B%D1%96%D0%BD%D0%B5%D0%B9%D0%BD%D0%B0%D1%8F_%D0%B0%D0%BB%D1%8C%D0%B3%D0%B5%D0%B1%D1%80%D0%B0" title="Лінейная альгебра – Belarusian (Taraškievica orthography)" lang="be-tarask" hreflang="be-tarask" data-title="Лінейная альгебра" data-language-autonym="Беларуская (тарашкевіца)" data-language-local-name="Belarusian (Taraškievica orthography)" class="interlanguage-link-target"><span>Беларуская (тарашкевіца)</span></a></li><li class="interlanguage-link interwiki-bg mw-list-item"><a href="https://bg.wikipedia.org/wiki/%D0%9B%D0%B8%D0%BD%D0%B5%D0%B9%D0%BD%D0%B0_%D0%B0%D0%BB%D0%B3%D0%B5%D0%B1%D1%80%D0%B0" title="Линейна алгебра – Bulgarian" lang="bg" hreflang="bg" data-title="Линейна алгебра" data-language-autonym="Български" data-language-local-name="Bulgarian" class="interlanguage-link-target"><span>Български</span></a></li><li class="interlanguage-link interwiki-bs mw-list-item"><a href="https://bs.wikipedia.org/wiki/Linearna_algebra" title="Linearna algebra – Bosnian" lang="bs" hreflang="bs" data-title="Linearna algebra" data-language-autonym="Bosanski" data-language-local-name="Bosnian" class="interlanguage-link-target"><span>Bosanski</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/%C3%80lgebra_lineal" title="Àlgebra lineal – Catalan" lang="ca" hreflang="ca" data-title="Àlgebra lineal" data-language-autonym="Català" data-language-local-name="Catalan" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-cv mw-list-item"><a href="https://cv.wikipedia.org/wiki/%D0%9B%D0%B8%D0%BD%D0%B8%D0%BB%D0%BB%D0%B5_%D0%B0%D0%BB%D0%B3%D0%B5%D0%B1%D1%80%D0%B0" title="Линилле алгебра – Chuvash" lang="cv" hreflang="cv" data-title="Линилле алгебра" data-language-autonym="Чӑвашла" data-language-local-name="Chuvash" class="interlanguage-link-target"><span>Чӑвашла</span></a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Line%C3%A1rn%C3%AD_algebra" title="Lineární algebra – Czech" lang="cs" hreflang="cs" data-title="Lineární algebra" data-language-autonym="Čeština" data-language-local-name="Czech" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-cy mw-list-item"><a href="https://cy.wikipedia.org/wiki/Algebra_llinol" title="Algebra llinol – Welsh" lang="cy" hreflang="cy" data-title="Algebra llinol" data-language-autonym="Cymraeg" data-language-local-name="Welsh" class="interlanguage-link-target"><span>Cymraeg</span></a></li><li class="interlanguage-link interwiki-da mw-list-item"><a href="https://da.wikipedia.org/wiki/Line%C3%A6r_algebra" title="Lineær algebra – Danish" lang="da" hreflang="da" data-title="Lineær algebra" data-language-autonym="Dansk" data-language-local-name="Danish" class="interlanguage-link-target"><span>Dansk</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Lineare_Algebra" title="Lineare Algebra – German" lang="de" hreflang="de" data-title="Lineare Algebra" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-et mw-list-item"><a href="https://et.wikipedia.org/wiki/Lineaaralgebra" title="Lineaaralgebra – Estonian" lang="et" hreflang="et" data-title="Lineaaralgebra" data-language-autonym="Eesti" data-language-local-name="Estonian" class="interlanguage-link-target"><span>Eesti</span></a></li><li class="interlanguage-link interwiki-el mw-list-item"><a href="https://el.wikipedia.org/wiki/%CE%93%CF%81%CE%B1%CE%BC%CE%BC%CE%B9%CE%BA%CE%AE_%CE%AC%CE%BB%CE%B3%CE%B5%CE%B2%CF%81%CE%B1" title="Γραμμική άλγεβρα – Greek" lang="el" hreflang="el" data-title="Γραμμική άλγεβρα" data-language-autonym="Ελληνικά" data-language-local-name="Greek" class="interlanguage-link-target"><span>Ελληνικά</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/%C3%81lgebra_lineal" title="Álgebra lineal – Spanish" lang="es" hreflang="es" data-title="Álgebra lineal" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-eo mw-list-item"><a href="https://eo.wikipedia.org/wiki/Lineara_algebro" title="Lineara algebro – Esperanto" lang="eo" hreflang="eo" data-title="Lineara algebro" data-language-autonym="Esperanto" data-language-local-name="Esperanto" class="interlanguage-link-target"><span>Esperanto</span></a></li><li class="interlanguage-link interwiki-eu mw-list-item"><a href="https://eu.wikipedia.org/wiki/Aljebra_lineal" title="Aljebra lineal – Basque" lang="eu" hreflang="eu" data-title="Aljebra lineal" data-language-autonym="Euskara" data-language-local-name="Basque" class="interlanguage-link-target"><span>Euskara</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D8%AC%D8%A8%D8%B1_%D8%AE%D8%B7%DB%8C" title="جبر خطی – Persian" lang="fa" hreflang="fa" data-title="جبر خطی" data-language-autonym="فارسی" data-language-local-name="Persian" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-hif mw-list-item"><a href="https://hif.wikipedia.org/wiki/Linear_algebra" title="Linear algebra – Fiji Hindi" lang="hif" hreflang="hif" data-title="Linear algebra" data-language-autonym="Fiji Hindi" data-language-local-name="Fiji Hindi" class="interlanguage-link-target"><span>Fiji Hindi</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Alg%C3%A8bre_lin%C3%A9aire" title="Algèbre linéaire – French" lang="fr" hreflang="fr" data-title="Algèbre linéaire" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-gl mw-list-item"><a href="https://gl.wikipedia.org/wiki/%C3%81lxebra_lineal" title="Álxebra lineal – Galician" lang="gl" hreflang="gl" data-title="Álxebra lineal" data-language-autonym="Galego" data-language-local-name="Galician" class="interlanguage-link-target"><span>Galego</span></a></li><li class="interlanguage-link interwiki-gan mw-list-item"><a href="https://gan.wikipedia.org/wiki/%E7%B7%9A%E6%80%A7%E4%BB%A3%E6%95%B8" title="線性代數 – Gan" lang="gan" hreflang="gan" data-title="線性代數" data-language-autonym="贛語" data-language-local-name="Gan" class="interlanguage-link-target"><span>贛語</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EC%84%A0%ED%98%95%EB%8C%80%EC%88%98%ED%95%99" title="선형대수학 – Korean" lang="ko" hreflang="ko" data-title="선형대수학" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-hy mw-list-item"><a href="https://hy.wikipedia.org/wiki/%D4%B3%D5%AE%D5%A1%D5%B5%D5%AB%D5%B6_%D5%B0%D5%A1%D5%B6%D6%80%D5%A1%D5%B0%D5%A1%D5%B7%D5%AB%D5%BE" title="Գծային հանրահաշիվ – Armenian" lang="hy" hreflang="hy" data-title="Գծային հանրահաշիվ" data-language-autonym="Հայերեն" data-language-local-name="Armenian" class="interlanguage-link-target"><span>Հայերեն</span></a></li><li class="interlanguage-link interwiki-hi mw-list-item"><a href="https://hi.wikipedia.org/wiki/%E0%A4%B0%E0%A5%88%E0%A4%96%E0%A4%BF%E0%A4%95_%E0%A4%AC%E0%A5%80%E0%A4%9C%E0%A4%97%E0%A4%A3%E0%A4%BF%E0%A4%A4" title="रैखिक बीजगणित – Hindi" lang="hi" hreflang="hi" data-title="रैखिक बीजगणित" data-language-autonym="हिन्दी" data-language-local-name="Hindi" class="interlanguage-link-target"><span>हिन्दी</span></a></li><li class="interlanguage-link interwiki-hr mw-list-item"><a href="https://hr.wikipedia.org/wiki/Linearna_algebra" title="Linearna algebra – Croatian" lang="hr" hreflang="hr" data-title="Linearna algebra" data-language-autonym="Hrvatski" data-language-local-name="Croatian" class="interlanguage-link-target"><span>Hrvatski</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Aljabar_linear" title="Aljabar linear – Indonesian" lang="id" hreflang="id" data-title="Aljabar linear" data-language-autonym="Bahasa Indonesia" data-language-local-name="Indonesian" class="interlanguage-link-target"><span>Bahasa Indonesia</span></a></li><li class="interlanguage-link interwiki-ia mw-list-item"><a href="https://ia.wikipedia.org/wiki/Algebra_linear" title="Algebra linear – Interlingua" lang="ia" hreflang="ia" data-title="Algebra linear" data-language-autonym="Interlingua" data-language-local-name="Interlingua" class="interlanguage-link-target"><span>Interlingua</span></a></li><li class="interlanguage-link interwiki-is mw-list-item"><a href="https://is.wikipedia.org/wiki/L%C3%ADnuleg_algebra" title="Línuleg algebra – Icelandic" lang="is" hreflang="is" data-title="Línuleg algebra" data-language-autonym="Íslenska" data-language-local-name="Icelandic" class="interlanguage-link-target"><span>Íslenska</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Algebra_lineare" title="Algebra lineare – Italian" lang="it" hreflang="it" data-title="Algebra lineare" data-language-autonym="Italiano" data-language-local-name="Italian" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%90%D7%9C%D7%92%D7%91%D7%A8%D7%94_%D7%9C%D7%99%D7%A0%D7%99%D7%90%D7%A8%D7%99%D7%AA" title="אלגברה ליניארית – Hebrew" lang="he" hreflang="he" data-title="אלגברה ליניארית" data-language-autonym="עברית" data-language-local-name="Hebrew" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-ka mw-list-item"><a href="https://ka.wikipedia.org/wiki/%E1%83%AC%E1%83%A0%E1%83%A4%E1%83%98%E1%83%95%E1%83%98_%E1%83%90%E1%83%9A%E1%83%92%E1%83%94%E1%83%91%E1%83%A0%E1%83%90" title="წრფივი ალგებრა – Georgian" lang="ka" hreflang="ka" data-title="წრფივი ალგებრა" data-language-autonym="ქართული" data-language-local-name="Georgian" class="interlanguage-link-target"><span>ქართული</span></a></li><li class="interlanguage-link interwiki-kk mw-list-item"><a href="https://kk.wikipedia.org/wiki/%D0%A1%D1%8B%D0%B7%D1%8B%D2%9B%D1%82%D1%8B%D2%9B_%D0%B0%D0%BB%D0%B3%D0%B5%D0%B1%D1%80%D0%B0" title="Сызықтық алгебра – Kazakh" lang="kk" hreflang="kk" data-title="Сызықтық алгебра" data-language-autonym="Қазақша" data-language-local-name="Kazakh" class="interlanguage-link-target"><span>Қазақша</span></a></li><li class="interlanguage-link interwiki-sw mw-list-item"><a href="https://sw.wikipedia.org/wiki/Aljebra_mstari" title="Aljebra mstari – Swahili" lang="sw" hreflang="sw" data-title="Aljebra mstari" data-language-autonym="Kiswahili" data-language-local-name="Swahili" class="interlanguage-link-target"><span>Kiswahili</span></a></li><li class="interlanguage-link interwiki-ht mw-list-item"><a href="https://ht.wikipedia.org/wiki/Alj%C3%A8b_liney%C3%A8" title="Aljèb lineyè – Haitian Creole" lang="ht" hreflang="ht" data-title="Aljèb lineyè" data-language-autonym="Kreyòl ayisyen" data-language-local-name="Haitian Creole" class="interlanguage-link-target"><span>Kreyòl ayisyen</span></a></li><li class="interlanguage-link interwiki-gcr mw-list-item"><a href="https://gcr.wikipedia.org/wiki/Alj%C3%A8b_lin%C3%A9y%C3%A8r" title="Aljèb linéyèr – Guianan Creole" lang="gcr" hreflang="gcr" data-title="Aljèb linéyèr" data-language-autonym="Kriyòl gwiyannen" data-language-local-name="Guianan Creole" class="interlanguage-link-target"><span>Kriyòl gwiyannen</span></a></li><li class="interlanguage-link interwiki-la mw-list-item"><a href="https://la.wikipedia.org/wiki/Algebra_linearis" title="Algebra linearis – Latin" lang="la" hreflang="la" data-title="Algebra linearis" data-language-autonym="Latina" data-language-local-name="Latin" class="interlanguage-link-target"><span>Latina</span></a></li><li class="interlanguage-link interwiki-lv mw-list-item"><a href="https://lv.wikipedia.org/wiki/Line%C4%81r%C4%81_algebra" title="Lineārā algebra – Latvian" lang="lv" hreflang="lv" data-title="Lineārā algebra" data-language-autonym="Latviešu" data-language-local-name="Latvian" class="interlanguage-link-target"><span>Latviešu</span></a></li><li class="interlanguage-link interwiki-lez mw-list-item"><a href="https://lez.wikipedia.org/wiki/%D0%A6%D3%80%D0%B0%D1%80%D1%86%D3%80%D0%B8%D0%BD_%D0%B0%D0%BB%D0%B3%D0%B5%D0%B1%D1%80%D0%B0" title="ЦӀарцӀин алгебра – Lezghian" lang="lez" hreflang="lez" data-title="ЦӀарцӀин алгебра" data-language-autonym="Лезги" data-language-local-name="Lezghian" class="interlanguage-link-target"><span>Лезги</span></a></li><li class="interlanguage-link interwiki-lt mw-list-item"><a href="https://lt.wikipedia.org/wiki/Tiesin%C4%97_algebra" title="Tiesinė algebra – Lithuanian" lang="lt" hreflang="lt" data-title="Tiesinė algebra" data-language-autonym="Lietuvių" data-language-local-name="Lithuanian" class="interlanguage-link-target"><span>Lietuvių</span></a></li><li class="interlanguage-link interwiki-lfn mw-list-item"><a href="https://lfn.wikipedia.org/wiki/Aljebra_linial" title="Aljebra linial – Lingua Franca Nova" lang="lfn" hreflang="lfn" data-title="Aljebra linial" data-language-autonym="Lingua Franca Nova" data-language-local-name="Lingua Franca Nova" class="interlanguage-link-target"><span>Lingua Franca Nova</span></a></li><li class="interlanguage-link interwiki-lmo mw-list-item"><a href="https://lmo.wikipedia.org/wiki/Algebra_lineara" title="Algebra lineara – Lombard" lang="lmo" hreflang="lmo" data-title="Algebra lineara" data-language-autonym="Lombard" data-language-local-name="Lombard" class="interlanguage-link-target"><span>Lombard</span></a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/Line%C3%A1ris_algebra" title="Lineáris algebra – Hungarian" lang="hu" hreflang="hu" data-title="Lineáris algebra" data-language-autonym="Magyar" data-language-local-name="Hungarian" class="interlanguage-link-target"><span>Magyar</span></a></li><li class="interlanguage-link interwiki-mk mw-list-item"><a href="https://mk.wikipedia.org/wiki/%D0%9B%D0%B8%D0%BD%D0%B5%D0%B0%D1%80%D0%BD%D0%B0_%D0%B0%D0%BB%D0%B3%D0%B5%D0%B1%D1%80%D0%B0" title="Линеарна алгебра – Macedonian" lang="mk" hreflang="mk" data-title="Линеарна алгебра" data-language-autonym="Македонски" data-language-local-name="Macedonian" class="interlanguage-link-target"><span>Македонски</span></a></li><li class="interlanguage-link interwiki-ml mw-list-item"><a href="https://ml.wikipedia.org/wiki/%E0%B4%B0%E0%B5%87%E0%B4%96%E0%B5%80%E0%B4%AF_%E0%B4%AC%E0%B5%80%E0%B4%9C%E0%B4%97%E0%B4%A3%E0%B4%BF%E0%B4%A4%E0%B4%82" title="രേഖീയ ബീജഗണിതം – Malayalam" lang="ml" hreflang="ml" data-title="രേഖീയ ബീജഗണിതം" data-language-autonym="മലയാളം" data-language-local-name="Malayalam" class="interlanguage-link-target"><span>മലയാളം</span></a></li><li class="interlanguage-link interwiki-ms mw-list-item"><a href="https://ms.wikipedia.org/wiki/Algebra_linear" title="Algebra linear – Malay" lang="ms" hreflang="ms" data-title="Algebra linear" data-language-autonym="Bahasa Melayu" data-language-local-name="Malay" class="interlanguage-link-target"><span>Bahasa Melayu</span></a></li><li class="interlanguage-link interwiki-mn mw-list-item"><a href="https://mn.wikipedia.org/wiki/%D0%A8%D1%83%D0%B3%D0%B0%D0%BC%D0%B0%D0%BD_%D0%B0%D0%BB%D0%B3%D0%B5%D0%B1%D1%80" title="Шугаман алгебр – Mongolian" lang="mn" hreflang="mn" data-title="Шугаман алгебр" data-language-autonym="Монгол" data-language-local-name="Mongolian" class="interlanguage-link-target"><span>Монгол</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Lineaire_algebra" title="Lineaire algebra – Dutch" lang="nl" hreflang="nl" data-title="Lineaire algebra" data-language-autonym="Nederlands" data-language-local-name="Dutch" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E7%B7%9A%E5%9E%8B%E4%BB%A3%E6%95%B0%E5%AD%A6" title="線型代数学 – Japanese" lang="ja" hreflang="ja" data-title="線型代数学" data-language-autonym="日本語" data-language-local-name="Japanese" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-frr mw-list-item"><a href="https://frr.wikipedia.org/wiki/Lineaar_algebra" title="Lineaar algebra – Northern Frisian" lang="frr" hreflang="frr" data-title="Lineaar algebra" data-language-autonym="Nordfriisk" data-language-local-name="Northern Frisian" class="interlanguage-link-target"><span>Nordfriisk</span></a></li><li class="interlanguage-link interwiki-no mw-list-item"><a href="https://no.wikipedia.org/wiki/Line%C3%A6r_algebra" title="Lineær algebra – Norwegian Bokmål" lang="nb" hreflang="nb" data-title="Lineær algebra" data-language-autonym="Norsk bokmål" data-language-local-name="Norwegian Bokmål" class="interlanguage-link-target"><span>Norsk bokmål</span></a></li><li class="interlanguage-link interwiki-nn mw-list-item"><a href="https://nn.wikipedia.org/wiki/Line%C3%A6r_algebra" title="Lineær algebra – Norwegian Nynorsk" lang="nn" hreflang="nn" data-title="Lineær algebra" data-language-autonym="Norsk nynorsk" data-language-local-name="Norwegian Nynorsk" class="interlanguage-link-target"><span>Norsk nynorsk</span></a></li><li class="interlanguage-link interwiki-oc mw-list-item"><a href="https://oc.wikipedia.org/wiki/Alg%C3%A8bra_lineara" title="Algèbra lineara – Occitan" lang="oc" hreflang="oc" data-title="Algèbra lineara" data-language-autonym="Occitan" data-language-local-name="Occitan" class="interlanguage-link-target"><span>Occitan</span></a></li><li class="interlanguage-link interwiki-uz mw-list-item"><a href="https://uz.wikipedia.org/wiki/Chiziqli_algebra" title="Chiziqli algebra – Uzbek" lang="uz" hreflang="uz" data-title="Chiziqli algebra" data-language-autonym="Oʻzbekcha / ўзбекча" data-language-local-name="Uzbek" class="interlanguage-link-target"><span>Oʻzbekcha / ўзбекча</span></a></li><li class="interlanguage-link interwiki-jam mw-list-item"><a href="https://jam.wikipedia.org/wiki/Linia_haljibra" title="Linia haljibra – Jamaican Creole English" lang="jam" hreflang="jam" data-title="Linia haljibra" data-language-autonym="Patois" data-language-local-name="Jamaican Creole English" class="interlanguage-link-target"><span>Patois</span></a></li><li class="interlanguage-link interwiki-pms mw-list-item"><a href="https://pms.wikipedia.org/wiki/%C3%80lgebra_linear" title="Àlgebra linear – Piedmontese" lang="pms" hreflang="pms" data-title="Àlgebra linear" data-language-autonym="Piemontèis" data-language-local-name="Piedmontese" class="interlanguage-link-target"><span>Piemontèis</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Algebra_liniowa" title="Algebra liniowa – Polish" lang="pl" hreflang="pl" data-title="Algebra liniowa" data-language-autonym="Polski" data-language-local-name="Polish" class="interlanguage-link-target"><span>Polski</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/%C3%81lgebra_linear" title="Álgebra linear – Portuguese" lang="pt" hreflang="pt" data-title="Álgebra linear" data-language-autonym="Português" data-language-local-name="Portuguese" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-ro mw-list-item"><a href="https://ro.wikipedia.org/wiki/Algebr%C4%83_liniar%C4%83" title="Algebră liniară – Romanian" lang="ro" hreflang="ro" data-title="Algebră liniară" data-language-autonym="Română" data-language-local-name="Romanian" class="interlanguage-link-target"><span>Română</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%9B%D0%B8%D0%BD%D0%B5%D0%B9%D0%BD%D0%B0%D1%8F_%D0%B0%D0%BB%D0%B3%D0%B5%D0%B1%D1%80%D0%B0" title="Линейная алгебра – Russian" lang="ru" hreflang="ru" data-title="Линейная алгебра" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-sco mw-list-item"><a href="https://sco.wikipedia.org/wiki/Linear_algebra" title="Linear algebra – Scots" lang="sco" hreflang="sco" data-title="Linear algebra" data-language-autonym="Scots" data-language-local-name="Scots" class="interlanguage-link-target"><span>Scots</span></a></li><li class="interlanguage-link interwiki-sq mw-list-item"><a href="https://sq.wikipedia.org/wiki/Algjebra_lineare" title="Algjebra lineare – Albanian" lang="sq" hreflang="sq" data-title="Algjebra lineare" data-language-autonym="Shqip" data-language-local-name="Albanian" class="interlanguage-link-target"><span>Shqip</span></a></li><li class="interlanguage-link interwiki-scn mw-list-item"><a href="https://scn.wikipedia.org/wiki/Algibbra_liniari" title="Algibbra liniari – Sicilian" lang="scn" hreflang="scn" data-title="Algibbra liniari" data-language-autonym="Sicilianu" data-language-local-name="Sicilian" class="interlanguage-link-target"><span>Sicilianu</span></a></li><li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://simple.wikipedia.org/wiki/Linear_algebra" title="Linear algebra – Simple English" lang="en-simple" hreflang="en-simple" data-title="Linear algebra" data-language-autonym="Simple English" data-language-local-name="Simple English" class="interlanguage-link-target"><span>Simple English</span></a></li><li class="interlanguage-link interwiki-sk mw-list-item"><a href="https://sk.wikipedia.org/wiki/Line%C3%A1rna_algebra" title="Lineárna algebra – Slovak" lang="sk" hreflang="sk" data-title="Lineárna algebra" 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href="https://uk.wikipedia.org/wiki/%D0%9B%D1%96%D0%BD%D1%96%D0%B9%D0%BD%D0%B0_%D0%B0%D0%BB%D0%B3%D0%B5%D0%B1%D1%80%D0%B0" title="Лінійна алгебра – Ukrainian" lang="uk" hreflang="uk" data-title="Лінійна алгебра" data-language-autonym="Українська" data-language-local-name="Ukrainian" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-ur mw-list-item"><a href="https://ur.wikipedia.org/wiki/%D8%AE%D8%B7%DB%8C_%D8%A7%D9%84%D8%AC%D8%A8%D8%B1%D8%A7" title="خطی الجبرا – Urdu" lang="ur" hreflang="ur" data-title="خطی الجبرا" data-language-autonym="اردو" data-language-local-name="Urdu" class="interlanguage-link-target"><span>اردو</span></a></li><li class="interlanguage-link interwiki-vi mw-list-item"><a href="https://vi.wikipedia.org/wiki/%C4%90%E1%BA%A1i_s%E1%BB%91_tuy%E1%BA%BFn_t%C3%ADnh" title="Đại số tuyến tính – Vietnamese" lang="vi" hreflang="vi" data-title="Đại số tuyến tính" data-language-autonym="Tiếng Việt" 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<button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-appearance.pin">move to sidebar</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-appearance.unpin">hide</button> </div> </div> </div> </nav> </div> </div> <div id="bodyContent" class="vector-body" aria-labelledby="firstHeading" data-mw-ve-target-container> <div class="vector-body-before-content"> <div class="mw-indicators"> </div> <div id="siteSub" class="noprint">From Wikipedia, the free encyclopedia</div> </div> <div id="contentSub"><div id="mw-content-subtitle"></div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr"><div class="shortdescription nomobile noexcerpt noprint searchaux" style="display:none">Branch of mathematics</div> <figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Linear_subspaces_with_shading.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/2/2f/Linear_subspaces_with_shading.svg/250px-Linear_subspaces_with_shading.svg.png" decoding="async" width="250" height="182" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/2f/Linear_subspaces_with_shading.svg/375px-Linear_subspaces_with_shading.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/2f/Linear_subspaces_with_shading.svg/500px-Linear_subspaces_with_shading.svg.png 2x" data-file-width="325" data-file-height="236" /></a><figcaption>In three-dimensional <a href="/wiki/Euclidean_space" title="Euclidean space">Euclidean space</a>, these three planes represent solutions to linear equations, and their intersection represents the set of common solutions: in this case, a unique point. The blue line is the common solution to two of these equations. </figcaption></figure> <p><b>Linear algebra</b> is the branch of <a href="/wiki/Mathematics" title="Mathematics">mathematics</a> concerning <a href="/wiki/Linear_equation" title="Linear equation">linear equations</a> such as: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{1}x_{1}+\cdots +a_{n}x_{n}=b,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <mo>⋯</mo> <mo>+</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>=</mo> <mi>b</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{1}x_{1}+\cdots +a_{n}x_{n}=b,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f4f0f2986d54c01f3bccf464d266dfac923c80f3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:22.811ex; height:2.509ex;" alt="{\displaystyle a_{1}x_{1}+\cdots +a_{n}x_{n}=b,}"></span></dd></dl> <p><a href="/wiki/Linear_map" title="Linear map">linear maps</a> such as: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (x_{1},\ldots ,x_{n})\mapsto a_{1}x_{1}+\cdots +a_{n}x_{n},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo stretchy="false">↦</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <mo>⋯</mo> <mo>+</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (x_{1},\ldots ,x_{n})\mapsto a_{1}x_{1}+\cdots +a_{n}x_{n},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/705c4efbe2eac03a9a36065cd900df1932f9f7d6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:34.249ex; height:2.843ex;" alt="{\displaystyle (x_{1},\ldots ,x_{n})\mapsto a_{1}x_{1}+\cdots +a_{n}x_{n},}"></span></dd></dl> <p>and their representations in <a href="/wiki/Vector_space" title="Vector space">vector spaces</a> and through <a href="/wiki/Matrix_(mathematics)" title="Matrix (mathematics)">matrices</a>.<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> </p><p>Linear algebra is central to almost all areas of mathematics. For instance, linear algebra is fundamental in modern presentations of <a href="/wiki/Geometry" title="Geometry">geometry</a>, including for defining basic objects such as <a href="/wiki/Line_(geometry)" title="Line (geometry)">lines</a>, <a href="/wiki/Plane_(geometry)" class="mw-redirect" title="Plane (geometry)">planes</a> and <a href="/wiki/Rotation_(mathematics)" title="Rotation (mathematics)">rotations</a>. Also, <a href="/wiki/Functional_analysis" title="Functional analysis">functional analysis</a>, a branch of <a href="/wiki/Mathematical_analysis" title="Mathematical analysis">mathematical analysis</a>, may be viewed as the application of linear algebra to <a href="/wiki/Space_of_functions" class="mw-redirect" title="Space of functions">function spaces</a>. </p><p>Linear algebra is also used in most sciences and fields of <a href="/wiki/Engineering" title="Engineering">engineering</a> because it allows <a href="/wiki/Mathematical_model" title="Mathematical model">modeling</a> many natural phenomena, and computing efficiently with such models. For <a href="/wiki/Nonlinear_system" title="Nonlinear system">nonlinear systems</a>, which cannot be modeled with linear algebra, it is often used for dealing with <a href="/wiki/First-order_approximation" class="mw-redirect" title="First-order approximation">first-order approximations</a>, using the fact that the <a href="/wiki/Differential_(mathematics)" title="Differential (mathematics)">differential</a> of a <a href="/wiki/Multivariate_function" class="mw-redirect" title="Multivariate function">multivariate function</a> at a point is the linear map that best approximates the function near that point. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="History">History</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Linear_algebra&action=edit&section=1" title="Edit section: History"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1236090951">.mw-parser-output .hatnote{font-style:italic}.mw-parser-output div.hatnote{padding-left:1.6em;margin-bottom:0.5em}.mw-parser-output .hatnote i{font-style:normal}.mw-parser-output .hatnote+link+.hatnote{margin-top:-0.5em}@media print{body.ns-0 .mw-parser-output .hatnote{display:none!important}}</style><div role="note" class="hatnote navigation-not-searchable">See also: <a href="/wiki/Determinant#History" title="Determinant">Determinant § History</a>, and <a href="/wiki/Gaussian_elimination#History" title="Gaussian elimination">Gaussian elimination § History</a></div> <p>The procedure (using counting rods) for solving simultaneous linear equations now called <a href="/wiki/Gaussian_elimination" title="Gaussian elimination">Gaussian elimination</a> appears in the ancient Chinese mathematical text <a href="/wiki/Rod_calculus#System_of_linear_equations" title="Rod calculus">Chapter Eight: <i>Rectangular Arrays</i></a> of <i><a href="/wiki/The_Nine_Chapters_on_the_Mathematical_Art" title="The Nine Chapters on the Mathematical Art">The Nine Chapters on the Mathematical Art</a></i>. Its use is illustrated in eighteen problems, with two to five equations.<sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> </p><p><a href="/wiki/Systems_of_linear_equations" class="mw-redirect" title="Systems of linear equations">Systems of linear equations</a> arose in Europe with the introduction in 1637 by <a href="/wiki/Ren%C3%A9_Descartes" title="René Descartes">René Descartes</a> of <a href="/wiki/Coordinates" class="mw-redirect" title="Coordinates">coordinates</a> in <a href="/wiki/Geometry" title="Geometry">geometry</a>. In fact, in this new geometry, now called <a href="/wiki/Cartesian_geometry" class="mw-redirect" title="Cartesian geometry">Cartesian geometry</a>, lines and planes are represented by linear equations, and computing their intersections amounts to solving systems of linear equations. </p><p>The first systematic methods for solving linear systems used <a href="/wiki/Determinant" title="Determinant">determinants</a> and were first considered by <a href="/wiki/Gottfried_Wilhelm_Leibniz" title="Gottfried Wilhelm Leibniz">Leibniz</a> in 1693. In 1750, <a href="/wiki/Gabriel_Cramer" title="Gabriel Cramer">Gabriel Cramer</a> used them for giving explicit solutions of linear systems, now called <a href="/wiki/Cramer%27s_rule" title="Cramer's rule">Cramer's rule</a>. Later, <a href="/wiki/Gauss" class="mw-redirect" title="Gauss">Gauss</a> further described the method of elimination, which was initially listed as an advancement in <a href="/wiki/Geodesy" title="Geodesy">geodesy</a>.<sup id="cite_ref-Vitulli,_Marie_5-0" class="reference"><a href="#cite_note-Vitulli,_Marie-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> </p><p>In 1844 <a href="/wiki/Hermann_Grassmann" title="Hermann Grassmann">Hermann Grassmann</a> published his "Theory of Extension" which included foundational new topics of what is today called linear algebra. In 1848, <a href="/wiki/James_Joseph_Sylvester" title="James Joseph Sylvester">James Joseph Sylvester</a> introduced the term <i>matrix</i>, which is Latin for <i>womb</i>. </p><p>Linear algebra grew with ideas noted in the <a href="/wiki/Complex_plane" title="Complex plane">complex plane</a>. For instance, two numbers <span class="texhtml mvar" style="font-style:italic;">w</span> and <span class="texhtml mvar" style="font-style:italic;">z</span> in <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {C} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {C} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f9add4085095b9b6d28d045fd9c92c2c09f549a7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {C} }"></span> have a difference <span class="texhtml"><i>w</i> – <i>z</i></span>, and the line segments <span class="texhtml"><span style="text-decoration:overline;"><i>wz</i></span></span> and <span class="texhtml"><span style="text-decoration:overline;">0(<i>w</i> − <i>z</i>)</span></span> are of the same length and direction. The segments are <a href="/wiki/Equipollence_(geometry)" title="Equipollence (geometry)">equipollent</a>. The four-dimensional system <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {H} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">H</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {H} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e050965453c42bcc6bd544546703c836bdafeac9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.808ex; height:2.176ex;" alt="{\displaystyle \mathbb {H} }"></span> of <a href="/wiki/Quaternion" title="Quaternion">quaternions</a> was discovered by <a href="/wiki/William_Rowan_Hamilton" title="William Rowan Hamilton">W.R. Hamilton</a> in 1843.<sup id="cite_ref-6" class="reference"><a href="#cite_note-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> The term <i>vector</i> was introduced as <span class="texhtml"><b>v</b> = <i>x</i><b>i</b> + <i>y</i><b>j</b> + <i>z</i><b>k</b></span> representing a point in space. The quaternion difference <span class="texhtml"><i>p</i> – <i>q</i></span> also produces a segment equipollent to <span class="texhtml"><span style="text-decoration:overline;"><i>pq</i></span></span>. Other <a href="/wiki/Hypercomplex_number" title="Hypercomplex number">hypercomplex number</a> systems also used the idea of a linear space with a <a href="/wiki/Basis_(linear_algebra)" title="Basis (linear algebra)">basis</a>. </p><p><a href="/wiki/Arthur_Cayley" title="Arthur Cayley">Arthur Cayley</a> introduced <a href="/wiki/Matrix_multiplication" title="Matrix multiplication">matrix multiplication</a> and the <a href="/wiki/Inverse_matrix" class="mw-redirect" title="Inverse matrix">inverse matrix</a> in 1856, making possible the <a href="/wiki/General_linear_group" title="General linear group">general linear group</a>. The mechanism of <a href="/wiki/Group_representation" title="Group representation">group representation</a> became available for describing complex and hypercomplex numbers. Crucially, Cayley used a single letter to denote a matrix, thus treating a matrix as an aggregate object. He also realized the connection between matrices and determinants and wrote "There would be many things to say about this theory of matrices which should, it seems to me, precede the theory of determinants".<sup id="cite_ref-Vitulli,_Marie_5-1" class="reference"><a href="#cite_note-Vitulli,_Marie-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> </p><p><a href="/wiki/Benjamin_Peirce" title="Benjamin Peirce">Benjamin Peirce</a> published his <i>Linear Associative Algebra</i> (1872), and his son <a href="/wiki/Charles_Sanders_Peirce" title="Charles Sanders Peirce">Charles Sanders Peirce</a> extended the work later.<sup id="cite_ref-7" class="reference"><a href="#cite_note-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> </p><p>The <a href="/wiki/Telegraph" class="mw-redirect" title="Telegraph">telegraph</a> required an explanatory system, and the 1873 publication by <a href="/wiki/James_Clerk_Maxwell" title="James Clerk Maxwell">James Clerk Maxwell</a> of <i><a href="/wiki/A_Treatise_on_Electricity_and_Magnetism" title="A Treatise on Electricity and Magnetism">A Treatise on Electricity and Magnetism</a></i> instituted a <a href="/wiki/Field_theory_(physics)" class="mw-redirect" title="Field theory (physics)">field theory</a> of forces and required <a href="/wiki/Differential_geometry" title="Differential geometry">differential geometry</a> for expression. Linear algebra is flat differential geometry and serves in tangent spaces to <a href="/wiki/Manifold" title="Manifold">manifolds</a>. Electromagnetic symmetries of spacetime are expressed by the <a href="/wiki/Lorentz_transformation" title="Lorentz transformation">Lorentz transformations</a>, and much of the history of linear algebra is the <a href="/wiki/History_of_Lorentz_transformations" title="History of Lorentz transformations">history of Lorentz transformations</a>. </p><p>The first modern and more precise definition of a vector space was introduced by <a href="/wiki/Peano" class="mw-redirect" title="Peano">Peano</a> in 1888;<sup id="cite_ref-Vitulli,_Marie_5-2" class="reference"><a href="#cite_note-Vitulli,_Marie-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> by 1900, a theory of linear transformations of finite-dimensional vector spaces had emerged. Linear algebra took its modern form in the first half of the twentieth century when many ideas and methods of previous centuries were generalized as <a href="/wiki/Abstract_algebra" title="Abstract algebra">abstract algebra</a>. The development of computers led to increased research in efficient <a href="/wiki/Algorithm" title="Algorithm">algorithms</a> for Gaussian elimination and matrix decompositions, and linear algebra became an essential tool for modeling and simulations.<sup id="cite_ref-Vitulli,_Marie_5-3" class="reference"><a href="#cite_note-Vitulli,_Marie-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Vector_spaces">Vector spaces</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Linear_algebra&action=edit&section=2" title="Edit section: Vector spaces"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Vector_space" title="Vector space">Vector space</a></div> <p>Until the 19th century, linear algebra was introduced through <a href="/wiki/Systems_of_linear_equations" class="mw-redirect" title="Systems of linear equations">systems of linear equations</a> and <a href="/wiki/Matrix_(mathematics)" title="Matrix (mathematics)">matrices</a>. In modern mathematics, the presentation through <i>vector spaces</i> is generally preferred, since it is more <a href="/wiki/Synthetic_geometry" title="Synthetic geometry">synthetic</a>, more general (not limited to the finite-dimensional case), and conceptually simpler, although more abstract. </p><p>A vector space over a <a href="/wiki/Field_(mathematics)" title="Field (mathematics)">field</a> <span class="texhtml"><i>F</i></span> (often the field of the <a href="/wiki/Real_number" title="Real number">real numbers</a>) is a <a href="/wiki/Set_(mathematics)" title="Set (mathematics)">set</a> <span class="texhtml"><i>V</i></span> equipped with two <a href="/wiki/Binary_operation" title="Binary operation">binary operations</a>. <a href="/wiki/Element_(mathematics)" title="Element (mathematics)">Elements</a> of <span class="texhtml"><i>V</i></span> are called <i>vectors</i>, and elements of <i>F</i> are called <i>scalars</i>. The first operation, <i><a href="/wiki/Vector_addition" class="mw-redirect" title="Vector addition">vector addition</a></i>, takes any two vectors <span class="texhtml"><b>v</b></span> and <span class="texhtml"><b>w</b></span> and outputs a third vector <span class="texhtml"><b>v</b> + <b>w</b></span>. The second operation, <i><a href="/wiki/Scalar_multiplication" title="Scalar multiplication">scalar multiplication</a></i>, takes any scalar <span class="texhtml"><i>a</i></span> and any vector <span class="texhtml"><b>v</b></span> and outputs a new <span class="nowrap">vector <span class="texhtml"><i>a</i><b>v</b></span></span>. The axioms that addition and scalar multiplication must satisfy are the following. (In the list below, <span class="texhtml"><b>u</b>, <b>v</b></span> and <span class="texhtml"><b>w</b></span> are arbitrary elements of <span class="texhtml"><i>V</i></span>, and <span class="texhtml"><i>a</i></span> and <span class="texhtml"><i>b</i></span> are arbitrary scalars in the field <span class="texhtml"><i>F</i></span>.)<sup id="cite_ref-8" class="reference"><a href="#cite_note-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> </p> <dl><dd><table border="0" style="width:100%;"> <tbody><tr> <td><b>Axiom</b></td> <td><b>Signification</b> </td></tr> <tr> <td><a href="/wiki/Associativity" class="mw-redirect" title="Associativity">Associativity</a> of addition</td> <td><span class="texhtml"><b>u</b> + (<b>v</b> + <b>w</b>) = (<b>u</b> + <b>v</b>) + <b>w</b></span> </td></tr> <tr style="background:#F8F4FF;"> <td><a href="/wiki/Commutativity" class="mw-redirect" title="Commutativity">Commutativity</a> of addition</td> <td><span class="texhtml"><b>u</b> + <b>v</b> = <b>v</b> + <b>u</b></span> </td></tr> <tr> <td><a href="/wiki/Identity_element" title="Identity element">Identity element</a> of addition</td> <td>There exists an element <span class="texhtml"><b>0</b></span> in <span class="texhtml"><i>V</i></span>, called the <i><a href="/wiki/Zero_vector" class="mw-redirect" title="Zero vector">zero vector</a></i> (or simply <i>zero</i>), such that <span class="texhtml"><b>v</b> + <b>0</b> = <b>v</b></span> for all <span class="texhtml"><b>v</b></span> in <span class="texhtml"><i>V</i></span>. </td></tr> <tr style="background:#F8F4FF;"> <td><a href="/wiki/Inverse_element" title="Inverse element">Inverse elements</a> of addition</td> <td>For every <span class="texhtml"><b>v</b></span> in <span class="texhtml"><i>V</i></span>, there exists an element <span class="texhtml">−<b>v</b></span> in <span class="texhtml"><i>V</i></span>, called the <i><a href="/wiki/Additive_inverse" title="Additive inverse">additive inverse</a></i> of <span class="texhtml"><b>v</b></span>, such that <span class="texhtml"><b>v</b> + (−<b>v</b>) = <b>0</b></span> </td></tr> <tr> <td><a href="/wiki/Distributivity" class="mw-redirect" title="Distributivity">Distributivity</a> of scalar multiplication with respect to vector addition</td> <td><span class="texhtml"><i>a</i>(<b>u</b> + <b>v</b>) = <i>a</i><b>u</b> + <i>a</i><b>v</b></span> </td></tr> <tr style="background:#F8F4FF;"> <td>Distributivity of scalar multiplication with respect to field addition</td> <td><span class="texhtml">(<i>a</i> + <i>b</i>)<b>v</b> = <i>a</i><b>v</b> + <i>b</i><b>v</b></span> </td></tr> <tr> <td>Compatibility of scalar multiplication with field multiplication</td> <td><span class="texhtml"><i>a</i>(<i>b</i><b>v</b>) = (<i>ab</i>)<b>v</b></span> <sup id="cite_ref-9" class="reference"><a href="#cite_note-9"><span class="cite-bracket">[</span>a<span class="cite-bracket">]</span></a></sup> </td></tr> <tr style="background:#F8F4FF;"> <td>Identity element of scalar multiplication</td> <td><span class="texhtml">1<b>v</b> = <b>v</b></span>, where <span class="texhtml">1</span> denotes the <a href="/wiki/Multiplicative_identity" class="mw-redirect" title="Multiplicative identity">multiplicative identity</a> of <span class="texhtml mvar" style="font-style:italic;">F</span>. </td></tr></tbody></table></dd></dl> <p>The first four axioms mean that <span class="texhtml"><i>V</i></span> is an <a href="/wiki/Abelian_group" title="Abelian group">abelian group</a> under addition. </p><p>An element of a specific vector space may have various natures; for example, it could be a <a href="/wiki/Sequence" title="Sequence">sequence</a>, a <a href="/wiki/Function_(mathematics)" title="Function (mathematics)">function</a>, a <a href="/wiki/Polynomial_ring" title="Polynomial ring">polynomial</a>, or a <a href="/wiki/Matrix_(mathematics)" title="Matrix (mathematics)">matrix</a>. Linear algebra is concerned with the properties of such objects that are common to all vector spaces. </p> <div class="mw-heading mw-heading3"><h3 id="Linear_maps">Linear maps</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Linear_algebra&action=edit&section=3" title="Edit section: Linear maps"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Linear_map" title="Linear map">Linear map</a></div> <p><b>Linear maps</b> are <a href="/wiki/Map_(mathematics)" title="Map (mathematics)">mappings</a> between vector spaces that preserve the vector-space structure. Given two vector spaces <span class="texhtml"><i>V</i></span> and <span class="texhtml"><i>W</i></span> over a field <span class="texhtml mvar" style="font-style:italic;">F</span>, a linear map (also called, in some contexts, linear transformation or linear mapping) is a <a href="/wiki/Map_(mathematics)" title="Map (mathematics)">map</a> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T:V\to W}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> <mo>:</mo> <mi>V</mi> <mo stretchy="false">→</mo> <mi>W</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T:V\to W}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4c59f606d4e06de82ae3016ab89884480356b3d6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:11.41ex; height:2.176ex;" alt="{\displaystyle T:V\to W}"></span></dd></dl> <p>That is compatible with addition and scalar multiplication, that is </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T(\mathbf {u} +\mathbf {v} )=T(\mathbf {u} )+T(\mathbf {v} ),\quad T(a\mathbf {v} )=aT(\mathbf {v} )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">u</mi> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <mi>T</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">u</mi> </mrow> <mo stretchy="false">)</mo> <mo>+</mo> <mi>T</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mo stretchy="false">)</mo> <mo>,</mo> <mspace width="1em" /> <mi>T</mi> <mo stretchy="false">(</mo> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <mi>a</mi> <mi>T</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T(\mathbf {u} +\mathbf {v} )=T(\mathbf {u} )+T(\mathbf {v} ),\quad T(a\mathbf {v} )=aT(\mathbf {v} )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/44b0a2c4d6ed55530894b9476f29aec608137744" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:43.536ex; height:2.843ex;" alt="{\displaystyle T(\mathbf {u} +\mathbf {v} )=T(\mathbf {u} )+T(\mathbf {v} ),\quad T(a\mathbf {v} )=aT(\mathbf {v} )}"></span></dd></dl> <p>for any vectors <span class="texhtml"><b>u</b>,<b>v</b></span> in <span class="texhtml"><i>V</i></span> and scalar <span class="texhtml"><i>a</i></span> in <span class="texhtml mvar" style="font-style:italic;">F</span>. </p><p>This implies that for any vectors <span class="texhtml"><b>u</b>, <b>v</b></span> in <span class="texhtml"><i>V</i></span> and scalars <span class="texhtml"><i>a</i>, <i>b</i></span> in <span class="texhtml mvar" style="font-style:italic;">F</span>, one has </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T(a\mathbf {u} +b\mathbf {v} )=T(a\mathbf {u} )+T(b\mathbf {v} )=aT(\mathbf {u} )+bT(\mathbf {v} )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> <mo stretchy="false">(</mo> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">u</mi> </mrow> <mo>+</mo> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <mi>T</mi> <mo stretchy="false">(</mo> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">u</mi> </mrow> <mo stretchy="false">)</mo> <mo>+</mo> <mi>T</mi> <mo stretchy="false">(</mo> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <mi>a</mi> <mi>T</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">u</mi> </mrow> <mo stretchy="false">)</mo> <mo>+</mo> <mi>b</mi> <mi>T</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T(a\mathbf {u} +b\mathbf {v} )=T(a\mathbf {u} )+T(b\mathbf {v} )=aT(\mathbf {u} )+bT(\mathbf {v} )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aa6fe34aeb6a1828372e6a90b4c6cbe84bb94ee9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:47.317ex; height:2.843ex;" alt="{\displaystyle T(a\mathbf {u} +b\mathbf {v} )=T(a\mathbf {u} )+T(b\mathbf {v} )=aT(\mathbf {u} )+bT(\mathbf {v} )}"></span></dd></dl> <p>When <span class="texhtml"><i>V</i> = <i>W</i></span> are the same vector space, a linear map <span class="texhtml"><i>T</i> : <i>V</i> → <i>V</i></span> is also known as a <i>linear operator</i> on <span class="texhtml mvar" style="font-style:italic;">V</span>. </p><p>A <a href="/wiki/Bijective" class="mw-redirect" title="Bijective">bijective</a> linear map between two vector spaces (that is, every vector from the second space is associated with exactly one in the first) is an <a href="/wiki/Isomorphism" title="Isomorphism">isomorphism</a>. Because an isomorphism preserves linear structure, two isomorphic vector spaces are "essentially the same" from the linear algebra point of view, in the sense that they cannot be distinguished by using vector space properties. An essential question in linear algebra is testing whether a linear map is an isomorphism or not, and, if it is not an isomorphism, finding its <a href="/wiki/Range_of_a_function" title="Range of a function">range</a> (or image) and the set of elements that are mapped to the zero vector, called the <a href="/wiki/Kernel_(linear_operator)" class="mw-redirect" title="Kernel (linear operator)">kernel</a> of the map. All these questions can be solved by using <a href="/wiki/Gaussian_elimination" title="Gaussian elimination">Gaussian elimination</a> or some variant of this <a href="/wiki/Algorithm" title="Algorithm">algorithm</a>. </p> <div class="mw-heading mw-heading3"><h3 id="Subspaces,_span,_and_basis"><span id="Subspaces.2C_span.2C_and_basis"></span>Subspaces, span, and basis</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Linear_algebra&action=edit&section=4" title="Edit section: Subspaces, span, and basis"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main articles: <a href="/wiki/Linear_subspace" title="Linear subspace">Linear subspace</a>, <a href="/wiki/Linear_span" title="Linear span">Linear span</a>, and <a href="/wiki/Basis_(linear_algebra)" title="Basis (linear algebra)">Basis (linear algebra)</a></div> <p>The study of those subsets of vector spaces that are in themselves vector spaces under the induced operations is fundamental, similarly as for many mathematical structures. These subsets are called <a href="/wiki/Linear_subspace" title="Linear subspace">linear subspaces</a>. More precisely, a linear subspace of a vector space <span class="texhtml mvar" style="font-style:italic;">V</span> over a field <span class="texhtml mvar" style="font-style:italic;">F</span> is a <a href="/wiki/Subset" title="Subset">subset</a> <span class="texhtml mvar" style="font-style:italic;">W</span> of <span class="texhtml mvar" style="font-style:italic;">V</span> such that <span class="texhtml"><b>u</b> + <b>v</b></span> and <span class="texhtml"><i>a</i><b>u</b></span> are in <span class="texhtml mvar" style="font-style:italic;">W</span>, for every <span class="texhtml"><b>u</b></span>, <span class="texhtml"><b>v</b></span> in <span class="texhtml mvar" style="font-style:italic;">W</span>, and every <span class="texhtml mvar" style="font-style:italic;">a</span> in <span class="texhtml mvar" style="font-style:italic;">F</span>. (These conditions suffice for implying that <span class="texhtml mvar" style="font-style:italic;">W</span> is a vector space.) </p><p>For example, given a linear map <span class="texhtml"><i>T</i> : <i>V</i> → <i>W</i></span>, the <a href="/wiki/Image_(function)" class="mw-redirect" title="Image (function)">image</a> <span class="texhtml"><i>T</i>(<i>V</i>)</span> of <span class="texhtml mvar" style="font-style:italic;">V</span>, and the <a href="/wiki/Inverse_image" class="mw-redirect" title="Inverse image">inverse image</a> <span class="texhtml"><i>T</i><sup>−1</sup>(<b>0</b>)</span> of <span class="texhtml"><b>0</b></span> (called <a href="/wiki/Kernel_(linear_algebra)" title="Kernel (linear algebra)">kernel</a> or null space), are linear subspaces of <span class="texhtml mvar" style="font-style:italic;">W</span> and <span class="texhtml mvar" style="font-style:italic;">V</span>, respectively. </p><p>Another important way of forming a subspace is to consider <a href="/wiki/Linear_combination" title="Linear combination">linear combinations</a> of a set <span class="texhtml mvar" style="font-style:italic;">S</span> of vectors: the set of all sums </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{1}\mathbf {v} _{1}+a_{2}\mathbf {v} _{2}+\cdots +a_{k}\mathbf {v} _{k},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>+</mo> <mo>⋯</mo> <mo>+</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{1}\mathbf {v} _{1}+a_{2}\mathbf {v} _{2}+\cdots +a_{k}\mathbf {v} _{k},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6879901e55d5c805997cd7e9c44581e9bf717274" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:26.208ex; height:2.343ex;" alt="{\displaystyle a_{1}\mathbf {v} _{1}+a_{2}\mathbf {v} _{2}+\cdots +a_{k}\mathbf {v} _{k},}"></span></dd></dl> <p>where <span class="texhtml"><b>v</b><sub>1</sub>, <b>v</b><sub>2</sub>, ..., <b>v</b><sub><i>k</i></sub></span> are in <span class="texhtml mvar" style="font-style:italic;">S</span>, and <span class="texhtml"><i>a</i><sub>1</sub>, <i>a</i><sub>2</sub>, ..., <i>a</i><sub><i>k</i></sub></span> are in <span class="texhtml mvar" style="font-style:italic;">F</span> form a linear subspace called the <a href="/wiki/Linear_span" title="Linear span">span</a> of <span class="texhtml mvar" style="font-style:italic;">S</span>. The span of <span class="texhtml mvar" style="font-style:italic;">S</span> is also the intersection of all linear subspaces containing <span class="texhtml mvar" style="font-style:italic;">S</span>. In other words, it is the smallest (for the inclusion relation) linear subspace containing <span class="texhtml mvar" style="font-style:italic;">S</span>. </p><p>A set of vectors is <a href="/wiki/Linearly_independent" class="mw-redirect" title="Linearly independent">linearly independent</a> if none is in the span of the others. Equivalently, a set <span class="texhtml mvar" style="font-style:italic;">S</span> of vectors is linearly independent if the only way to express the zero vector as a linear combination of elements of <span class="texhtml mvar" style="font-style:italic;">S</span> is to take zero for every coefficient <span class="texhtml mvar" style="font-style:italic;">a<sub>i</sub></span>. </p><p>A set of vectors that spans a vector space is called a <a href="/wiki/Spanning_set" class="mw-redirect" title="Spanning set">spanning set</a> or <a href="/wiki/Generating_set" class="mw-redirect" title="Generating set">generating set</a>. If a spanning set <span class="texhtml mvar" style="font-style:italic;">S</span> is <i>linearly dependent</i> (that is not linearly independent), then some element <span class="texhtml"><b>w</b></span> of <span class="texhtml mvar" style="font-style:italic;">S</span> is in the span of the other elements of <span class="texhtml mvar" style="font-style:italic;">S</span>, and the span would remain the same if one were to remove <span class="texhtml"><b>w</b></span> from <span class="texhtml mvar" style="font-style:italic;">S</span>. One may continue to remove elements of <span class="texhtml mvar" style="font-style:italic;">S</span> until getting a <i>linearly independent spanning set</i>. Such a linearly independent set that spans a vector space <span class="texhtml mvar" style="font-style:italic;">V</span> is called a <a href="/wiki/Basis_(linear_algebra)" title="Basis (linear algebra)">basis</a> of <span class="texhtml"><i>V</i></span>. The importance of bases lies in the fact that they are simultaneously minimal-generating sets and maximal independent sets. More precisely, if <span class="texhtml mvar" style="font-style:italic;">S</span> is a linearly independent set, and <span class="texhtml mvar" style="font-style:italic;">T</span> is a spanning set such that <span class="texhtml"><i>S</i> ⊆ <i>T</i></span>, then there is a basis <span class="texhtml mvar" style="font-style:italic;">B</span> such that <span class="texhtml"><i>S</i> ⊆ <i>B</i> ⊆ <i>T</i></span>. </p><p>Any two bases of a vector space <span class="texhtml"><i>V</i></span> have the same <a href="/wiki/Cardinality" title="Cardinality">cardinality</a>, which is called the <a href="/wiki/Dimension_(vector_space)" title="Dimension (vector space)">dimension</a> of <span class="texhtml"><i>V</i></span>; this is the <a href="/wiki/Dimension_theorem_for_vector_spaces" title="Dimension theorem for vector spaces">dimension theorem for vector spaces</a>. Moreover, two vector spaces over the same field <span class="texhtml mvar" style="font-style:italic;">F</span> are <a href="/wiki/Isomorphic" class="mw-redirect" title="Isomorphic">isomorphic</a> if and only if they have the same dimension.<sup id="cite_ref-10" class="reference"><a href="#cite_note-10"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup> </p><p>If any basis of <span class="texhtml"><i>V</i></span> (and therefore every basis) has a finite number of elements, <span class="texhtml"><i>V</i></span> is a <i>finite-dimensional vector space</i>. If <span class="texhtml"><i>U</i></span> is a subspace of <span class="texhtml"><i>V</i></span>, then <span class="texhtml">dim <i>U</i> ≤ dim <i>V</i></span>. In the case where <span class="texhtml"><i>V</i></span> is finite-dimensional, the equality of the dimensions implies <span class="texhtml"><i>U</i> = <i>V</i></span>. </p><p>If <span class="texhtml"><i>U</i><sub>1</sub></span> and <span class="texhtml"><i>U</i><sub>2</sub></span> are subspaces of <span class="texhtml"><i>V</i></span>, then </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \dim(U_{1}+U_{2})=\dim U_{1}+\dim U_{2}-\dim(U_{1}\cap U_{2}),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>dim</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <msub> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <mi>dim</mi> <mo>⁡</mo> <msub> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <mi>dim</mi> <mo>⁡</mo> <msub> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>−</mo> <mi>dim</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <msub> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>∩</mo> <msub> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \dim(U_{1}+U_{2})=\dim U_{1}+\dim U_{2}-\dim(U_{1}\cap U_{2}),}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b2f0694e66688bc88f285b1ae33a68d046cb3df0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:50.593ex; height:2.843ex;" alt="{\displaystyle \dim(U_{1}+U_{2})=\dim U_{1}+\dim U_{2}-\dim(U_{1}\cap U_{2}),}"></span></dd></dl> <p>where <span class="texhtml"><i>U</i><sub>1</sub> + <i>U</i><sub>2</sub></span> denotes the span of <span class="texhtml"><i>U</i><sub>1</sub> ∪ <i>U</i><sub>2</sub></span>.<sup id="cite_ref-11" class="reference"><a href="#cite_note-11"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Matrices">Matrices</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Linear_algebra&action=edit&section=5" title="Edit section: Matrices"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Matrix_(mathematics)" title="Matrix (mathematics)">Matrix (mathematics)</a></div> <p>Matrices allow explicit manipulation of finite-dimensional vector spaces and <a href="/wiki/Linear_map" title="Linear map">linear maps</a>. Their theory is thus an essential part of linear algebra. </p><p>Let <span class="texhtml mvar" style="font-style:italic;">V</span> be a finite-dimensional vector space over a field <span class="texhtml"><i>F</i></span>, and <span class="texhtml">(<b>v</b><sub>1</sub>, <b>v</b><sub>2</sub>, ..., <b>v</b><sub><i>m</i></sub>)</span> be a basis of <span class="texhtml"><i>V</i></span> (thus <span class="texhtml mvar" style="font-style:italic;">m</span> is the dimension of <span class="texhtml"><i>V</i></span>). By definition of a basis, the map </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}(a_{1},\ldots ,a_{m})&\mapsto a_{1}\mathbf {v} _{1}+\cdots a_{m}\mathbf {v} _{m}\\F^{m}&\to V\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mo stretchy="false">(</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo stretchy="false">↦</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <mo>⋯</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msup> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msup> </mtd> <mtd> <mi></mi> <mo stretchy="false">→</mo> <mi>V</mi> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}(a_{1},\ldots ,a_{m})&\mapsto a_{1}\mathbf {v} _{1}+\cdots a_{m}\mathbf {v} _{m}\\F^{m}&\to V\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/46ee9985b8cbfdc38e8485c759129f7f0f953db8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:33.233ex; height:5.843ex;" alt="{\displaystyle {\begin{aligned}(a_{1},\ldots ,a_{m})&\mapsto a_{1}\mathbf {v} _{1}+\cdots a_{m}\mathbf {v} _{m}\\F^{m}&\to V\end{aligned}}}"></span></dd></dl> <p>is a <a href="/wiki/Bijection" title="Bijection">bijection</a> from <span class="texhtml"><i>F<sup>m</sup></i></span>, the set of the <a href="/wiki/Sequence_(mathematics)" class="mw-redirect" title="Sequence (mathematics)">sequences</a> of <span class="texhtml mvar" style="font-style:italic;">m</span> elements of <span class="texhtml mvar" style="font-style:italic;">F</span>, onto <span class="texhtml mvar" style="font-style:italic;">V</span>. This is an <a href="/wiki/Isomorphism" title="Isomorphism">isomorphism</a> of vector spaces, if <span class="texhtml"><i>F<sup>m</sup></i></span> is equipped with its standard structure of vector space, where vector addition and scalar multiplication are done component by component. </p><p>This isomorphism allows representing a vector by its <a href="/wiki/Inverse_image" class="mw-redirect" title="Inverse image">inverse image</a> under this isomorphism, that is by the <a href="/wiki/Coordinate_vector" title="Coordinate vector">coordinate vector</a> <span class="texhtml">(<i>a</i><sub>1</sub>, ..., <i>a<sub>m</sub></i>)</span> or by the <a href="/wiki/Column_matrix" class="mw-redirect" title="Column matrix">column matrix</a> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{bmatrix}a_{1}\\\vdots \\a_{m}\end{bmatrix}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <mo>⋮</mo> </mtd> </mtr> <mtr> <mtd> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{bmatrix}a_{1}\\\vdots \\a_{m}\end{bmatrix}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/70a031c92741d94ef93fa6613a3f993940c41943" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.671ex; width:7.404ex; height:10.509ex;" alt="{\displaystyle {\begin{bmatrix}a_{1}\\\vdots \\a_{m}\end{bmatrix}}.}"></span></dd></dl> <p>If <span class="texhtml mvar" style="font-style:italic;">W</span> is another finite dimensional vector space (possibly the same), with a basis <span class="texhtml">(<b>w</b><sub>1</sub>, ..., <b>w</b><sub><i>n</i></sub>)</span>, a linear map <span class="texhtml mvar" style="font-style:italic;">f</span> from <span class="texhtml mvar" style="font-style:italic;">W</span> to <span class="texhtml mvar" style="font-style:italic;">V</span> is well defined by its values on the basis elements, that is <span class="texhtml">(<i>f</i>(<b>w</b><sub>1</sub>), ..., <i>f</i>(<b>w</b><sub><i>n</i></sub>))</span>. Thus, <span class="texhtml mvar" style="font-style:italic;">f</span> is well represented by the list of the corresponding column matrices. That is, if </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(w_{j})=a_{1,j}v_{1}+\cdots +a_{m,j}v_{m},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <msub> <mi>w</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mo>,</mo> <mi>j</mi> </mrow> </msub> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <mo>⋯</mo> <mo>+</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> <mo>,</mo> <mi>j</mi> </mrow> </msub> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(w_{j})=a_{1,j}v_{1}+\cdots +a_{m,j}v_{m},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/866d92a12b845daa665a5d1ea59a0052137b1a37" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:30.254ex; height:3.009ex;" alt="{\displaystyle f(w_{j})=a_{1,j}v_{1}+\cdots +a_{m,j}v_{m},}"></span></dd></dl> <p>for <span class="texhtml"><i>j</i> = 1, ..., <i>n</i></span>, then <span class="texhtml mvar" style="font-style:italic;">f</span> is represented by the matrix </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{bmatrix}a_{1,1}&\cdots &a_{1,n}\\\vdots &\ddots &\vdots \\a_{m,1}&\cdots &a_{m,n}\end{bmatrix}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mo>,</mo> <mn>1</mn> </mrow> </msub> </mtd> <mtd> <mo>⋯</mo> </mtd> <mtd> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mo>,</mo> <mi>n</mi> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <mo>⋮</mo> </mtd> <mtd> <mo>⋱</mo> </mtd> <mtd> <mo>⋮</mo> </mtd> </mtr> <mtr> <mtd> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> <mo>,</mo> <mn>1</mn> </mrow> </msub> </mtd> <mtd> <mo>⋯</mo> </mtd> <mtd> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> <mo>,</mo> <mi>n</mi> </mrow> </msub> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{bmatrix}a_{1,1}&\cdots &a_{1,n}\\\vdots &\ddots &\vdots \\a_{m,1}&\cdots &a_{m,n}\end{bmatrix}},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5384246603a7c16c37ce4c5a430c50d590f8e0c4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.171ex; width:20.656ex; height:11.509ex;" alt="{\displaystyle {\begin{bmatrix}a_{1,1}&\cdots &a_{1,n}\\\vdots &\ddots &\vdots \\a_{m,1}&\cdots &a_{m,n}\end{bmatrix}},}"></span></dd></dl> <p>with <span class="texhtml mvar" style="font-style:italic;">m</span> rows and <span class="texhtml mvar" style="font-style:italic;">n</span> columns. </p><p><a href="/wiki/Matrix_multiplication" title="Matrix multiplication">Matrix multiplication</a> is defined in such a way that the product of two matrices is the matrix of the <a href="/wiki/Function_composition" title="Function composition">composition</a> of the corresponding linear maps, and the product of a matrix and a column matrix is the column matrix representing the result of applying the represented linear map to the represented vector. It follows that the theory of finite-dimensional vector spaces and the theory of matrices are two different languages for expressing the same concepts. </p><p>Two matrices that encode the same linear transformation in different bases are called <a href="/wiki/Similar_(linear_algebra)" class="mw-redirect" title="Similar (linear algebra)">similar</a>. It can be proved that two matrices are similar if and only if one can transform one into the other by <a href="/wiki/Elementary_matrix" title="Elementary matrix">elementary row and column operations</a>. For a matrix representing a linear map from <span class="texhtml mvar" style="font-style:italic;">W</span> to <span class="texhtml mvar" style="font-style:italic;">V</span>, the row operations correspond to change of bases in <span class="texhtml mvar" style="font-style:italic;">V</span> and the column operations correspond to change of bases in <span class="texhtml mvar" style="font-style:italic;">W</span>. Every matrix is similar to an <a href="/wiki/Identity_matrix" title="Identity matrix">identity matrix</a> possibly bordered by zero rows and zero columns. In terms of vector spaces, this means that, for any linear map from <span class="texhtml mvar" style="font-style:italic;">W</span> to <span class="texhtml mvar" style="font-style:italic;">V</span>, there are bases such that a part of the basis of <span class="texhtml mvar" style="font-style:italic;">W</span> is mapped bijectively on a part of the basis of <span class="texhtml mvar" style="font-style:italic;">V</span>, and that the remaining basis elements of <span class="texhtml mvar" style="font-style:italic;">W</span>, if any, are mapped to zero. <a href="/wiki/Gaussian_elimination" title="Gaussian elimination">Gaussian elimination</a> is the basic algorithm for finding these elementary operations, and proving these results. </p> <div class="mw-heading mw-heading2"><h2 id="Linear_systems">Linear systems</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Linear_algebra&action=edit&section=6" title="Edit section: Linear systems"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/System_of_linear_equations" title="System of linear equations">System of linear equations</a></div> <p>A finite set of linear equations in a finite set of variables, for example, <span class="texhtml"><i>x</i><sub>1</sub>, <i>x</i><sub>2</sub>, ..., <i>x<sub>n</sub></i></span>, or <span class="texhtml"><i>x</i>, <i>y</i>, ..., <i>z</i></span> is called a <b> system of linear equations</b> or a <b>linear system</b>.<sup id="cite_ref-12" class="reference"><a href="#cite_note-12"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-13" class="reference"><a href="#cite_note-13"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-14" class="reference"><a href="#cite_note-14"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-15" class="reference"><a href="#cite_note-15"><span class="cite-bracket">[</span>14<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-16" class="reference"><a href="#cite_note-16"><span class="cite-bracket">[</span>15<span class="cite-bracket">]</span></a></sup> </p><p>Systems of linear equations form a fundamental part of linear algebra. Historically, linear algebra and matrix theory have been developed for solving such systems. In the modern presentation of linear algebra through vector spaces and matrices, many problems may be interpreted in terms of linear systems. </p><p>For example, let </p> <style data-mw-deduplicate="TemplateStyles:r1266403038">.mw-parser-output table.numblk{border-collapse:collapse;border:none;margin-top:0;margin-right:0;margin-bottom:0}.mw-parser-output table.numblk>tbody>tr>td{vertical-align:middle;padding:0}.mw-parser-output table.numblk>tbody>tr>td:nth-child(2){width:99%}.mw-parser-output table.numblk>tbody>tr>td:nth-child(2)>table{border-collapse:collapse;margin:0;border:none;width:100%}.mw-parser-output table.numblk>tbody>tr>td:nth-child(2)>table>tbody>tr:first-child>td:first-child,.mw-parser-output table.numblk>tbody>tr>td:nth-child(2)>table>tbody>tr:first-child>td:last-child{padding:0 0.4ex}.mw-parser-output table.numblk>tbody>tr>td:nth-child(2)>table>tbody>tr:first-child>td:nth-child(2){width:100%;padding:0}.mw-parser-output table.numblk>tbody>tr>td:nth-child(2)>table>tbody>tr:last-child>td{padding:0}.mw-parser-output table.numblk>tbody>tr>td:last-child{font-weight:bold}.mw-parser-output table.numblk.numblk-raw-n>tbody>tr>td:last-child{font-weight:unset}.mw-parser-output table.numblk>tbody>tr>td:last-child::before{content:"("}.mw-parser-output table.numblk>tbody>tr>td:last-child::after{content:")"}.mw-parser-output table.numblk.numblk-raw-n>tbody>tr>td:last-child::before,.mw-parser-output table.numblk.numblk-raw-n>tbody>tr>td:last-child::after{content:none}.mw-parser-output table.numblk>tbody>tr>td{border:none}.mw-parser-output table.numblk.numblk-border>tbody>tr>td{border:thin solid}.mw-parser-output table.numblk>tbody>tr>td:nth-child(2)>table>tbody>tr:first-child>td{border:none}.mw-parser-output table.numblk.numblk-border>tbody>tr>td:nth-child(2)>table>tbody>tr:first-child>td{border:thin solid}.mw-parser-output table.numblk>tbody>tr>td:nth-child(2)>table>tbody>tr:last-child>td{border-left:none;border-right:none;border-bottom:none}.mw-parser-output table.numblk.numblk-border>tbody>tr>td:nth-child(2)>table>tbody>tr:last-child>td{border-left:thin solid;border-right:thin solid;border-bottom:thin solid}.mw-parser-output table.numblk:target{color:var(--color-base,#202122);background-color:#cfe8fd}@media screen{html.skin-theme-clientpref-night .mw-parser-output table.numblk:target{color:var(--color-base,#eaecf0);background-color:#301702}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output table.numblk:target{color:var(--color-base,#eaecf0);background-color:#301702}}</style><table role="presentation" class="numblk" id="math_S" style="margin-left: 1.6em;"><tbody><tr><td class="nowrap"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{alignedat}{7}2x&&\;+\;&&y&&\;-\;&&z&&\;=\;&&8\\-3x&&\;-\;&&y&&\;+\;&&2z&&\;=\;&&-11\\-2x&&\;+\;&&y&&\;+\;&&2z&&\;=\;&&-3\end{alignedat}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 0em 0em 0em 0em 0em 0em 0em 0em 0em 0em 0em 0em 0em" displaystyle="true"> <mtr> <mtd> <mn>2</mn> <mi>x</mi> </mtd> <mtd /> <mtd> <mspace width="thickmathspace" /> <mo>+</mo> <mspace width="thickmathspace" /> </mtd> <mtd /> <mtd> <mi>y</mi> </mtd> <mtd /> <mtd> <mspace width="thickmathspace" /> <mo>−</mo> <mspace width="thickmathspace" /> </mtd> <mtd /> <mtd> <mi>z</mi> </mtd> <mtd /> <mtd> <mspace width="thickmathspace" /> <mo>=</mo> <mspace width="thickmathspace" /> </mtd> <mtd /> <mtd> <mn>8</mn> </mtd> </mtr> <mtr> <mtd> <mo>−</mo> <mn>3</mn> <mi>x</mi> </mtd> <mtd /> <mtd> <mspace width="thickmathspace" /> <mo>−</mo> <mspace width="thickmathspace" /> </mtd> <mtd /> <mtd> <mi>y</mi> </mtd> <mtd /> <mtd> <mspace width="thickmathspace" /> <mo>+</mo> <mspace width="thickmathspace" /> </mtd> <mtd /> <mtd> <mn>2</mn> <mi>z</mi> </mtd> <mtd /> <mtd> <mspace width="thickmathspace" /> <mo>=</mo> <mspace width="thickmathspace" /> </mtd> <mtd /> <mtd> <mo>−</mo> <mn>11</mn> </mtd> </mtr> <mtr> <mtd> <mo>−</mo> <mn>2</mn> <mi>x</mi> </mtd> <mtd /> <mtd> <mspace width="thickmathspace" /> <mo>+</mo> <mspace width="thickmathspace" /> </mtd> <mtd /> <mtd> <mi>y</mi> </mtd> <mtd /> <mtd> <mspace width="thickmathspace" /> <mo>+</mo> <mspace width="thickmathspace" /> </mtd> <mtd /> <mtd> <mn>2</mn> <mi>z</mi> </mtd> <mtd /> <mtd> <mspace width="thickmathspace" /> <mo>=</mo> <mspace width="thickmathspace" /> </mtd> <mtd /> <mtd> <mo>−</mo> <mn>3</mn> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{alignedat}{7}2x&&\;+\;&&y&&\;-\;&&z&&\;=\;&&8\\-3x&&\;-\;&&y&&\;+\;&&2z&&\;=\;&&-11\\-2x&&\;+\;&&y&&\;+\;&&2z&&\;=\;&&-3\end{alignedat}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/12b1907474800554685121168dfb4d56ba7fb9a0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.838ex; width:21.886ex; height:8.843ex;" alt="{\displaystyle {\begin{alignedat}{7}2x&&\;+\;&&y&&\;-\;&&z&&\;=\;&&8\\-3x&&\;-\;&&y&&\;+\;&&2z&&\;=\;&&-11\\-2x&&\;+\;&&y&&\;+\;&&2z&&\;=\;&&-3\end{alignedat}}}"></span></td> <td></td> <td class="nowrap"><a href="#math_S">S</a></td></tr></tbody></table> <p>be a linear system. </p><p>To such a system, one may associate its matrix </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M=\left[{\begin{array}{rrr}2&1&-1\\-3&-1&2\\-2&1&2\end{array}}\right].}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>M</mi> <mo>=</mo> <mrow> <mo>[</mo> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right right right" rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>2</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mo>−</mo> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mo>−</mo> <mn>3</mn> </mtd> <mtd> <mo>−</mo> <mn>1</mn> </mtd> <mtd> <mn>2</mn> </mtd> </mtr> <mtr> <mtd> <mo>−</mo> <mn>2</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mn>2</mn> </mtd> </mtr> </mtable> </mrow> <mo>]</mo> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M=\left[{\begin{array}{rrr}2&1&-1\\-3&-1&2\\-2&1&2\end{array}}\right].}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fc5140ca13be8f555186b8d57b6223a4ef2e1b0b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.005ex; width:23.984ex; height:9.176ex;" alt="{\displaystyle M=\left[{\begin{array}{rrr}2&1&-1\\-3&-1&2\\-2&1&2\end{array}}\right].}"></span></dd></dl> <p>and its right member vector </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {v} ={\begin{bmatrix}8\\-11\\-3\end{bmatrix}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>8</mn> </mtd> </mtr> <mtr> <mtd> <mo>−</mo> <mn>11</mn> </mtd> </mtr> <mtr> <mtd> <mo>−</mo> <mn>3</mn> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {v} ={\begin{bmatrix}8\\-11\\-3\end{bmatrix}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0a40dba1d9901273a39c9ba1bae9eb4c0cada2ef" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.005ex; width:13.141ex; height:9.176ex;" alt="{\displaystyle \mathbf {v} ={\begin{bmatrix}8\\-11\\-3\end{bmatrix}}.}"></span></dd></dl> <p>Let <span class="texhtml mvar" style="font-style:italic;">T</span> be the linear transformation associated with the matrix <span class="texhtml mvar" style="font-style:italic;">M</span>. A solution of the system (<b><a href="#math_S">S</a></b>) is a vector </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {X} ={\begin{bmatrix}x\\y\\z\end{bmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">X</mi> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mi>x</mi> </mtd> </mtr> <mtr> <mtd> <mi>y</mi> </mtd> </mtr> <mtr> <mtd> <mi>z</mi> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {X} ={\begin{bmatrix}x\\y\\z\end{bmatrix}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e1ff89f5145cb10c907b9d8bf628579b19734c0f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.171ex; width:10.3ex; height:9.509ex;" alt="{\displaystyle \mathbf {X} ={\begin{bmatrix}x\\y\\z\end{bmatrix}}}"></span></dd></dl> <p>such that </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T(\mathbf {X} )=\mathbf {v} ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">X</mi> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T(\mathbf {X} )=\mathbf {v} ,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5adab660fcaac2d62105be89cae62af17c0900a9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.621ex; height:2.843ex;" alt="{\displaystyle T(\mathbf {X} )=\mathbf {v} ,}"></span></dd></dl> <p>that is an element of the <a href="/wiki/Preimage" class="mw-redirect" title="Preimage">preimage</a> of <span class="texhtml mvar" style="font-style:italic;">v</span> by <span class="texhtml mvar" style="font-style:italic;">T</span>. </p><p>Let (<b><a href="#math_S′">S′</a></b>) be the associated <a href="/wiki/Homogeneous_system_of_linear_equations" class="mw-redirect" title="Homogeneous system of linear equations">homogeneous system</a>, where the right-hand sides of the equations are put to zero: </p> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1266403038"><table role="presentation" class="numblk" id="math_S′" style="margin-left: 1.6em;"><tbody><tr><td class="nowrap"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{alignedat}{7}2x&&\;+\;&&y&&\;-\;&&z&&\;=\;&&0\\-3x&&\;-\;&&y&&\;+\;&&2z&&\;=\;&&0\\-2x&&\;+\;&&y&&\;+\;&&2z&&\;=\;&&0\end{alignedat}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 0em 0em 0em 0em 0em 0em 0em 0em 0em 0em 0em 0em 0em" displaystyle="true"> <mtr> <mtd> <mn>2</mn> <mi>x</mi> </mtd> <mtd /> <mtd> <mspace width="thickmathspace" /> <mo>+</mo> <mspace width="thickmathspace" /> </mtd> <mtd /> <mtd> <mi>y</mi> </mtd> <mtd /> <mtd> <mspace width="thickmathspace" /> <mo>−</mo> <mspace width="thickmathspace" /> </mtd> <mtd /> <mtd> <mi>z</mi> </mtd> <mtd /> <mtd> <mspace width="thickmathspace" /> <mo>=</mo> <mspace width="thickmathspace" /> </mtd> <mtd /> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mo>−</mo> <mn>3</mn> <mi>x</mi> </mtd> <mtd /> <mtd> <mspace width="thickmathspace" /> <mo>−</mo> <mspace width="thickmathspace" /> </mtd> <mtd /> <mtd> <mi>y</mi> </mtd> <mtd /> <mtd> <mspace width="thickmathspace" /> <mo>+</mo> <mspace width="thickmathspace" /> </mtd> <mtd /> <mtd> <mn>2</mn> <mi>z</mi> </mtd> <mtd /> <mtd> <mspace width="thickmathspace" /> <mo>=</mo> <mspace width="thickmathspace" /> </mtd> <mtd /> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mo>−</mo> <mn>2</mn> <mi>x</mi> </mtd> <mtd /> <mtd> <mspace width="thickmathspace" /> <mo>+</mo> <mspace width="thickmathspace" /> </mtd> <mtd /> <mtd> <mi>y</mi> </mtd> <mtd /> <mtd> <mspace width="thickmathspace" /> <mo>+</mo> <mspace width="thickmathspace" /> </mtd> <mtd /> <mtd> <mn>2</mn> <mi>z</mi> </mtd> <mtd /> <mtd> <mspace width="thickmathspace" /> <mo>=</mo> <mspace width="thickmathspace" /> </mtd> <mtd /> <mtd> <mn>0</mn> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{alignedat}{7}2x&&\;+\;&&y&&\;-\;&&z&&\;=\;&&0\\-3x&&\;-\;&&y&&\;+\;&&2z&&\;=\;&&0\\-2x&&\;+\;&&y&&\;+\;&&2z&&\;=\;&&0\end{alignedat}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eff28cf9e7d6eafa9961ebfb94299338385a0628" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.838ex; width:18.916ex; height:8.843ex;" alt="{\displaystyle {\begin{alignedat}{7}2x&&\;+\;&&y&&\;-\;&&z&&\;=\;&&0\\-3x&&\;-\;&&y&&\;+\;&&2z&&\;=\;&&0\\-2x&&\;+\;&&y&&\;+\;&&2z&&\;=\;&&0\end{alignedat}}}"></span></td> <td></td> <td class="nowrap"><a href="#math_S′">S′</a></td></tr></tbody></table> <p>The solutions of (<b><a href="#math_S′">S′</a></b>) are exactly the elements of the <a href="/wiki/Kernel_(linear_algebra)" title="Kernel (linear algebra)">kernel</a> of <span class="texhtml mvar" style="font-style:italic;">T</span> or, equivalently, <span class="texhtml mvar" style="font-style:italic;">M</span>. </p><p>The <a href="/wiki/Gaussian_elimination" title="Gaussian elimination">Gaussian-elimination</a> consists of performing <a href="/wiki/Elementary_row_operation" class="mw-redirect" title="Elementary row operation">elementary row operations</a> on the <a href="/wiki/Augmented_matrix" title="Augmented matrix">augmented matrix</a> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left[\!{\begin{array}{c|c}M&\mathbf {v} \end{array}}\!\right]=\left[{\begin{array}{rrr|r}2&1&-1&8\\-3&-1&2&-11\\-2&1&2&-3\end{array}}\right]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>[</mo> <mrow> <mspace width="negativethinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="center center" rowspacing="4pt" columnspacing="1em" columnlines="solid"> <mtr> <mtd> <mi>M</mi> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> </mtd> </mtr> </mtable> </mrow> <mspace width="negativethinmathspace" /> </mrow> <mo>]</mo> </mrow> <mo>=</mo> <mrow> <mo>[</mo> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right right right right" rowspacing="4pt" columnspacing="1em" columnlines="none none solid"> <mtr> <mtd> <mn>2</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mo>−</mo> <mn>1</mn> </mtd> <mtd> <mn>8</mn> </mtd> </mtr> <mtr> <mtd> <mo>−</mo> <mn>3</mn> </mtd> <mtd> <mo>−</mo> <mn>1</mn> </mtd> <mtd> <mn>2</mn> </mtd> <mtd> <mo>−</mo> <mn>11</mn> </mtd> </mtr> <mtr> <mtd> <mo>−</mo> <mn>2</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mn>2</mn> </mtd> <mtd> <mo>−</mo> <mn>3</mn> </mtd> </mtr> </mtable> </mrow> <mo>]</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left[\!{\begin{array}{c|c}M&\mathbf {v} \end{array}}\!\right]=\left[{\begin{array}{rrr|r}2&1&-1&8\\-3&-1&2&-11\\-2&1&2&-3\end{array}}\right]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6a94bf6a98576cd520287535af3a6587376f9be8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.505ex; width:39.027ex; height:10.176ex;" alt="{\displaystyle \left[\!{\begin{array}{c|c}M&\mathbf {v} \end{array}}\!\right]=\left[{\begin{array}{rrr|r}2&1&-1&8\\-3&-1&2&-11\\-2&1&2&-3\end{array}}\right]}"></span></dd></dl> <p>for putting it in <a href="/wiki/Reduced_row_echelon_form" class="mw-redirect" title="Reduced row echelon form">reduced row echelon form</a>. These row operations do not change the set of solutions of the system of equations. In the example, the reduced echelon form is </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left[\!{\begin{array}{c|c}M&\mathbf {v} \end{array}}\!\right]=\left[{\begin{array}{rrr|r}1&0&0&2\\0&1&0&3\\0&0&1&-1\end{array}}\right],}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>[</mo> <mrow> <mspace width="negativethinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="center center" rowspacing="4pt" columnspacing="1em" columnlines="solid"> <mtr> <mtd> <mi>M</mi> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> </mtd> </mtr> </mtable> </mrow> <mspace width="negativethinmathspace" /> </mrow> <mo>]</mo> </mrow> <mo>=</mo> <mrow> <mo>[</mo> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right right right right" rowspacing="4pt" columnspacing="1em" columnlines="none none solid"> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>2</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>3</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mo>−</mo> <mn>1</mn> </mtd> </mtr> </mtable> </mrow> <mo>]</mo> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left[\!{\begin{array}{c|c}M&\mathbf {v} \end{array}}\!\right]=\left[{\begin{array}{rrr|r}1&0&0&2\\0&1&0&3\\0&0&1&-1\end{array}}\right],}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a6a99163495ae1cf328208d89cadcdf23397fba0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.505ex; width:33.474ex; height:10.176ex;" alt="{\displaystyle \left[\!{\begin{array}{c|c}M&\mathbf {v} \end{array}}\!\right]=\left[{\begin{array}{rrr|r}1&0&0&2\\0&1&0&3\\0&0&1&-1\end{array}}\right],}"></span></dd></dl> <p>showing that the system (<b><a href="#math_S">S</a></b>) has the unique solution </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}x&=2\\y&=3\\z&=-1.\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mi>x</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <mn>2</mn> </mtd> </mtr> <mtr> <mtd> <mi>y</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <mn>3</mn> </mtd> </mtr> <mtr> <mtd> <mi>z</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <mo>−</mo> <mn>1.</mn> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}x&=2\\y&=3\\z&=-1.\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9995c77c5af33f793e99b15e577480243041c6c5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.838ex; width:8.797ex; height:8.843ex;" alt="{\displaystyle {\begin{aligned}x&=2\\y&=3\\z&=-1.\end{aligned}}}"></span></dd></dl> <p>It follows from this matrix interpretation of linear systems that the same methods can be applied for solving linear systems and for many operations on matrices and linear transformations, which include the computation of the <a href="/wiki/Rank_of_a_matrix" class="mw-redirect" title="Rank of a matrix">ranks</a>, <a href="/wiki/Kernel_(linear_algebra)" title="Kernel (linear algebra)">kernels</a>, <a href="/wiki/Matrix_inverse" class="mw-redirect" title="Matrix inverse">matrix inverses</a>. </p> <div class="mw-heading mw-heading2"><h2 id="Endomorphisms_and_square_matrices">Endomorphisms and square matrices</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Linear_algebra&action=edit&section=7" title="Edit section: Endomorphisms and square matrices"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Square_matrix" title="Square matrix">Square matrix</a></div> <p>A linear <a href="/wiki/Endomorphism" title="Endomorphism">endomorphism</a> is a linear map that maps a vector space <span class="texhtml mvar" style="font-style:italic;">V</span> to itself. If <span class="texhtml mvar" style="font-style:italic;">V</span> has a basis of <span class="texhtml mvar" style="font-style:italic;">n</span> elements, such an endomorphism is represented by a square matrix of size <span class="texhtml mvar" style="font-style:italic;">n</span>. </p><p>Concerning general linear maps, linear endomorphisms, and square matrices have some specific properties that make their study an important part of linear algebra, which is used in many parts of mathematics, including <a href="/wiki/Geometric_transformation" title="Geometric transformation">geometric transformations</a>, <a href="/wiki/Coordinate_change" class="mw-redirect" title="Coordinate change">coordinate changes</a>, <a href="/wiki/Quadratic_form" title="Quadratic form">quadratic forms</a>, and many other parts of mathematics. </p> <div class="mw-heading mw-heading3"><h3 id="Determinant">Determinant</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Linear_algebra&action=edit&section=8" title="Edit section: Determinant"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Determinant" title="Determinant">Determinant</a></div> <p>The <i>determinant</i> of a square matrix <span class="texhtml mvar" style="font-style:italic;">A</span> is defined to be<sup id="cite_ref-17" class="reference"><a href="#cite_note-17"><span class="cite-bracket">[</span>16<span class="cite-bracket">]</span></a></sup> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sum _{\sigma \in S_{n}}(-1)^{\sigma }a_{1\sigma (1)}\cdots a_{n\sigma (n)},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>σ</mi> <mo>∈</mo> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mrow> </munder> <mo stretchy="false">(</mo> <mo>−</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>σ</mi> </mrow> </msup> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mi>σ</mi> <mo stretchy="false">(</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </msub> <mo>⋯</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mi>σ</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mrow> </msub> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum _{\sigma \in S_{n}}(-1)^{\sigma }a_{1\sigma (1)}\cdots a_{n\sigma (n)},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ad2255d7cf0d7ad8e05b318ec9b67b136a13d796" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:25.085ex; height:5.843ex;" alt="{\displaystyle \sum _{\sigma \in S_{n}}(-1)^{\sigma }a_{1\sigma (1)}\cdots a_{n\sigma (n)},}"></span></dd></dl> <p>where <span class="texhtml"><i>S<sub>n</sub></i></span> is the <a href="/wiki/Symmetric_group" title="Symmetric group">group of all permutations</a> of <span class="texhtml mvar" style="font-style:italic;">n</span> elements, <span class="texhtml mvar" style="font-style:italic;">σ</span> is a permutation, and <span class="texhtml">(−1)<sup><i>σ</i></sup></span> the <a href="/wiki/Parity_of_a_permutation" title="Parity of a permutation">parity</a> of the permutation. A matrix is <a href="/wiki/Invertible_matrix" title="Invertible matrix">invertible</a> if and only if the determinant is invertible (i.e., nonzero if the scalars belong to a field). </p><p><a href="/wiki/Cramer%27s_rule" title="Cramer's rule">Cramer's rule</a> is a <a href="/wiki/Closed-form_expression" title="Closed-form expression">closed-form expression</a>, in terms of determinants, of the solution of a <a href="/wiki/System_of_linear_equations" title="System of linear equations">system of <span class="texhtml mvar" style="font-style:italic;">n</span> linear equations in <span class="texhtml mvar" style="font-style:italic;">n</span> unknowns</a>. Cramer's rule is useful for reasoning about the solution, but, except for <span class="texhtml"><i>n</i> = 2</span> or <span class="texhtml">3</span>, it is rarely used for computing a solution, since <a href="/wiki/Gaussian_elimination" title="Gaussian elimination">Gaussian elimination</a> is a faster algorithm. </p><p>The <i>determinant of an endomorphism</i> is the determinant of the matrix representing the endomorphism in terms of some ordered basis. This definition makes sense since this determinant is independent of the choice of the basis. </p> <div class="mw-heading mw-heading3"><h3 id="Eigenvalues_and_eigenvectors">Eigenvalues and eigenvectors</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Linear_algebra&action=edit&section=9" title="Edit section: Eigenvalues and eigenvectors"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Eigenvalues_and_eigenvectors" title="Eigenvalues and eigenvectors">Eigenvalues and eigenvectors</a></div> <p>If <span class="texhtml mvar" style="font-style:italic;">f</span> is a linear endomorphism of a vector space <span class="texhtml mvar" style="font-style:italic;">V</span> over a field <span class="texhtml mvar" style="font-style:italic;">F</span>, an <i>eigenvector</i> of <span class="texhtml mvar" style="font-style:italic;">f</span> is a nonzero vector <span class="texhtml mvar" style="font-style:italic;">v</span> of <span class="texhtml mvar" style="font-style:italic;">V</span> such that <span class="texhtml"><i>f</i>(<i>v</i>) = <i>av</i></span> for some scalar <span class="texhtml mvar" style="font-style:italic;">a</span> in <span class="texhtml mvar" style="font-style:italic;">F</span>. This scalar <span class="texhtml mvar" style="font-style:italic;">a</span> is an <i>eigenvalue</i> of <span class="texhtml mvar" style="font-style:italic;">f</span>. </p><p>If the dimension of <span class="texhtml mvar" style="font-style:italic;">V</span> is finite, and a basis has been chosen, <span class="texhtml mvar" style="font-style:italic;">f</span> and <span class="texhtml mvar" style="font-style:italic;">v</span> may be represented, respectively, by a square matrix <span class="texhtml mvar" style="font-style:italic;">M</span> and a column matrix <span class="texhtml mvar" style="font-style:italic;">z</span>; the equation defining eigenvectors and eigenvalues becomes </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Mz=az.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>M</mi> <mi>z</mi> <mo>=</mo> <mi>a</mi> <mi>z</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Mz=az.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9256a6bf21c144488d5efa785da63950abd46958" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:9.594ex; height:2.176ex;" alt="{\displaystyle Mz=az.}"></span></dd></dl> <p>Using the <a href="/wiki/Identity_matrix" title="Identity matrix">identity matrix</a> <span class="texhtml mvar" style="font-style:italic;">I</span>, whose entries are all zero, except those of the main diagonal, which are equal to one, this may be rewritten </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (M-aI)z=0.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>M</mi> <mo>−</mo> <mi>a</mi> <mi>I</mi> <mo stretchy="false">)</mo> <mi>z</mi> <mo>=</mo> <mn>0.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (M-aI)z=0.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/22d42e0d9e0b62bc4e78cfd102b21e4a7217ac6c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.489ex; height:2.843ex;" alt="{\displaystyle (M-aI)z=0.}"></span></dd></dl> <p>As <span class="texhtml mvar" style="font-style:italic;">z</span> is supposed to be nonzero, this means that <span class="texhtml"><i>M</i> – <i>aI</i></span> is a <a href="/wiki/Singular_matrix" class="mw-redirect" title="Singular matrix">singular matrix</a>, and thus that its determinant <span class="texhtml">det (<i>M</i> − <i>aI</i>)</span> equals zero. The eigenvalues are thus the <a href="/wiki/Root_of_a_function" class="mw-redirect" title="Root of a function">roots</a> of the <a href="/wiki/Polynomial" title="Polynomial">polynomial</a> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \det(xI-M).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo movablelimits="true" form="prefix">det</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mi>I</mi> <mo>−</mo> <mi>M</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \det(xI-M).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/29c19c2b35265004add12e16a97d849ba94764d0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.47ex; height:2.843ex;" alt="{\displaystyle \det(xI-M).}"></span></dd></dl> <p>If <span class="texhtml mvar" style="font-style:italic;">V</span> is of dimension <span class="texhtml mvar" style="font-style:italic;">n</span>, this is a <a href="/wiki/Monic_polynomial" title="Monic polynomial">monic polynomial</a> of degree <span class="texhtml mvar" style="font-style:italic;">n</span>, called the <a href="/wiki/Characteristic_polynomial" title="Characteristic polynomial">characteristic polynomial</a> of the matrix (or of the endomorphism), and there are, at most, <span class="texhtml mvar" style="font-style:italic;">n</span> eigenvalues. </p><p>If a basis exists that consists only of eigenvectors, the matrix of <span class="texhtml mvar" style="font-style:italic;">f</span> on this basis has a very simple structure: it is a <a href="/wiki/Diagonal_matrix" title="Diagonal matrix">diagonal matrix</a> such that the entries on the <a href="/wiki/Main_diagonal" title="Main diagonal">main diagonal</a> are eigenvalues, and the other entries are zero. In this case, the endomorphism and the matrix are said to be <a href="/wiki/Diagonalizable_matrix" title="Diagonalizable matrix">diagonalizable</a>. More generally, an endomorphism and a matrix are also said diagonalizable, if they become diagonalizable after <a href="/wiki/Field_extension" title="Field extension">extending</a> the field of scalars. In this extended sense, if the characteristic polynomial is <a href="/wiki/Square-free_polynomial" title="Square-free polynomial">square-free</a>, then the matrix is diagonalizable. </p><p>A <a href="/wiki/Symmetric_matrix" title="Symmetric matrix">symmetric matrix</a> is always diagonalizable. There are non-diagonalizable matrices, the simplest being </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{bmatrix}0&1\\0&0\end{bmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{bmatrix}0&1\\0&0\end{bmatrix}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eb69343e096235ec162a5f08f9a59c684687c041" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:7.854ex; height:6.176ex;" alt="{\displaystyle {\begin{bmatrix}0&1\\0&0\end{bmatrix}}}"></span></dd></dl> <p>(it cannot be diagonalizable since its square is the <a href="/wiki/Zero_matrix" title="Zero matrix">zero matrix</a>, and the square of a nonzero diagonal matrix is never zero). </p><p>When an endomorphism is not diagonalizable, there are bases on which it has a simple form, although not as simple as the diagonal form. The <a href="/wiki/Frobenius_normal_form" title="Frobenius normal form">Frobenius normal form</a> does not need to extend the field of scalars and makes the characteristic polynomial immediately readable on the matrix. The <a href="/wiki/Jordan_normal_form" title="Jordan normal form">Jordan normal form</a> requires to extension of the field of scalar for containing all eigenvalues and differs from the diagonal form only by some entries that are just above the main diagonal and are equal to 1. </p> <div class="mw-heading mw-heading2"><h2 id="Duality">Duality</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Linear_algebra&action=edit&section=10" title="Edit section: Duality"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Dual_space" title="Dual space">Dual space</a></div> <p>A <a href="/wiki/Linear_form" title="Linear form">linear form</a> is a linear map from a vector space <span class="texhtml mvar" style="font-style:italic;">V</span> over a field <span class="texhtml mvar" style="font-style:italic;">F</span> to the field of scalars <span class="texhtml mvar" style="font-style:italic;">F</span>, viewed as a vector space over itself. Equipped by <a href="/wiki/Pointwise" title="Pointwise">pointwise</a> addition and multiplication by a scalar, the linear forms form a vector space, called the <b>dual space</b> of <span class="texhtml mvar" style="font-style:italic;">V</span>, and usually denoted <span class="texhtml mvar" style="font-style:italic;">V*</span><sup id="cite_ref-18" class="reference"><a href="#cite_note-18"><span class="cite-bracket">[</span>17<span class="cite-bracket">]</span></a></sup> or <span class="texhtml mvar" style="font-style:italic;">V<span class="nowrap" style="padding-left:0.15em;">′</span></span>.<sup id="cite_ref-19" class="reference"><a href="#cite_note-19"><span class="cite-bracket">[</span>18<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-20" class="reference"><a href="#cite_note-20"><span class="cite-bracket">[</span>19<span class="cite-bracket">]</span></a></sup> </p><p>If <span class="texhtml"><b>v</b><sub>1</sub>, ..., <b>v</b><sub><i>n</i></sub></span> is a basis of <span class="texhtml mvar" style="font-style:italic;">V</span> (this implies that <span class="texhtml mvar" style="font-style:italic;">V</span> is finite-dimensional), then one can define, for <span class="texhtml"><i>i</i> = 1, ..., <i>n</i></span>, a linear map <span class="texhtml"><i>v<sub>i</sub></i>*</span> such that <span class="texhtml"><i>v<sub>i</sub></i>*(<b>v</b><sub><i>i</i></sub>) = 1</span> and <span class="texhtml"><i>v<sub>i</sub></i>*(<b>v</b><sub><i>j</i></sub>) = 0</span> if <span class="texhtml"><i>j</i> ≠ <i>i</i></span>. These linear maps form a basis of <span class="texhtml"><i>V</i>*</span>, called the <a href="/wiki/Dual_basis" title="Dual basis">dual basis</a> of <span class="texhtml"><b>v</b><sub>1</sub>, ..., <b>v</b><sub><i>n</i></sub></span>. (If <span class="texhtml mvar" style="font-style:italic;">V</span> is not finite-dimensional, the <span class="texhtml"><i>v<sub>i</sub></i>*</span> may be defined similarly; they are linearly independent, but do not form a basis.) </p><p>For <span class="texhtml"><b>v</b></span> in <span class="texhtml mvar" style="font-style:italic;">V</span>, the map </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f\to f(\mathbf {v} )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">→</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f\to f(\mathbf {v} )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5bfba9654a09f3f65a2a150432f9e27a9c0fbdb4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.392ex; height:2.843ex;" alt="{\displaystyle f\to f(\mathbf {v} )}"></span></dd></dl> <p>is a linear form on <span class="texhtml mvar" style="font-style:italic;">V*</span>. This defines the <a href="/wiki/Canonical_map" title="Canonical map">canonical linear map</a> from <span class="texhtml mvar" style="font-style:italic;">V</span> into <span class="texhtml">(<i>V</i>*)*</span>, the dual of <span class="texhtml mvar" style="font-style:italic;">V*</span>, called the <b><a href="/wiki/Double_dual" class="mw-redirect" title="Double dual">double dual</a></b> or <b><a href="/wiki/Bidual" class="mw-redirect" title="Bidual">bidual</a></b> of <span class="texhtml mvar" style="font-style:italic;">V</span>. This canonical map is an <a href="/wiki/Isomorphism" title="Isomorphism">isomorphism</a> if <span class="texhtml mvar" style="font-style:italic;">V</span> is finite-dimensional, and this allows identifying <span class="texhtml mvar" style="font-style:italic;">V</span> with its bidual. (In the infinite-dimensional case, the canonical map is injective, but not surjective.) </p><p>There is thus a complete symmetry between a finite-dimensional vector space and its dual. This motivates the frequent use, in this context, of the <a href="/wiki/Bra%E2%80%93ket_notation" title="Bra–ket notation">bra–ket notation</a> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \langle f,\mathbf {x} \rangle }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">⟨</mo> <mi>f</mi> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mo fence="false" stretchy="false">⟩</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \langle f,\mathbf {x} \rangle }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6e11176c78659642becbfb2d3d65abe2e4a34d39" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.533ex; height:2.843ex;" alt="{\displaystyle \langle f,\mathbf {x} \rangle }"></span></dd></dl> <p>for denoting <span class="texhtml"><i>f</i>(<b>x</b>)</span>. </p> <div class="mw-heading mw-heading3"><h3 id="Dual_map">Dual map</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Linear_algebra&action=edit&section=11" title="Edit section: Dual map"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Transpose_of_a_linear_map" title="Transpose of a linear map">Transpose of a linear map</a></div> <p>Let </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f:V\to W}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>:</mo> <mi>V</mi> <mo stretchy="false">→</mo> <mi>W</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f:V\to W}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/574dffa1c85efaef6b6ef553ebd8ad9cf7f87fd6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.052ex; height:2.509ex;" alt="{\displaystyle f:V\to W}"></span></dd></dl> <p>be a linear map. For every linear form <span class="texhtml mvar" style="font-style:italic;">h</span> on <span class="texhtml mvar" style="font-style:italic;">W</span>, the <a href="/wiki/Composite_function" class="mw-redirect" title="Composite function">composite function</a> <span class="texhtml"><i>h</i> ∘ <i>f</i></span> is a linear form on <span class="texhtml mvar" style="font-style:italic;">V</span>. This defines a linear map </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f^{*}:W^{*}\to V^{*}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> <mo>:</mo> <msup> <mi>W</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> <mo stretchy="false">→</mo> <msup> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f^{*}:W^{*}\to V^{*}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d73927ebc5f7b531d76d7fa92c13722386219041" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:14.459ex; height:2.676ex;" alt="{\displaystyle f^{*}:W^{*}\to V^{*}}"></span></dd></dl> <p>between the dual spaces, which is called the <b>dual</b> or the <b>transpose</b> of <span class="texhtml mvar" style="font-style:italic;">f</span>. </p><p>If <span class="texhtml mvar" style="font-style:italic;">V</span> and <span class="texhtml mvar" style="font-style:italic;">W</span> are finite-dimensional, and <span class="texhtml mvar" style="font-style:italic;">M</span> is the matrix of <span class="texhtml mvar" style="font-style:italic;">f</span> in terms of some ordered bases, then the matrix of <span class="texhtml mvar" style="font-style:italic;">f*</span> over the dual bases is the <a href="/wiki/Transpose" title="Transpose">transpose</a> <span class="texhtml"><i>M</i><sup>T</sup></span> of <span class="texhtml mvar" style="font-style:italic;">M</span>, obtained by exchanging rows and columns. </p><p>If elements of vector spaces and their duals are represented by column vectors, this duality may be expressed in <a href="/wiki/Bra%E2%80%93ket_notation" title="Bra–ket notation">bra–ket notation</a> by </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \langle h^{\mathsf {T}},M\mathbf {v} \rangle =\langle h^{\mathsf {T}}M,\mathbf {v} \rangle .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">⟨</mo> <msup> <mi>h</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">T</mi> </mrow> </mrow> </msup> <mo>,</mo> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mo fence="false" stretchy="false">⟩</mo> <mo>=</mo> <mo fence="false" stretchy="false">⟨</mo> <msup> <mi>h</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">T</mi> </mrow> </mrow> </msup> <mi>M</mi> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mo fence="false" stretchy="false">⟩</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \langle h^{\mathsf {T}},M\mathbf {v} \rangle =\langle h^{\mathsf {T}}M,\mathbf {v} \rangle .}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/982967567f1a808c805fa4e7ce60a71fe9084498" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.519ex; height:3.176ex;" alt="{\displaystyle \langle h^{\mathsf {T}},M\mathbf {v} \rangle =\langle h^{\mathsf {T}}M,\mathbf {v} \rangle .}"></span></dd></dl> <p>To highlight this symmetry, the two members of this equality are sometimes written </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \langle h^{\mathsf {T}}\mid M\mid \mathbf {v} \rangle .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">⟨</mo> <msup> <mi>h</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">T</mi> </mrow> </mrow> </msup> <mo>∣</mo> <mi>M</mi> <mo>∣</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mo fence="false" stretchy="false">⟩</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \langle h^{\mathsf {T}}\mid M\mid \mathbf {v} \rangle .}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/037596ed4087a884c3e4b0227a07ce7b1085cebe" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.874ex; height:3.176ex;" alt="{\displaystyle \langle h^{\mathsf {T}}\mid M\mid \mathbf {v} \rangle .}"></span></dd></dl> <div class="mw-heading mw-heading3"><h3 id="Inner-product_spaces">Inner-product spaces</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Linear_algebra&action=edit&section=12" title="Edit section: Inner-product spaces"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Inner_product_space" title="Inner product space">Inner product space</a></div> <p>Besides these basic concepts, linear algebra also studies vector spaces with additional structure, such as an <a href="/wiki/Inner_product" class="mw-redirect" title="Inner product">inner product</a>. The inner product is an example of a <a href="/wiki/Bilinear_form" title="Bilinear form">bilinear form</a>, and it gives the vector space a geometric structure by allowing for the definition of length and angles. Formally, an <i>inner product</i> is a map. </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \langle \cdot ,\cdot \rangle :V\times V\to F}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">⟨</mo> <mo>⋅</mo> <mo>,</mo> <mo>⋅</mo> <mo fence="false" stretchy="false">⟩</mo> <mo>:</mo> <mi>V</mi> <mo>×</mo> <mi>V</mi> <mo stretchy="false">→</mo> <mi>F</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \langle \cdot ,\cdot \rangle :V\times V\to F}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c7871e0b592252e518983a32886d33692d7b4614" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.844ex; height:2.843ex;" alt="{\displaystyle \langle \cdot ,\cdot \rangle :V\times V\to F}"></span></dd></dl> <p>that satisfies the following three <a href="/wiki/Axiom" title="Axiom">axioms</a> for all vectors <span class="texhtml"><b>u</b>, <b>v</b>, <b>w</b></span> in <span class="texhtml"><i>V</i></span> and all scalars <span class="texhtml"><i>a</i></span> in <span class="texhtml"><i>F</i></span>:<sup id="cite_ref-Jain_21-0" class="reference"><a href="#cite_note-Jain-21"><span class="cite-bracket">[</span>20<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Prugovec̆ki_22-0" class="reference"><a href="#cite_note-Prugovec̆ki-22"><span class="cite-bracket">[</span>21<span class="cite-bracket">]</span></a></sup> </p> <ul><li><a href="/wiki/Complex_conjugate" title="Complex conjugate">Conjugate</a> symmetry: <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \langle \mathbf {u} ,\mathbf {v} \rangle ={\overline {\langle \mathbf {v} ,\mathbf {u} \rangle }}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">⟨</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">u</mi> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mo fence="false" stretchy="false">⟩</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mo fence="false" stretchy="false">⟨</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">u</mi> </mrow> <mo fence="false" stretchy="false">⟩</mo> </mrow> <mo accent="false">¯</mo> </mover> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \langle \mathbf {u} ,\mathbf {v} \rangle ={\overline {\langle \mathbf {v} ,\mathbf {u} \rangle }}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/80733e7e51ef049170d29ce6f4e85fd47feee80f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.339ex; height:3.676ex;" alt="{\displaystyle \langle \mathbf {u} ,\mathbf {v} \rangle ={\overline {\langle \mathbf {v} ,\mathbf {u} \rangle }}.}"></span></dd></dl></li></ul> <dl><dd>In <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/786849c765da7a84dbc3cce43e96aad58a5868dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {R} }"></span>, it is symmetric.</dd></dl> <ul><li><a href="/wiki/Linear" class="mw-redirect" title="Linear">Linearity</a> in the first argument: <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}\langle a\mathbf {u} ,\mathbf {v} \rangle &=a\langle \mathbf {u} ,\mathbf {v} \rangle .\\\langle \mathbf {u} +\mathbf {v} ,\mathbf {w} \rangle &=\langle \mathbf {u} ,\mathbf {w} \rangle +\langle \mathbf {v} ,\mathbf {w} \rangle .\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mo fence="false" stretchy="false">⟨</mo> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">u</mi> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mo fence="false" stretchy="false">⟩</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>a</mi> <mo fence="false" stretchy="false">⟨</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">u</mi> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mo fence="false" stretchy="false">⟩</mo> <mo>.</mo> </mtd> </mtr> <mtr> <mtd> <mo fence="false" stretchy="false">⟨</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">u</mi> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">w</mi> </mrow> <mo fence="false" stretchy="false">⟩</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <mo fence="false" stretchy="false">⟨</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">u</mi> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">w</mi> </mrow> <mo fence="false" stretchy="false">⟩</mo> <mo>+</mo> <mo fence="false" stretchy="false">⟨</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">w</mi> </mrow> <mo fence="false" stretchy="false">⟩</mo> <mo>.</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\langle a\mathbf {u} ,\mathbf {v} \rangle &=a\langle \mathbf {u} ,\mathbf {v} \rangle .\\\langle \mathbf {u} +\mathbf {v} ,\mathbf {w} \rangle &=\langle \mathbf {u} ,\mathbf {w} \rangle +\langle \mathbf {v} ,\mathbf {w} \rangle .\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/84f68af739c798566ebac4437cda440741f1b728" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:30.294ex; height:6.176ex;" alt="{\displaystyle {\begin{aligned}\langle a\mathbf {u} ,\mathbf {v} \rangle &=a\langle \mathbf {u} ,\mathbf {v} \rangle .\\\langle \mathbf {u} +\mathbf {v} ,\mathbf {w} \rangle &=\langle \mathbf {u} ,\mathbf {w} \rangle +\langle \mathbf {v} ,\mathbf {w} \rangle .\end{aligned}}}"></span></dd></dl></li> <li><a href="/wiki/Definite_bilinear_form" class="mw-redirect" title="Definite bilinear form">Positive-definiteness</a>: <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \langle \mathbf {v} ,\mathbf {v} \rangle \geq 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">⟨</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mo fence="false" stretchy="false">⟩</mo> <mo>≥</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \langle \mathbf {v} ,\mathbf {v} \rangle \geq 0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/870574fdeec6fd7c7c383e109ec1bf4afcd9a9fa" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.926ex; height:2.843ex;" alt="{\displaystyle \langle \mathbf {v} ,\mathbf {v} \rangle \geq 0}"></span></dd></dl></li></ul> <dl><dd>with equality only for <span class="texhtml"><b>v</b> = 0</span>.</dd></dl> <p>We can define the length of a vector <b>v</b> in <i>V</i> by </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \|\mathbf {v} \|^{2}=\langle \mathbf {v} ,\mathbf {v} \rangle ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <msup> <mo fence="false" stretchy="false">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mo fence="false" stretchy="false">⟨</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mo fence="false" stretchy="false">⟩</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \|\mathbf {v} \|^{2}=\langle \mathbf {v} ,\mathbf {v} \rangle ,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b1d3513b25d826a8f6c02269655edf6517c29c67" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.201ex; height:3.176ex;" alt="{\displaystyle \|\mathbf {v} \|^{2}=\langle \mathbf {v} ,\mathbf {v} \rangle ,}"></span></dd></dl> <p>and we can prove the <a href="/wiki/Cauchy%E2%80%93Schwarz_inequality" title="Cauchy–Schwarz inequality">Cauchy–Schwarz inequality</a>: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |\langle \mathbf {u} ,\mathbf {v} \rangle |\leq \|\mathbf {u} \|\cdot \|\mathbf {v} \|.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo fence="false" stretchy="false">⟨</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">u</mi> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mo fence="false" stretchy="false">⟩</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>≤</mo> <mo fence="false" stretchy="false">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">u</mi> </mrow> <mo fence="false" stretchy="false">‖</mo> <mo>⋅</mo> <mo fence="false" stretchy="false">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mo fence="false" stretchy="false">‖</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |\langle \mathbf {u} ,\mathbf {v} \rangle |\leq \|\mathbf {u} \|\cdot \|\mathbf {v} \|.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8c1449df451ce2c043c47b58132bb97d9d91da9f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.004ex; height:2.843ex;" alt="{\displaystyle |\langle \mathbf {u} ,\mathbf {v} \rangle |\leq \|\mathbf {u} \|\cdot \|\mathbf {v} \|.}"></span></dd></dl> <p>In particular, the quantity </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {|\langle \mathbf {u} ,\mathbf {v} \rangle |}{\|\mathbf {u} \|\cdot \|\mathbf {v} \|}}\leq 1,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo fence="false" stretchy="false">⟨</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">u</mi> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mo fence="false" stretchy="false">⟩</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mrow> <mrow> <mo fence="false" stretchy="false">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">u</mi> </mrow> <mo fence="false" stretchy="false">‖</mo> <mo>⋅</mo> <mo fence="false" stretchy="false">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mo fence="false" stretchy="false">‖</mo> </mrow> </mfrac> </mrow> <mo>≤</mo> <mn>1</mn> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {|\langle \mathbf {u} ,\mathbf {v} \rangle |}{\|\mathbf {u} \|\cdot \|\mathbf {v} \|}}\leq 1,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6851acc32facee4a5b99a4645e55da2d33fe4fcd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:14.969ex; height:6.509ex;" alt="{\displaystyle {\frac {|\langle \mathbf {u} ,\mathbf {v} \rangle |}{\|\mathbf {u} \|\cdot \|\mathbf {v} \|}}\leq 1,}"></span></dd></dl> <p>and so we can call this quantity the cosine of the angle between the two vectors. </p><p>Two vectors are orthogonal if <span class="texhtml">⟨<b>u</b>, <b>v</b>⟩ = 0</span>. An orthonormal basis is a basis where all basis vectors have length 1 and are orthogonal to each other. Given any finite-dimensional vector space, an orthonormal basis could be found by the <a href="/wiki/Gram%E2%80%93Schmidt" class="mw-redirect" title="Gram–Schmidt">Gram–Schmidt</a> procedure. Orthonormal bases are particularly easy to deal with, since if <span class="nowrap"><b>v</b> = <i>a</i><sub>1</sub> <b>v</b><sub>1</sub> + ⋯ + <i>a<sub>n</sub></i> <b>v</b><sub><i>n</i></sub></span>, then </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{i}=\langle \mathbf {v} ,\mathbf {v} _{i}\rangle .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>=</mo> <mo fence="false" stretchy="false">⟨</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mo>,</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo fence="false" stretchy="false">⟩</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{i}=\langle \mathbf {v} ,\mathbf {v} _{i}\rangle .}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/79e9249c7723dc4eff39ecf5dc91c1e5f2989c04" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.24ex; height:2.843ex;" alt="{\displaystyle a_{i}=\langle \mathbf {v} ,\mathbf {v} _{i}\rangle .}"></span></dd></dl> <p>The inner product facilitates the construction of many useful concepts. For instance, given a transform <span class="texhtml"><i>T</i></span>, we can define its <a href="/wiki/Hermitian_conjugate" class="mw-redirect" title="Hermitian conjugate">Hermitian conjugate</a> <span class="texhtml"><i>T*</i></span> as the linear transform satisfying </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \langle T\mathbf {u} ,\mathbf {v} \rangle =\langle \mathbf {u} ,T^{*}\mathbf {v} \rangle .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">⟨</mo> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">u</mi> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mo fence="false" stretchy="false">⟩</mo> <mo>=</mo> <mo fence="false" stretchy="false">⟨</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">u</mi> </mrow> <mo>,</mo> <msup> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mo fence="false" stretchy="false">⟩</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \langle T\mathbf {u} ,\mathbf {v} \rangle =\langle \mathbf {u} ,T^{*}\mathbf {v} \rangle .}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fb316df0527a9010dffd4beecdee839037111c9c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.635ex; height:2.843ex;" alt="{\displaystyle \langle T\mathbf {u} ,\mathbf {v} \rangle =\langle \mathbf {u} ,T^{*}\mathbf {v} \rangle .}"></span></dd></dl> <p>If <span class="texhtml"><i>T</i></span> satisfies <span class="texhtml"><i>TT*</i> = <i>T*T</i></span>, we call <span class="texhtml"><i>T</i></span> <a href="/wiki/Normal_matrix" title="Normal matrix">normal</a>. It turns out that normal matrices are precisely the matrices that have an orthonormal system of eigenvectors that span <span class="texhtml"><i>V</i></span>. </p> <div class="mw-heading mw-heading2"><h2 id="Relationship_with_geometry">Relationship with geometry</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Linear_algebra&action=edit&section=13" title="Edit section: Relationship with geometry"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>There is a strong relationship between linear algebra and <a href="/wiki/Geometry" title="Geometry">geometry</a>, which started with the introduction by <a href="/wiki/Ren%C3%A9_Descartes" title="René Descartes">René Descartes</a>, in 1637, of <a href="/wiki/Cartesian_coordinates" class="mw-redirect" title="Cartesian coordinates">Cartesian coordinates</a>. In this new (at that time) geometry, now called <a href="/wiki/Cartesian_geometry" class="mw-redirect" title="Cartesian geometry">Cartesian geometry</a>, points are represented by <a href="/wiki/Cartesian_coordinates" class="mw-redirect" title="Cartesian coordinates">Cartesian coordinates</a>, which are sequences of three real numbers (in the case of the usual <a href="/wiki/Three-dimensional_space" title="Three-dimensional space">three-dimensional space</a>). The basic objects of geometry, which are <a href="/wiki/Line_(geometry)" title="Line (geometry)">lines</a> and <a href="/wiki/Plane_(geometry)" class="mw-redirect" title="Plane (geometry)">planes</a> are represented by linear equations. Thus, computing intersections of lines and planes amounts to solving systems of linear equations. This was one of the main motivations for developing linear algebra. </p><p>Most <a href="/wiki/Geometric_transformation" title="Geometric transformation">geometric transformation</a>, such as <a href="/wiki/Translation_(geometry)" title="Translation (geometry)">translations</a>, <a href="/wiki/Rotation" title="Rotation">rotations</a>, <a href="/wiki/Reflection_(mathematics)" title="Reflection (mathematics)">reflections</a>, <a href="/wiki/Rigid_motion" class="mw-redirect" title="Rigid motion">rigid motions</a>, <a href="/wiki/Isometry" title="Isometry">isometries</a>, and <a href="/wiki/Projection_(mathematics)" title="Projection (mathematics)">projections</a> transform lines into lines. It follows that they can be defined, specified, and studied in terms of linear maps. This is also the case of <a href="/wiki/Homography" title="Homography">homographies</a> and <a href="/wiki/M%C3%B6bius_transformation" title="Möbius transformation">Möbius transformations</a> when considered as transformations of a <a href="/wiki/Projective_space" title="Projective space">projective space</a>. </p><p>Until the end of the 19th century, geometric spaces were defined by <a href="/wiki/Axiom" title="Axiom">axioms</a> relating points, lines, and planes (<a href="/wiki/Synthetic_geometry" title="Synthetic geometry">synthetic geometry</a>). Around this date, it appeared that one may also define geometric spaces by constructions involving vector spaces (see, for example, <a href="/wiki/Projective_space" title="Projective space">Projective space</a> and <a href="/wiki/Affine_space" title="Affine space">Affine space</a>). It has been shown that the two approaches are essentially equivalent.<sup id="cite_ref-23" class="reference"><a href="#cite_note-23"><span class="cite-bracket">[</span>22<span class="cite-bracket">]</span></a></sup> In classical geometry, the involved vector spaces are vector spaces over the reals, but the constructions may be extended to vector spaces over any field, allowing considering geometry over arbitrary fields, including <a href="/wiki/Finite_field" title="Finite field">finite fields</a>. </p><p>Presently, most textbooks introduce geometric spaces from linear algebra, and geometry is often presented, at the elementary level, as a subfield of linear algebra. </p> <div class="mw-heading mw-heading2"><h2 id="Usage_and_applications">Usage and applications<span class="anchor" id="Applications"></span></h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Linear_algebra&action=edit&section=14" title="Edit section: Usage and applications"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Linear algebra is used in almost all areas of mathematics, thus making it relevant in almost all scientific domains that use mathematics. These applications may be divided into several wide categories. </p> <div class="mw-heading mw-heading3"><h3 id="Functional_analysis">Functional analysis</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Linear_algebra&action=edit&section=15" title="Edit section: Functional analysis"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><a href="/wiki/Functional_analysis" title="Functional analysis">Functional analysis</a> studies <a href="/wiki/Function_space" title="Function space">function spaces</a>. These are vector spaces with additional structure, such as <a href="/wiki/Hilbert_space" title="Hilbert space">Hilbert spaces</a>. Linear algebra is thus a fundamental part of functional analysis and its applications, which include, in particular, <a href="/wiki/Quantum_mechanics" title="Quantum mechanics">quantum mechanics</a> (<a href="/wiki/Wave_function" title="Wave function">wave functions</a>) and <a href="/wiki/Fourier_analysis" title="Fourier analysis">Fourier analysis</a> (<a href="/wiki/Orthogonal_basis" title="Orthogonal basis">orthogonal basis</a>). </p> <div class="mw-heading mw-heading3"><h3 id="Scientific_computation">Scientific computation</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Linear_algebra&action=edit&section=16" title="Edit section: Scientific computation"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Nearly all <a href="/wiki/Scientific_computation" class="mw-redirect" title="Scientific computation">scientific computations</a> involve linear algebra. Consequently, linear algebra algorithms have been highly optimized. <a href="/wiki/Basic_Linear_Algebra_Subprograms" title="Basic Linear Algebra Subprograms">BLAS</a> and <a href="/wiki/LAPACK" title="LAPACK">LAPACK</a> are the best known implementations. For improving efficiency, some of them configure the algorithms automatically, at run time, to adapt them to the specificities of the computer (<a href="/wiki/Cache_(computing)" title="Cache (computing)">cache</a> size, number of available <a href="/wiki/Multi-core_processor" title="Multi-core processor">cores</a>, ...). </p><p>Since the 1960s there have been processors with specialized instructions<sup id="cite_ref-24" class="reference"><a href="#cite_note-24"><span class="cite-bracket">[</span>23<span class="cite-bracket">]</span></a></sup> for optimizing the operations of linear algebra, optional array processors<sup id="cite_ref-25" class="reference"><a href="#cite_note-25"><span class="cite-bracket">[</span>24<span class="cite-bracket">]</span></a></sup> under the control of a conventional processor, supercomputers<sup id="cite_ref-26" class="reference"><a href="#cite_note-26"><span class="cite-bracket">[</span>25<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Star100HW_27-0" class="reference"><a href="#cite_note-Star100HW-27"><span class="cite-bracket">[</span>26<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-cray1hw_28-0" class="reference"><a href="#cite_note-cray1hw-28"><span class="cite-bracket">[</span>27<span class="cite-bracket">]</span></a></sup> designed for array processing and conventional processors augmented<sup id="cite_ref-29" class="reference"><a href="#cite_note-29"><span class="cite-bracket">[</span>28<span class="cite-bracket">]</span></a></sup> with vector registers. </p><p>Some contemporary <a href="/wiki/Processor_(computing)" title="Processor (computing)">processors</a>, typically <a href="/wiki/Graphics_processing_units" class="mw-redirect" title="Graphics processing units">graphics processing units</a> (GPU), are designed with a matrix structure, for optimizing the operations of linear algebra.<sup id="cite_ref-30" class="reference"><a href="#cite_note-30"><span class="cite-bracket">[</span>29<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Geometry_of_ambient_space">Geometry of ambient space</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Linear_algebra&action=edit&section=17" title="Edit section: Geometry of ambient space"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The <a href="/wiki/Mathematical_model" title="Mathematical model">modeling</a> of <a href="/wiki/Ambient_space" class="mw-redirect" title="Ambient space">ambient space</a> is based on <a href="/wiki/Geometry" title="Geometry">geometry</a>. Sciences concerned with this space use geometry widely. This is the case with <a href="/wiki/Mechanics" title="Mechanics">mechanics</a> and <a href="/wiki/Robotics" title="Robotics">robotics</a>, for describing <a href="/wiki/Rigid_body_dynamics" title="Rigid body dynamics">rigid body dynamics</a>; <a href="/wiki/Geodesy" title="Geodesy">geodesy</a> for describing <a href="/wiki/Earth_shape" class="mw-redirect" title="Earth shape">Earth shape</a>; <a href="/wiki/Perspectivity" title="Perspectivity">perspectivity</a>, <a href="/wiki/Computer_vision" title="Computer vision">computer vision</a>, and <a href="/wiki/Computer_graphics" title="Computer graphics">computer graphics</a>, for describing the relationship between a scene and its plane representation; and many other scientific domains. </p><p>In all these applications, <a href="/wiki/Synthetic_geometry" title="Synthetic geometry">synthetic geometry</a> is often used for general descriptions and a qualitative approach, but for the study of explicit situations, one must compute with <a href="/wiki/Coordinates" class="mw-redirect" title="Coordinates">coordinates</a>. This requires the heavy use of linear algebra. </p> <div class="mw-heading mw-heading3"><h3 id="Study_of_complex_systems">Study of complex systems</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Linear_algebra&action=edit&section=18" title="Edit section: Study of complex systems"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">See also: <a href="/wiki/Complex_system" title="Complex system">Complex system</a></div> <p>Most physical phenomena are modeled by <a href="/wiki/Partial_differential_equation" title="Partial differential equation">partial differential equations</a>. To solve them, one usually decomposes the space in which the solutions are searched into small, mutually interacting <a href="/wiki/Discretization" title="Discretization">cells</a>. For <a href="/wiki/Linear_system" title="Linear system">linear systems</a> this interaction involves <a href="/wiki/Linear_function" title="Linear function">linear functions</a>. For <a href="/wiki/Nonlinear_systems" class="mw-redirect" title="Nonlinear systems">nonlinear systems</a>, this interaction is often approximated by linear functions.<sup id="cite_ref-31" class="reference"><a href="#cite_note-31"><span class="cite-bracket">[</span>b<span class="cite-bracket">]</span></a></sup>This is called a linear model or first-order approximation. Linear models are frequently used for complex nonlinear real-world systems because they make <a href="/wiki/Parametrization_(geometry)" title="Parametrization (geometry)">parametrization</a> more manageable.<sup id="cite_ref-32" class="reference"><a href="#cite_note-32"><span class="cite-bracket">[</span>30<span class="cite-bracket">]</span></a></sup> In both cases, very large matrices are generally involved. <a href="/wiki/Weather_forecasting" title="Weather forecasting">Weather forecasting</a> (or more specifically, <a href="/wiki/Parametrization_(atmospheric_modeling)" class="mw-redirect" title="Parametrization (atmospheric modeling)">parametrization for atmospheric modeling</a>) is a typical example of a real-world application, where the whole Earth <a href="/wiki/Atmosphere" title="Atmosphere">atmosphere</a> is divided into cells of, say, 100 km of width and 100 km of height. </p> <div class="mw-heading mw-heading3"><h3 id="Fluid_mechanics,_fluid_dynamics,_and_thermal_energy_systems"><span id="Fluid_mechanics.2C_fluid_dynamics.2C_and_thermal_energy_systems"></span>Fluid mechanics, fluid dynamics, and thermal energy systems</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Linear_algebra&action=edit&section=19" title="Edit section: Fluid mechanics, fluid dynamics, and thermal energy systems"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><sup id="cite_ref-33" class="reference"><a href="#cite_note-33"><span class="cite-bracket">[</span>31<span class="cite-bracket">]</span></a></sup> <sup id="cite_ref-34" class="reference"><a href="#cite_note-34"><span class="cite-bracket">[</span>32<span class="cite-bracket">]</span></a></sup> <sup id="cite_ref-35" class="reference"><a href="#cite_note-35"><span class="cite-bracket">[</span>33<span class="cite-bracket">]</span></a></sup> </p><p>Linear algebra, a branch of mathematics dealing with <a href="/wiki/Vector_spaces" class="mw-redirect" title="Vector spaces">vector spaces</a> and <a href="/wiki/Linear_mapping" class="mw-redirect" title="Linear mapping">linear mappings</a> between these spaces, plays a critical role in various engineering disciplines, including <a href="/wiki/Fluid_mechanics" title="Fluid mechanics">fluid mechanics</a>, <a href="/wiki/Fluid_dynamics" title="Fluid dynamics">fluid dynamics</a>, and <a href="/wiki/Thermal_energy" title="Thermal energy">thermal energy</a> systems. Its application in these fields is multifaceted and indispensable for solving complex problems. </p><p>In <a href="/wiki/Fluid_mechanics" title="Fluid mechanics">fluid mechanics</a>, linear algebra is integral to understanding and solving problems related to the behavior of fluids. It assists in the modeling and simulation of fluid flow, providing essential tools for the analysis of <a href="/wiki/Fluid_dynamics" title="Fluid dynamics">fluid dynamics</a> problems. For instance, linear algebraic techniques are used to solve systems of <a href="/wiki/Differential_equations" class="mw-redirect" title="Differential equations">differential equations</a> that describe fluid motion. These equations, often complex and <a href="/wiki/Non-linear" class="mw-redirect" title="Non-linear">non-linear</a>, can be linearized using linear algebra methods, allowing for simpler solutions and analyses. </p><p>In the field of fluid dynamics, linear algebra finds its application in <a href="/wiki/Computational_fluid_dynamics" title="Computational fluid dynamics">computational fluid dynamics</a> (CFD), a branch that uses <a href="/wiki/Numerical_analysis" title="Numerical analysis">numerical analysis</a> and <a href="/wiki/Data_structure" title="Data structure">data structures</a> to solve and analyze problems involving fluid flows. CFD relies heavily on linear algebra for the computation of fluid flow and <a href="/wiki/Heat_transfer" title="Heat transfer">heat transfer</a> in various applications. For example, the <a href="/wiki/Navier%E2%80%93Stokes_equations" title="Navier–Stokes equations">Navier–Stokes equations</a>, fundamental in <a href="/wiki/Fluid_dynamics" title="Fluid dynamics">fluid dynamics</a>, are often solved using techniques derived from linear algebra. This includes the use of <a href="/wiki/Matrix_(mathematics)" title="Matrix (mathematics)">matrices</a> and <a href="/wiki/Vector_(mathematics_and_physics)" title="Vector (mathematics and physics)">vectors</a> to represent and manipulate fluid flow fields. </p><p>Furthermore, linear algebra plays a crucial role in <a href="/wiki/Thermal_energy" title="Thermal energy">thermal energy</a> systems, particularly in <a href="/wiki/Power_systems" class="mw-redirect" title="Power systems">power systems</a> analysis. It is used to model and optimize the generation, <a href="/wiki/Electric_power_transmission" title="Electric power transmission">transmission</a>, and <a href="/wiki/Electric_power_distribution" title="Electric power distribution">distribution</a> of electric power. Linear algebraic concepts such as matrix operations and <a href="/wiki/Eigenvalue" class="mw-redirect" title="Eigenvalue">eigenvalue</a> problems are employed to enhance the efficiency, reliability, and economic performance of <a href="/wiki/Power_systems" class="mw-redirect" title="Power systems">power systems</a>. The application of linear algebra in this context is vital for the design and operation of modern <a href="/wiki/Power_systems" class="mw-redirect" title="Power systems">power systems</a>, including <a href="/wiki/Renewable_energy" title="Renewable energy">renewable energy</a> sources and <a href="/wiki/Smart_grid" title="Smart grid">smart grids</a>. </p><p>Overall, the application of linear algebra in <a href="/wiki/Fluid_mechanics" title="Fluid mechanics">fluid mechanics</a>, <a href="/wiki/Fluid_dynamics" title="Fluid dynamics">fluid dynamics</a>, and <a href="/wiki/Thermal_energy" title="Thermal energy">thermal energy</a> systems is an example of the profound interconnection between <a href="/wiki/Mathematics" title="Mathematics">mathematics</a> and <a href="/wiki/Engineering" title="Engineering">engineering</a>. It provides engineers with the necessary tools to model, analyze, and solve complex problems in these domains, leading to advancements in technology and industry. </p> <div class="mw-heading mw-heading2"><h2 id="Extensions_and_generalizations">Extensions and generalizations</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Linear_algebra&action=edit&section=20" title="Edit section: Extensions and generalizations"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>This section presents several related topics that do not appear generally in elementary textbooks on linear algebra but are commonly considered, in advanced mathematics, as parts of linear algebra. </p> <div class="mw-heading mw-heading3"><h3 id="Module_theory">Module theory</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Linear_algebra&action=edit&section=21" title="Edit section: Module theory"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Module_(mathematics)" title="Module (mathematics)">Module (mathematics)</a></div> <p>The existence of multiplicative inverses in fields is not involved in the axioms defining a vector space. One may thus replace the field of scalars by a <a href="/wiki/Ring_(mathematics)" title="Ring (mathematics)">ring</a> <span class="texhtml mvar" style="font-style:italic;">R</span>, and this gives the structure called a <b>module</b> over <span class="texhtml mvar" style="font-style:italic;">R</span>, or <span class="texhtml mvar" style="font-style:italic;">R</span>-module. </p><p>The concepts of linear independence, span, basis, and linear maps (also called <a href="/wiki/Module_homomorphism" title="Module homomorphism">module homomorphisms</a>) are defined for modules exactly as for vector spaces, with the essential difference that, if <span class="texhtml mvar" style="font-style:italic;">R</span> is not a field, there are modules that do not have any basis. The modules that have a basis are the <a href="/wiki/Free_module" title="Free module">free modules</a>, and those that are spanned by a finite set are the <a href="/wiki/Finitely_generated_module" title="Finitely generated module">finitely generated modules</a>. Module homomorphisms between finitely generated free modules may be represented by matrices. The theory of matrices over a ring is similar to that of matrices over a field, except that <a href="/wiki/Determinant" title="Determinant">determinants</a> exist only if the ring is <a href="/wiki/Commutative_ring" title="Commutative ring">commutative</a>, and that a square matrix over a commutative ring is <a href="/wiki/Invertible_matrix" title="Invertible matrix">invertible</a> only if its determinant has a <a href="/wiki/Multiplicative_inverse" title="Multiplicative inverse">multiplicative inverse</a> in the ring. </p><p>Vector spaces are completely characterized by their dimension (up to an isomorphism). In general, there is not such a complete classification for modules, even if one restricts oneself to finitely generated modules. However, every module is a <a href="/wiki/Cokernel" title="Cokernel">cokernel</a> of a homomorphism of free modules. </p><p>Modules over the integers can be identified with <a href="/wiki/Abelian_group" title="Abelian group">abelian groups</a>, since the multiplication by an integer may be identified as a repeated addition. Most of the theory of abelian groups may be extended to modules over a <a href="/wiki/Principal_ideal_domain" title="Principal ideal domain">principal ideal domain</a>. In particular, over a principal ideal domain, every submodule of a free module is free, and the <a href="/wiki/Fundamental_theorem_of_finitely_generated_abelian_groups" class="mw-redirect" title="Fundamental theorem of finitely generated abelian groups">fundamental theorem of finitely generated abelian groups</a> may be extended straightforwardly to finitely generated modules over a principal ring. </p><p>There are many rings for which there are algorithms for solving linear equations and systems of linear equations. However, these algorithms have generally a <a href="/wiki/Computational_complexity" title="Computational complexity">computational complexity</a> that is much higher than similar algorithms over a field. For more details, see <a href="/wiki/Linear_equation_over_a_ring" title="Linear equation over a ring">Linear equation over a ring</a>. </p> <div class="mw-heading mw-heading3"><h3 id="Multilinear_algebra_and_tensors">Multilinear algebra and tensors</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Linear_algebra&action=edit&section=22" title="Edit section: Multilinear algebra and tensors"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1251242444">.mw-parser-output .ambox{border:1px solid #a2a9b1;border-left:10px solid #36c;background-color:#fbfbfb;box-sizing:border-box}.mw-parser-output .ambox+link+.ambox,.mw-parser-output .ambox+link+style+.ambox,.mw-parser-output .ambox+link+link+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+style+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+link+.ambox{margin-top:-1px}html body.mediawiki .mw-parser-output .ambox.mbox-small-left{margin:4px 1em 4px 0;overflow:hidden;width:238px;border-collapse:collapse;font-size:88%;line-height:1.25em}.mw-parser-output .ambox-speedy{border-left:10px solid #b32424;background-color:#fee7e6}.mw-parser-output .ambox-delete{border-left:10px solid #b32424}.mw-parser-output .ambox-content{border-left:10px solid #f28500}.mw-parser-output .ambox-style{border-left:10px solid #fc3}.mw-parser-output .ambox-move{border-left:10px solid #9932cc}.mw-parser-output .ambox-protection{border-left:10px solid #a2a9b1}.mw-parser-output .ambox .mbox-text{border:none;padding:0.25em 0.5em;width:100%}.mw-parser-output .ambox .mbox-image{border:none;padding:2px 0 2px 0.5em;text-align:center}.mw-parser-output .ambox .mbox-imageright{border:none;padding:2px 0.5em 2px 0;text-align:center}.mw-parser-output .ambox .mbox-empty-cell{border:none;padding:0;width:1px}.mw-parser-output .ambox .mbox-image-div{width:52px}@media(min-width:720px){.mw-parser-output .ambox{margin:0 10%}}@media print{body.ns-0 .mw-parser-output .ambox{display:none!important}}</style><table class="box-Cleanup plainlinks metadata ambox ambox-style ambox-Cleanup" role="presentation"><tbody><tr><td class="mbox-image"><div class="mbox-image-div"><span typeof="mw:File"><span><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/f/f2/Edit-clear.svg/40px-Edit-clear.svg.png" decoding="async" width="40" height="40" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/f/f2/Edit-clear.svg/60px-Edit-clear.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/f/f2/Edit-clear.svg/80px-Edit-clear.svg.png 2x" data-file-width="48" data-file-height="48" /></span></span></div></td><td class="mbox-text"><div class="mbox-text-span">This section may <b>require <a href="/wiki/Wikipedia:Cleanup" title="Wikipedia:Cleanup">cleanup</a></b> to meet Wikipedia's <a href="/wiki/Wikipedia:Manual_of_Style" title="Wikipedia:Manual of Style">quality standards</a>. The specific problem is: <b>The dual space is considered above, and the section must be rewritten to give an understandable summary of this subject.</b><span class="hide-when-compact"> Please help <a href="/wiki/Special:EditPage/Linear_algebra" title="Special:EditPage/Linear algebra">improve this section</a> if you can.</span> <span class="date-container"><i>(<span class="date">September 2018</span>)</i></span><span class="hide-when-compact"><i> (<small><a href="/wiki/Help:Maintenance_template_removal" title="Help:Maintenance template removal">Learn how and when to remove this message</a></small>)</i></span></div></td></tr></tbody></table> <p>In <a href="/wiki/Multilinear_algebra" title="Multilinear algebra">multilinear algebra</a>, one considers multivariable linear transformations, that is, mappings that are linear in each of several different variables. This line of inquiry naturally leads to the idea of the <a href="/wiki/Dual_space" title="Dual space">dual space</a>, the vector space <span class="texhtml"><i>V*</i></span> consisting of linear maps <span class="texhtml"><i>f</i> : <i>V</i> → <i>F</i></span> where <i>F</i> is the field of scalars. Multilinear maps <span class="texhtml"><i>T</i> : <i>V<sup>n</sup></i> → <i>F</i></span> can be described via <a href="/wiki/Tensor_product" title="Tensor product">tensor products</a> of elements of <span class="texhtml"><i>V*</i></span>. </p><p>If, in addition to vector addition and scalar multiplication, there is a bilinear vector product <span class="texhtml"><i>V</i> × <i>V</i> → <i>V</i></span>, the vector space is called an <a href="/wiki/Algebra_over_a_field" title="Algebra over a field">algebra</a>; for instance, associative algebras are algebras with an associate vector product (like the algebra of square matrices, or the algebra of polynomials). </p> <div class="mw-heading mw-heading3"><h3 id="Topological_vector_spaces">Topological vector spaces</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Linear_algebra&action=edit&section=23" title="Edit section: Topological vector spaces"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main articles: <a href="/wiki/Topological_vector_space" title="Topological vector space">Topological vector space</a>, <a href="/wiki/Normed_vector_space" title="Normed vector space">Normed vector space</a>, and <a href="/wiki/Hilbert_space" title="Hilbert space">Hilbert space</a></div> <p>Vector spaces that are not finite-dimensional often require additional structure to be tractable. A <a href="/wiki/Normed_vector_space" title="Normed vector space">normed vector space</a> is a vector space along with a function called a <a href="/wiki/Norm_(mathematics)" title="Norm (mathematics)">norm</a>, which measures the "size" of elements. The norm induces a <a href="/wiki/Metric_(mathematics)" class="mw-redirect" title="Metric (mathematics)">metric</a>, which measures the distance between elements, and induces a <a href="/wiki/Topological_space" title="Topological space">topology</a>, which allows for a definition of continuous maps. The metric also allows for a definition of <a href="/wiki/Limit_(mathematics)" title="Limit (mathematics)">limits</a> and <a href="/wiki/Complete_metric_space" title="Complete metric space">completeness</a> – a normed vector space that is complete is known as a <a href="/wiki/Banach_space" title="Banach space">Banach space</a>. A complete metric space along with the additional structure of an <a href="/wiki/Inner_product_space" title="Inner product space">inner product</a> (a conjugate symmetric <a href="/wiki/Sesquilinear_form" title="Sesquilinear form">sesquilinear form</a>) is known as a <a href="/wiki/Hilbert_space" title="Hilbert space">Hilbert space</a>, which is in some sense a particularly well-behaved Banach space. <a href="/wiki/Functional_analysis" title="Functional analysis">Functional analysis</a> applies the methods of linear algebra alongside those of <a href="/wiki/Mathematical_analysis" title="Mathematical analysis">mathematical analysis</a> to study various function spaces; the central objects of study in functional analysis are <a href="/wiki/Lp_space" title="Lp space"><span class="texhtml mvar" style="font-style:italic;">L<sup>p</sup></span> spaces</a>, which are Banach spaces, and especially the <span class="texhtml"><i>L</i><sup>2</sup></span> space of square-integrable functions, which is the only Hilbert space among them. Functional analysis is of particular importance to quantum mechanics, the theory of partial differential equations, digital signal processing, and electrical engineering. It also provides the foundation and theoretical framework that underlies the Fourier transform and related methods. </p> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Linear_algebra&action=edit&section=24" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Fundamental_matrix_(computer_vision)" title="Fundamental matrix (computer vision)">Fundamental matrix (computer vision)</a></li> <li><a href="/wiki/Geometric_algebra" title="Geometric algebra">Geometric algebra</a></li> <li><a href="/wiki/Linear_programming" title="Linear programming">Linear programming</a></li> <li><a href="/wiki/Linear_regression" title="Linear regression">Linear regression</a>, a statistical estimation method</li> <li><a href="/wiki/Numerical_linear_algebra" title="Numerical linear algebra">Numerical linear algebra</a></li> <li><a href="/wiki/Outline_of_linear_algebra" title="Outline of linear algebra">Outline of linear algebra</a></li> <li><a href="/wiki/Transformation_matrix" title="Transformation matrix">Transformation matrix</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="Explanatory_notes">Explanatory notes</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Linear_algebra&action=edit&section=25" title="Edit section: Explanatory notes"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist reflist-lower-alpha"> <div class="mw-references-wrap"><ol class="references"> <li id="cite_note-9"><span class="mw-cite-backlink"><b><a href="#cite_ref-9">^</a></b></span> <span class="reference-text">This axiom is not asserting the associativity of an operation, since there are two operations in question, scalar multiplication <span class="texhtml"><i>b</i><b>v</b></span>; and field multiplication: <span class="texhtml"><i>ab</i></span>.</span> </li> <li id="cite_note-31"><span class="mw-cite-backlink"><b><a href="#cite_ref-31">^</a></b></span> <span class="reference-text">This may have the consequence that some physically interesting solutions are omitted.</span> </li> </ol></div></div> <div class="mw-heading mw-heading2"><h2 id="Citations">Citations</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Linear_algebra&action=edit&section=26" title="Edit section: Citations"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239543626"><div class="reflist reflist-columns references-column-width" style="column-width: 30em;"> <ol class="references"> <li id="cite_note-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-1">^</a></b></span> <span class="reference-text"><style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite id="CITEREFBanerjeeRoy2014" class="citation book cs1">Banerjee, Sudipto; Roy, Anindya (2014). <i>Linear Algebra and Matrix Analysis for Statistics</i>. Texts in Statistical Science (1st ed.). Chapman and Hall/CRC. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1420095388" title="Special:BookSources/978-1420095388"><bdi>978-1420095388</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Linear+Algebra+and+Matrix+Analysis+for+Statistics&rft.series=Texts+in+Statistical+Science&rft.edition=1st&rft.pub=Chapman+and+Hall%2FCRC&rft.date=2014&rft.isbn=978-1420095388&rft.aulast=Banerjee&rft.aufirst=Sudipto&rft.au=Roy%2C+Anindya&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALinear+algebra" class="Z3988"></span></span> </li> <li id="cite_note-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-2">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFStrang2005" class="citation book cs1">Strang, Gilbert (July 19, 2005). <i>Linear Algebra and Its Applications</i> (4th ed.). Brooks Cole. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-03-010567-8" title="Special:BookSources/978-0-03-010567-8"><bdi>978-0-03-010567-8</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Linear+Algebra+and+Its+Applications&rft.edition=4th&rft.pub=Brooks+Cole&rft.date=2005-07-19&rft.isbn=978-0-03-010567-8&rft.aulast=Strang&rft.aufirst=Gilbert&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALinear+algebra" class="Z3988"></span></span> </li> <li id="cite_note-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-3">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWeisstein" class="citation web cs1">Weisstein, Eric. <a rel="nofollow" class="external text" href="http://mathworld.wolfram.com/LinearAlgebra.html">"Linear Algebra"</a>. <i><a href="/wiki/MathWorld" title="MathWorld">MathWorld</a></i>. Wolfram<span class="reference-accessdate">. Retrieved <span class="nowrap">16 April</span> 2012</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=MathWorld&rft.atitle=Linear+Algebra&rft.aulast=Weisstein&rft.aufirst=Eric&rft_id=http%3A%2F%2Fmathworld.wolfram.com%2FLinearAlgebra.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALinear+algebra" class="Z3988"></span></span> </li> <li id="cite_note-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-4">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHart2010" class="citation book cs1">Hart, Roger (2010). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=zLPm3xE2qWgC"><i>The Chinese Roots of Linear Algebra</i></a>. <a href="/wiki/JHU_Press" class="mw-redirect" title="JHU Press">JHU Press</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9780801899584" title="Special:BookSources/9780801899584"><bdi>9780801899584</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Chinese+Roots+of+Linear+Algebra&rft.pub=JHU+Press&rft.date=2010&rft.isbn=9780801899584&rft.aulast=Hart&rft.aufirst=Roger&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DzLPm3xE2qWgC&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALinear+algebra" class="Z3988"></span></span> </li> <li id="cite_note-Vitulli,_Marie-5"><span class="mw-cite-backlink">^ <a href="#cite_ref-Vitulli,_Marie_5-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Vitulli,_Marie_5-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-Vitulli,_Marie_5-2"><sup><i><b>c</b></i></sup></a> <a href="#cite_ref-Vitulli,_Marie_5-3"><sup><i><b>d</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFVitulli" class="citation web cs1"><a href="/wiki/Marie_A._Vitulli" title="Marie A. Vitulli">Vitulli, Marie</a>. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20120910034016/http://darkwing.uoregon.edu/~vitulli/441.sp04/LinAlgHistory.html">"A Brief History of Linear Algebra and Matrix Theory"</a>. <i>Department of Mathematics</i>. University of Oregon. Archived from <a rel="nofollow" class="external text" href="http://darkwing.uoregon.edu/~vitulli/441.sp04/LinAlgHistory.html">the original</a> on 2012-09-10<span class="reference-accessdate">. Retrieved <span class="nowrap">2014-07-08</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=Department+of+Mathematics&rft.atitle=A+Brief+History+of+Linear+Algebra+and+Matrix+Theory&rft.aulast=Vitulli&rft.aufirst=Marie&rft_id=http%3A%2F%2Fdarkwing.uoregon.edu%2F~vitulli%2F441.sp04%2FLinAlgHistory.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALinear+algebra" class="Z3988"></span></span> </li> <li id="cite_note-6"><span class="mw-cite-backlink"><b><a href="#cite_ref-6">^</a></b></span> <span class="reference-text">Koecher, M., Remmert, R. (1991). Hamilton’s Quaternions. In: Numbers. Graduate Texts in Mathematics, vol 123. Springer, New York, NY. <a rel="nofollow" class="external free" href="https://doi.org/10.1007/978-1-4612-1005-4_10">https://doi.org/10.1007/978-1-4612-1005-4_10</a></span> </li> <li id="cite_note-7"><span class="mw-cite-backlink"><b><a href="#cite_ref-7">^</a></b></span> <span class="reference-text"><a href="/wiki/Benjamin_Peirce" title="Benjamin Peirce">Benjamin Peirce</a> (1872) <i>Linear Associative Algebra</i>, lithograph, new edition with corrections, notes, and an added 1875 paper by Peirce, plus notes by his son <a href="/wiki/Charles_Sanders_Peirce" title="Charles Sanders Peirce">Charles Sanders Peirce</a>, published in the <i>American Journal of Mathematics</i> v. 4, 1881, Johns Hopkins University, pp. 221–226, <i>Google</i> <a rel="nofollow" class="external text" href="https://books.google.com/books?id=LQgPAAAAIAAJ&pg=PA221">Eprint</a> and as an extract, D. Van Nostrand, 1882, <i>Google</i> <a rel="nofollow" class="external text" href="https://archive.org/details/bub_gb_De0GAAAAYAAJ">Eprint</a>.</span> </li> <li id="cite_note-8"><span class="mw-cite-backlink"><b><a href="#cite_ref-8">^</a></b></span> <span class="reference-text"><a href="#CITEREFRoman2005">Roman (2005</a>, ch. 1, p. 27)</span> </li> <li id="cite_note-10"><span class="mw-cite-backlink"><b><a href="#cite_ref-10">^</a></b></span> <span class="reference-text"><a href="#CITEREFAxler2015">Axler (2015)</a> p. 82, §3.59</span> </li> <li id="cite_note-11"><span class="mw-cite-backlink"><b><a href="#cite_ref-11">^</a></b></span> <span class="reference-text"><a href="#CITEREFAxler2015">Axler (2015)</a> p. 23, §1.45</span> </li> <li id="cite_note-12"><span class="mw-cite-backlink"><b><a href="#cite_ref-12">^</a></b></span> <span class="reference-text"><a href="#CITEREFAnton1987">Anton (1987</a>, p. 2)</span> </li> <li id="cite_note-13"><span class="mw-cite-backlink"><b><a href="#cite_ref-13">^</a></b></span> <span class="reference-text"><a href="#CITEREFBeauregardFraleigh1973">Beauregard & Fraleigh (1973</a>, p. 65)</span> </li> <li id="cite_note-14"><span class="mw-cite-backlink"><b><a href="#cite_ref-14">^</a></b></span> <span class="reference-text"><a href="#CITEREFBurdenFaires1993">Burden & Faires (1993</a>, p. 324)</span> </li> <li id="cite_note-15"><span class="mw-cite-backlink"><b><a href="#cite_ref-15">^</a></b></span> <span class="reference-text"><a href="#CITEREFGolubVan_Loan1996">Golub & Van Loan (1996</a>, p. 87)</span> </li> <li id="cite_note-16"><span class="mw-cite-backlink"><b><a href="#cite_ref-16">^</a></b></span> <span class="reference-text"><a href="#CITEREFHarper1976">Harper (1976</a>, p. 57)</span> </li> <li id="cite_note-17"><span class="mw-cite-backlink"><b><a href="#cite_ref-17">^</a></b></span> <span class="reference-text"><a href="#CITEREFKatznelsonKatznelson2008">Katznelson & Katznelson (2008)</a> pp. 76–77, § 4.4.1–4.4.6</span> </li> <li id="cite_note-18"><span class="mw-cite-backlink"><b><a href="#cite_ref-18">^</a></b></span> <span class="reference-text"><a href="#CITEREFKatznelsonKatznelson2008">Katznelson & Katznelson (2008)</a> p. 37 §2.1.3</span> </li> <li id="cite_note-19"><span class="mw-cite-backlink"><b><a href="#cite_ref-19">^</a></b></span> <span class="reference-text"><a href="#CITEREFHalmos1974">Halmos (1974)</a> p. 20, §13</span> </li> <li id="cite_note-20"><span class="mw-cite-backlink"><b><a href="#cite_ref-20">^</a></b></span> <span class="reference-text"><a href="#CITEREFAxler2015">Axler (2015)</a> p. 101, §3.94</span> </li> <li id="cite_note-Jain-21"><span class="mw-cite-backlink"><b><a href="#cite_ref-Jain_21-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFP._K._Jain,_Khalil_Ahmad1995" class="citation book cs1">P. K. Jain, Khalil Ahmad (1995). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=yZ68h97pnAkC&pg=PA203">"5.1 Definitions and basic properties of inner product spaces and Hilbert spaces"</a>. <i>Functional analysis</i> (2nd ed.). New Age International. p. 203. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/81-224-0801-X" title="Special:BookSources/81-224-0801-X"><bdi>81-224-0801-X</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=5.1+Definitions+and+basic+properties+of+inner+product+spaces+and+Hilbert+spaces&rft.btitle=Functional+analysis&rft.pages=203&rft.edition=2nd&rft.pub=New+Age+International&rft.date=1995&rft.isbn=81-224-0801-X&rft.au=P.+K.+Jain%2C+Khalil+Ahmad&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DyZ68h97pnAkC%26pg%3DPA203&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALinear+algebra" class="Z3988"></span></span> </li> <li id="cite_note-Prugovec̆ki-22"><span class="mw-cite-backlink"><b><a href="#cite_ref-Prugovec̆ki_22-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFEduard_Prugovec̆ki1981" class="citation book cs1">Eduard Prugovec̆ki (1981). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=GxmQxn2PF3IC&pg=PA18">"Definition 2.1"</a>. <i>Quantum mechanics in Hilbert space</i> (2nd ed.). Academic Press. pp. 18 <i>ff</i>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-12-566060-X" title="Special:BookSources/0-12-566060-X"><bdi>0-12-566060-X</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Definition+2.1&rft.btitle=Quantum+mechanics+in+Hilbert+space&rft.pages=18+%27%27ff%27%27&rft.edition=2nd&rft.pub=Academic+Press&rft.date=1981&rft.isbn=0-12-566060-X&rft.au=Eduard+Prugovec%CC%86ki&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DGxmQxn2PF3IC%26pg%3DPA18&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALinear+algebra" class="Z3988"></span></span> </li> <li id="cite_note-23"><span class="mw-cite-backlink"><b><a href="#cite_ref-23">^</a></b></span> <span class="reference-text"><a href="/wiki/Emil_Artin" title="Emil Artin">Emil Artin</a> (1957) <i><a href="/wiki/Geometric_Algebra_(book)" title="Geometric Algebra (book)">Geometric Algebra</a></i> <a href="/wiki/Interscience_Publishers" class="mw-redirect" title="Interscience Publishers">Interscience Publishers</a></span> </li> <li id="cite_note-24"><span class="mw-cite-backlink"><b><a href="#cite_ref-24">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation book cs1"><i>IBM System/36O Model 40 - Sum of Products Instruction-RPQ W12561 - Special Systems Feature</i>. <a href="/wiki/IBM" title="IBM">IBM</a>. L22-6902.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=IBM+System%2F36O+Model+40+-+Sum+of+Products+Instruction-RPQ+W12561+-+Special+Systems+Feature&rft.pub=IBM&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALinear+algebra" class="Z3988"></span></span> </li> <li id="cite_note-25"><span class="mw-cite-backlink"><b><a href="#cite_ref-25">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation book cs1"><i>IBM System/360 Custom Feature Description: 2938 Array Processor Model 1, - RPQ W24563; Model 2, RPQ 815188</i>. <a href="/wiki/IBM" title="IBM">IBM</a>. A24-3519.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=IBM+System%2F360+Custom+Feature+Description%3A+2938+Array+Processor+Model+1%2C+-+RPQ+W24563%3B+Model+2%2C+RPQ+815188&rft.pub=IBM&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALinear+algebra" class="Z3988"></span></span> </li> <li id="cite_note-26"><span class="mw-cite-backlink"><b><a href="#cite_ref-26">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBarnesBrownKatoKuck1968" class="citation journal cs1">Barnes, George; Brown, Richard; Kato, Maso; Kuck, David; Slotnick, Daniel; Stokes, Richard (August 1968). <a rel="nofollow" class="external text" href="http://gordonbell.azurewebsites.net/cgb%20files/computer%20structures%20readings%20and%20examples%201971.pdf">"The ILLIAC IV Computer"</a> <span class="cs1-format">(PDF)</span>. <i><a href="/wiki/IEEE_Transactions_on_Computers" title="IEEE Transactions on Computers">IEEE Transactions on Computers</a></i>. <b>C.17</b> (8): <span class="nowrap">746–</span>757. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1109%2Ftc.1968.229158">10.1109/tc.1968.229158</a>. <a href="/wiki/ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0018-9340">0018-9340</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:206617237">206617237</a><span class="reference-accessdate">. Retrieved <span class="nowrap">October 31,</span> 2024</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=IEEE+Transactions+on+Computers&rft.atitle=The+ILLIAC+IV+Computer&rft.volume=C.17&rft.issue=8&rft.pages=%3Cspan+class%3D%22nowrap%22%3E746-%3C%2Fspan%3E757&rft.date=1968-08&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A206617237%23id-name%3DS2CID&rft.issn=0018-9340&rft_id=info%3Adoi%2F10.1109%2Ftc.1968.229158&rft.aulast=Barnes&rft.aufirst=George&rft.au=Brown%2C+Richard&rft.au=Kato%2C+Maso&rft.au=Kuck%2C+David&rft.au=Slotnick%2C+Daniel&rft.au=Stokes%2C+Richard&rft_id=http%3A%2F%2Fgordonbell.azurewebsites.net%2Fcgb%2520files%2Fcomputer%2520structures%2520readings%2520and%2520examples%25201971.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALinear+algebra" class="Z3988"></span></span> </li> <li id="cite_note-Star100HW-27"><span class="mw-cite-backlink"><b><a href="#cite_ref-Star100HW_27-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation book cs1"><a rel="nofollow" class="external text" href="http://bitsavers.trailing-edge.com/pdf/cdc/cyber/cyber_200/60256000_STAR-100hw_Dec75.pdf"><i>Star-100 - Hardware Reference Manual</i></a> <span class="cs1-format">(PDF)</span>. Revision 9. <a href="/wiki/Control_Data_Corporation" title="Control Data Corporation">Control Data Corporation</a>. December 15, 1975. 60256000<span class="reference-accessdate">. Retrieved <span class="nowrap">October 31,</span> 2024</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Star-100+-+Hardware+Reference+Manual&rft.series=Revision+9&rft.pub=Control+Data+Corporation&rft.date=1975-12-15&rft_id=http%3A%2F%2Fbitsavers.trailing-edge.com%2Fpdf%2Fcdc%2Fcyber%2Fcyber_200%2F60256000_STAR-100hw_Dec75.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALinear+algebra" class="Z3988"></span></span> </li> <li id="cite_note-cray1hw-28"><span class="mw-cite-backlink"><b><a href="#cite_ref-cray1hw_28-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation book cs1"><a rel="nofollow" class="external text" href="http://bitsavers.trailing-edge.com/pdf/cray/CRAY-1/2240004C_CRAY-1_Hardware_Reference_Nov77.pdf"><i>Cray-1 - Computer System - Hardware Reference Manual</i></a> <span class="cs1-format">(PDF)</span>. Rev. C. <a href="/wiki/Cray_Research,_Inc." class="mw-redirect" title="Cray Research, Inc.">Cray Research, Inc.</a> November 4, 1977. 2240004<span class="reference-accessdate">. Retrieved <span class="nowrap">October 31,</span> 2024</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Cray-1+-+Computer+System+-+Hardware+Reference+Manual&rft.series=Rev.+C&rft.pub=Cray+Research%2C+Inc.&rft.date=1977-11-04&rft_id=http%3A%2F%2Fbitsavers.trailing-edge.com%2Fpdf%2Fcray%2FCRAY-1%2F2240004C_CRAY-1_Hardware_Reference_Nov77.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALinear+algebra" class="Z3988"></span></span> </li> <li id="cite_note-29"><span class="mw-cite-backlink"><b><a href="#cite_ref-29">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation book cs1"><a rel="nofollow" class="external text" href="http://bitsavers.org/pdf/ibm/370/vectorFacility/SA22-7125-3_Vector_Operations_Aug88.pdf"><i>IBM Enterprise Systems Architecture/370 and System/370 Vector Operations</i></a> <span class="cs1-format">(PDF)</span> (Fourth ed.). <a href="/wiki/IBM" title="IBM">IBM</a>. August 1988. SA22-7125-3<span class="reference-accessdate">. Retrieved <span class="nowrap">October 31,</span> 2024</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=IBM+Enterprise+Systems+Architecture%2F370+and+System%2F370+Vector+Operations&rft.edition=Fourth&rft.pub=IBM&rft.date=1988-08&rft_id=http%3A%2F%2Fbitsavers.org%2Fpdf%2Fibm%2F370%2FvectorFacility%2FSA22-7125-3_Vector_Operations_Aug88.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALinear+algebra" class="Z3988"></span></span> </li> <li id="cite_note-30"><span class="mw-cite-backlink"><b><a href="#cite_ref-30">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://docs.nvidia.com/deeplearning/performance/dl-performance-gpu-background/index.html">"GPU Performance Background User's Guide"</a>. <i>NVIDIA Docs</i><span class="reference-accessdate">. Retrieved <span class="nowrap">2024-10-29</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=NVIDIA+Docs&rft.atitle=GPU+Performance+Background+User%27s+Guide&rft_id=https%3A%2F%2Fdocs.nvidia.com%2Fdeeplearning%2Fperformance%2Fdl-performance-gpu-background%2Findex.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALinear+algebra" class="Z3988"></span></span> </li> <li id="cite_note-32"><span class="mw-cite-backlink"><b><a href="#cite_ref-32">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSavov2017" class="citation book cs1">Savov, Ivan (2017). <i>No Bullshit Guide to Linear Algebra</i>. MinireferenceCo. pp. <span class="nowrap">150–</span>155. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9780992001025" title="Special:BookSources/9780992001025"><bdi>9780992001025</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=No+Bullshit+Guide+to+Linear+Algebra&rft.pages=%3Cspan+class%3D%22nowrap%22%3E150-%3C%2Fspan%3E155&rft.pub=MinireferenceCo.&rft.date=2017&rft.isbn=9780992001025&rft.aulast=Savov&rft.aufirst=Ivan&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALinear+algebra" class="Z3988"></span></span> </li> <li id="cite_note-33"><span class="mw-cite-backlink"><b><a href="#cite_ref-33">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://ocw.mit.edu/courses/2-035-special-topics-in-mathematics-with-applications-linear-algebra-and-the-calculus-of-variations-spring-2007/">"MIT OpenCourseWare. Special Topics in Mathematics with Applications: Linear Algebra and the Calculus of Variations - Mechanical Engineering"</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=MIT+OpenCourseWare.+Special+Topics+in+Mathematics+with+Applications%3A+Linear+Algebra+and+the+Calculus+of+Variations+-+Mechanical+Engineering&rft_id=https%3A%2F%2Focw.mit.edu%2Fcourses%2F2-035-special-topics-in-mathematics-with-applications-linear-algebra-and-the-calculus-of-variations-spring-2007%2F&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALinear+algebra" class="Z3988"></span></span> </li> <li id="cite_note-34"><span class="mw-cite-backlink"><b><a href="#cite_ref-34">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://engineering.ucdenver.edu/electrical-engineering/research/energy-and-power-systems#:~:text=Power%20systems%20analysis%20deals%20with,the%20analysis%20of%20power%20systems">"FAMU-FSU College of Engineering. ME Undergraduate Curriculum"</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=FAMU-FSU+College+of+Engineering.+ME+Undergraduate+Curriculum&rft_id=https%3A%2F%2Fengineering.ucdenver.edu%2Felectrical-engineering%2Fresearch%2Fenergy-and-power-systems%23%3A~%3Atext%3DPower%2520systems%2520analysis%2520deals%2520with%2Cthe%2520analysis%2520of%2520power%2520systems&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALinear+algebra" class="Z3988"></span></span> </li> <li id="cite_note-35"><span class="mw-cite-backlink"><b><a href="#cite_ref-35">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://eng.famu.fsu.edu/me/undergraduate-curriculum#:~:text=MAS%203105%20Linear%20Algebra%20%283%29,and%20eigenvectors%2C%20linear%20transformations%2C%20applications)">"University of Colorado Denver. Energy and Power Systems"</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=University+of+Colorado+Denver.+Energy+and+Power+Systems&rft_id=https%3A%2F%2Feng.famu.fsu.edu%2Fme%2Fundergraduate-curriculum%23%3A~%3Atext%3DMAS%25203105%2520Linear%2520Algebra%2520%25283%2529%2Cand%2520eigenvectors%252C%2520linear%2520transformations%252C%2520applications%29&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALinear+algebra" class="Z3988"></span></span> </li> </ol></div> <div class="mw-heading mw-heading2"><h2 id="General_and_cited_sources">General and cited sources</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Linear_algebra&action=edit&section=27" title="Edit section: General and cited sources"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239549316">.mw-parser-output .refbegin{margin-bottom:0.5em}.mw-parser-output .refbegin-hanging-indents>ul{margin-left:0}.mw-parser-output .refbegin-hanging-indents>ul>li{margin-left:0;padding-left:3.2em;text-indent:-3.2em}.mw-parser-output .refbegin-hanging-indents ul,.mw-parser-output .refbegin-hanging-indents ul li{list-style:none}@media(max-width:720px){.mw-parser-output .refbegin-hanging-indents>ul>li{padding-left:1.6em;text-indent:-1.6em}}.mw-parser-output .refbegin-columns{margin-top:0.3em}.mw-parser-output .refbegin-columns ul{margin-top:0}.mw-parser-output .refbegin-columns li{page-break-inside:avoid;break-inside:avoid-column}@media screen{.mw-parser-output .refbegin{font-size:90%}}</style><div class="refbegin" style=""> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFAnton1987" class="citation cs2">Anton, Howard (1987), <i>Elementary Linear Algebra</i> (5th ed.), New York: <a href="/wiki/John_Wiley_%26_Sons" class="mw-redirect" title="John Wiley & Sons">Wiley</a>, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-471-84819-0" title="Special:BookSources/0-471-84819-0"><bdi>0-471-84819-0</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Elementary+Linear+Algebra&rft.place=New+York&rft.edition=5th&rft.pub=Wiley&rft.date=1987&rft.isbn=0-471-84819-0&rft.aulast=Anton&rft.aufirst=Howard&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALinear+algebra" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFAxler2015" class="citation cs2"><a href="/wiki/Sheldon_Axler" title="Sheldon Axler">Axler, Sheldon</a> (2024), <a rel="nofollow" class="external text" href="https://link.springer.com/book/10.1007/978-3-031-41026-0"><i>Linear Algebra Done Right</i></a>, <a href="/wiki/Undergraduate_Texts_in_Mathematics" title="Undergraduate Texts in Mathematics">Undergraduate Texts in Mathematics</a> (4th ed.), <a href="/wiki/Springer_Publishing" title="Springer Publishing">Springer Publishing</a>, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-031-41026-0" title="Special:BookSources/978-3-031-41026-0"><bdi>978-3-031-41026-0</bdi></a>, <a href="/wiki/MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=3308468">3308468</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Linear+Algebra+Done+Right&rft.series=Undergraduate+Texts+in+Mathematics&rft.edition=4th&rft.pub=Springer+Publishing&rft.date=2024&rft.isbn=978-3-031-41026-0&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D3308468%23id-name%3DMR&rft.aulast=Axler&rft.aufirst=Sheldon&rft_id=https%3A%2F%2Flink.springer.com%2Fbook%2F10.1007%2F978-3-031-41026-0&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALinear+algebra" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBeauregardFraleigh1973" class="citation cs2">Beauregard, Raymond A.; Fraleigh, John B. (1973), <span class="id-lock-registration" title="Free registration required"><a rel="nofollow" class="external text" href="https://archive.org/details/firstcourseinlin0000beau"><i>A First Course In Linear Algebra: with Optional Introduction to Groups, Rings, and Fields</i></a></span>, Boston: <a href="/wiki/Houghton_Mifflin_Company" class="mw-redirect" title="Houghton Mifflin Company">Houghton Mifflin Company</a>, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-395-14017-X" title="Special:BookSources/0-395-14017-X"><bdi>0-395-14017-X</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=A+First+Course+In+Linear+Algebra%3A+with+Optional+Introduction+to+Groups%2C+Rings%2C+and+Fields&rft.place=Boston&rft.pub=Houghton+Mifflin+Company&rft.date=1973&rft.isbn=0-395-14017-X&rft.aulast=Beauregard&rft.aufirst=Raymond+A.&rft.au=Fraleigh%2C+John+B.&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Ffirstcourseinlin0000beau&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALinear+algebra" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBurdenFaires1993" class="citation cs2">Burden, Richard L.; Faires, J. Douglas (1993), <span class="id-lock-registration" title="Free registration required"><a rel="nofollow" class="external text" href="https://archive.org/details/numericalanalysi00burd"><i>Numerical Analysis</i></a></span> (5th ed.), Boston: <a href="/w/index.php?title=Prindle,_Weber_and_Schmidt&action=edit&redlink=1" class="new" title="Prindle, Weber and Schmidt (page does not exist)">Prindle, Weber and Schmidt</a>, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-534-93219-3" title="Special:BookSources/0-534-93219-3"><bdi>0-534-93219-3</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Numerical+Analysis&rft.place=Boston&rft.edition=5th&rft.pub=Prindle%2C+Weber+and+Schmidt&rft.date=1993&rft.isbn=0-534-93219-3&rft.aulast=Burden&rft.aufirst=Richard+L.&rft.au=Faires%2C+J.+Douglas&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fnumericalanalysi00burd&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALinear+algebra" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGolubVan_Loan1996" class="citation cs2"><a href="/wiki/Gene_H._Golub" title="Gene H. Golub">Golub, Gene H.</a>; <a href="/wiki/Charles_F._Van_Loan" title="Charles F. Van Loan">Van Loan, Charles F.</a> (1996), <i>Matrix Computations</i>, Johns Hopkins Studies in Mathematical Sciences (3rd ed.), Baltimore: <a href="/wiki/Johns_Hopkins_University_Press" title="Johns Hopkins University Press">Johns Hopkins University Press</a>, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-8018-5414-9" title="Special:BookSources/978-0-8018-5414-9"><bdi>978-0-8018-5414-9</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Matrix+Computations&rft.place=Baltimore&rft.series=Johns+Hopkins+Studies+in+Mathematical+Sciences&rft.edition=3rd&rft.pub=Johns+Hopkins+University+Press&rft.date=1996&rft.isbn=978-0-8018-5414-9&rft.aulast=Golub&rft.aufirst=Gene+H.&rft.au=Van+Loan%2C+Charles+F.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALinear+algebra" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHalmos1974" class="citation cs2"><a href="/wiki/Paul_Halmos" title="Paul Halmos">Halmos, Paul Richard</a> (1974), <a rel="nofollow" class="external text" href="https://www.worldcat.org/oclc/1251216"><i>Finite-Dimensional Vector Spaces</i></a>, <a href="/wiki/Undergraduate_Texts_in_Mathematics" title="Undergraduate Texts in Mathematics">Undergraduate Texts in Mathematics</a> (1958 2nd ed.), <a href="/wiki/Springer_Publishing" title="Springer Publishing">Springer Publishing</a>, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-387-90093-4" title="Special:BookSources/0-387-90093-4"><bdi>0-387-90093-4</bdi></a>, <a href="/wiki/OCLC_(identifier)" class="mw-redirect" title="OCLC (identifier)">OCLC</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/oclc/1251216">1251216</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Finite-Dimensional+Vector+Spaces&rft.series=Undergraduate+Texts+in+Mathematics&rft.edition=1958+2nd&rft.pub=Springer+Publishing&rft.date=1974&rft_id=info%3Aoclcnum%2F1251216&rft.isbn=0-387-90093-4&rft.aulast=Halmos&rft.aufirst=Paul+Richard&rft_id=https%3A%2F%2Fwww.worldcat.org%2Foclc%2F1251216&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALinear+algebra" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHarper1976" class="citation cs2">Harper, Charlie (1976), <i>Introduction to Mathematical Physics</i>, New Jersey: <a href="/wiki/Prentice-Hall" class="mw-redirect" title="Prentice-Hall">Prentice-Hall</a>, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-13-487538-9" title="Special:BookSources/0-13-487538-9"><bdi>0-13-487538-9</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Introduction+to+Mathematical+Physics&rft.place=New+Jersey&rft.pub=Prentice-Hall&rft.date=1976&rft.isbn=0-13-487538-9&rft.aulast=Harper&rft.aufirst=Charlie&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALinear+algebra" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKatznelsonKatznelson2008" class="citation cs2"><a href="/wiki/Yitzhak_Katznelson" title="Yitzhak Katznelson">Katznelson, Yitzhak</a>; Katznelson, Yonatan R. (2008), <i>A (Terse) Introduction to Linear Algebra</i>, <a href="/wiki/American_Mathematical_Society" title="American Mathematical Society">American Mathematical Society</a>, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-8218-4419-9" title="Special:BookSources/978-0-8218-4419-9"><bdi>978-0-8218-4419-9</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=A+%28Terse%29+Introduction+to+Linear+Algebra&rft.pub=American+Mathematical+Society&rft.date=2008&rft.isbn=978-0-8218-4419-9&rft.aulast=Katznelson&rft.aufirst=Yitzhak&rft.au=Katznelson%2C+Yonatan+R.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALinear+algebra" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFRoman2005" class="citation cs2"><a href="/wiki/Steven_Roman" title="Steven Roman">Roman, Steven</a> (March 22, 2005), <i>Advanced Linear Algebra</i>, <a href="/wiki/Graduate_Texts_in_Mathematics" title="Graduate Texts in Mathematics">Graduate Texts in Mathematics</a> (2nd ed.), Springer, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-387-24766-3" title="Special:BookSources/978-0-387-24766-3"><bdi>978-0-387-24766-3</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Advanced+Linear+Algebra&rft.series=Graduate+Texts+in+Mathematics&rft.edition=2nd&rft.pub=Springer&rft.date=2005-03-22&rft.isbn=978-0-387-24766-3&rft.aulast=Roman&rft.aufirst=Steven&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALinear+algebra" class="Z3988"></span></li></ul> </div> <div class="mw-heading mw-heading2"><h2 id="Further_reading">Further reading</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Linear_algebra&action=edit&section=28" title="Edit section: Further reading"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="History_2">History</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Linear_algebra&action=edit&section=29" title="Edit section: History"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li>Fearnley-Sander, Desmond, "<a rel="nofollow" class="external text" href="https://www.jstor.org/stable/pdf/2320145.pdf">Hermann Grassmann and the Creation of Linear Algebra</a>", American Mathematical Monthly <b>86</b> (1979), pp. 809–817.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGrassmann1844" class="citation cs2"><a href="/wiki/Hermann_Grassmann" title="Hermann Grassmann">Grassmann, Hermann</a> (1844), <i>Die lineale Ausdehnungslehre ein neuer Zweig der Mathematik: dargestellt und durch Anwendungen auf die übrigen Zweige der Mathematik, wie auch auf die Statik, Mechanik, die Lehre vom Magnetismus und die Krystallonomie erläutert</i>, Leipzig: O. Wigand</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Die+lineale+Ausdehnungslehre+ein+neuer+Zweig+der+Mathematik%3A+dargestellt+und+durch+Anwendungen+auf+die+%C3%BCbrigen+Zweige+der+Mathematik%2C+wie+auch+auf+die+Statik%2C+Mechanik%2C+die+Lehre+vom+Magnetismus+und+die+Krystallonomie+erl%C3%A4utert&rft.place=Leipzig&rft.pub=O.+Wigand&rft.date=1844&rft.aulast=Grassmann&rft.aufirst=Hermann&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALinear+algebra" class="Z3988"></span></li></ul> <div class="mw-heading mw-heading3"><h3 id="Introductory_textbooks">Introductory textbooks</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Linear_algebra&action=edit&section=30" title="Edit section: Introductory textbooks"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFAnton2005" class="citation cs2">Anton, Howard (2005), <i>Elementary Linear Algebra (Applications Version)</i> (9th ed.), Wiley International</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Elementary+Linear+Algebra+%28Applications+Version%29&rft.edition=9th&rft.pub=Wiley+International&rft.date=2005&rft.aulast=Anton&rft.aufirst=Howard&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALinear+algebra" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBanerjeeRoy2014" class="citation cs2">Banerjee, Sudipto; Roy, Anindya (2014), <i>Linear Algebra and Matrix Analysis for Statistics</i>, Texts in Statistical Science (1st ed.), Chapman and Hall/CRC, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1420095388" title="Special:BookSources/978-1420095388"><bdi>978-1420095388</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Linear+Algebra+and+Matrix+Analysis+for+Statistics&rft.series=Texts+in+Statistical+Science&rft.edition=1st&rft.pub=Chapman+and+Hall%2FCRC&rft.date=2014&rft.isbn=978-1420095388&rft.aulast=Banerjee&rft.aufirst=Sudipto&rft.au=Roy%2C+Anindya&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALinear+algebra" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBretscher2004" class="citation cs2">Bretscher, Otto (2004), <i>Linear Algebra with Applications</i> (3rd ed.), Prentice Hall, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-13-145334-0" title="Special:BookSources/978-0-13-145334-0"><bdi>978-0-13-145334-0</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Linear+Algebra+with+Applications&rft.edition=3rd&rft.pub=Prentice+Hall&rft.date=2004&rft.isbn=978-0-13-145334-0&rft.aulast=Bretscher&rft.aufirst=Otto&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALinear+algebra" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFFarinHansford2004" class="citation cs2">Farin, Gerald; <a href="/wiki/Dianne_Hansford" title="Dianne Hansford">Hansford, Dianne</a> (2004), <i>Practical Linear Algebra: A Geometry Toolbox</i>, AK Peters, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-56881-234-2" title="Special:BookSources/978-1-56881-234-2"><bdi>978-1-56881-234-2</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Practical+Linear+Algebra%3A+A+Geometry+Toolbox&rft.pub=AK+Peters&rft.date=2004&rft.isbn=978-1-56881-234-2&rft.aulast=Farin&rft.aufirst=Gerald&rft.au=Hansford%2C+Dianne&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALinear+algebra" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHefferon2020" class="citation book cs1"><a href="/wiki/Jim_Hefferon" title="Jim Hefferon">Hefferon, Jim</a> (2020). <a rel="nofollow" class="external text" href="https://hefferon.net/linearalgebra/"><i>Linear Algebra</i></a> (4th ed.). <a href="/wiki/Ann_Arbor,_Michigan" title="Ann Arbor, Michigan">Ann Arbor, Michigan</a>: Orthogonal Publishing. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-944325-11-4" title="Special:BookSources/978-1-944325-11-4"><bdi>978-1-944325-11-4</bdi></a>. <a href="/wiki/OCLC_(identifier)" class="mw-redirect" title="OCLC (identifier)">OCLC</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/oclc/1178900366">1178900366</a>. <a href="/wiki/OL_(identifier)" class="mw-redirect" title="OL (identifier)">OL</a> <a rel="nofollow" class="external text" href="https://openlibrary.org/books/OL30872051M">30872051M</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Linear+Algebra&rft.place=Ann+Arbor%2C+Michigan&rft.edition=4th&rft.pub=Orthogonal+Publishing&rft.date=2020&rft_id=info%3Aoclcnum%2F1178900366&rft_id=https%3A%2F%2Fopenlibrary.org%2Fbooks%2FOL30872051M%23id-name%3DOL&rft.isbn=978-1-944325-11-4&rft.aulast=Hefferon&rft.aufirst=Jim&rft_id=https%3A%2F%2Fhefferon.net%2Flinearalgebra%2F&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALinear+algebra" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKolmanHill2007" class="citation cs2">Kolman, Bernard; Hill, David R. (2007), <i>Elementary Linear Algebra with Applications</i> (9th ed.), Prentice Hall, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-13-229654-0" title="Special:BookSources/978-0-13-229654-0"><bdi>978-0-13-229654-0</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Elementary+Linear+Algebra+with+Applications&rft.edition=9th&rft.pub=Prentice+Hall&rft.date=2007&rft.isbn=978-0-13-229654-0&rft.aulast=Kolman&rft.aufirst=Bernard&rft.au=Hill%2C+David+R.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALinear+algebra" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFLay2005" class="citation cs2">Lay, David C. (2005), <i>Linear Algebra and Its Applications</i> (3rd ed.), Addison Wesley, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-321-28713-7" title="Special:BookSources/978-0-321-28713-7"><bdi>978-0-321-28713-7</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Linear+Algebra+and+Its+Applications&rft.edition=3rd&rft.pub=Addison+Wesley&rft.date=2005&rft.isbn=978-0-321-28713-7&rft.aulast=Lay&rft.aufirst=David+C.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALinear+algebra" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFLeon2006" class="citation cs2">Leon, Steven J. (2006), <span class="id-lock-registration" title="Free registration required"><a rel="nofollow" class="external text" href="https://archive.org/details/linearalgebrawit00leon"><i>Linear Algebra With Applications</i></a></span> (7th ed.), Pearson Prentice Hall, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-13-185785-8" title="Special:BookSources/978-0-13-185785-8"><bdi>978-0-13-185785-8</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Linear+Algebra+With+Applications&rft.edition=7th&rft.pub=Pearson+Prentice+Hall&rft.date=2006&rft.isbn=978-0-13-185785-8&rft.aulast=Leon&rft.aufirst=Steven+J.&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Flinearalgebrawit00leon&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALinear+algebra" class="Z3988"></span></li> <li>Murty, Katta G. (2014) <i><a rel="nofollow" class="external text" href="http://www.worldscientific.com/worldscibooks/10.1142/8261">Computational and Algorithmic Linear Algebra and n-Dimensional Geometry</a></i>, World Scientific Publishing, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-981-4366-62-5" title="Special:BookSources/978-981-4366-62-5">978-981-4366-62-5</a>. <i><a rel="nofollow" class="external text" href="http://www.worldscientific.com/doi/suppl/10.1142/8261/suppl_file/8261_chap01.pdf">Chapter 1: Systems of Simultaneous Linear Equations</a></i></li> <li>Noble, B. & Daniel, J.W. (2nd Ed. 1977) <i><a rel="nofollow" class="external autonumber" href="https://www.pearson.com/us/higher-education/program/Noble-Applied-Linear-Algebra-3rd-Edition/PGM17768.html">[1]</a></i>, Pearson Higher Education, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0130413437" title="Special:BookSources/978-0130413437">978-0130413437</a>.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFPoole2010" class="citation cs2">Poole, David (2010), <i>Linear Algebra: A Modern Introduction</i> (3rd ed.), Cengage – Brooks/Cole, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-538-73545-2" title="Special:BookSources/978-0-538-73545-2"><bdi>978-0-538-73545-2</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Linear+Algebra%3A+A+Modern+Introduction&rft.edition=3rd&rft.pub=Cengage+%E2%80%93+Brooks%2FCole&rft.date=2010&rft.isbn=978-0-538-73545-2&rft.aulast=Poole&rft.aufirst=David&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALinear+algebra" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFRicardo2010" class="citation cs2">Ricardo, Henry (2010), <i>A Modern Introduction To Linear Algebra</i> (1st ed.), CRC Press, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-4398-0040-9" title="Special:BookSources/978-1-4398-0040-9"><bdi>978-1-4398-0040-9</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=A+Modern+Introduction+To+Linear+Algebra&rft.edition=1st&rft.pub=CRC+Press&rft.date=2010&rft.isbn=978-1-4398-0040-9&rft.aulast=Ricardo&rft.aufirst=Henry&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALinear+algebra" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSadun2008" class="citation cs2">Sadun, Lorenzo (2008), <i>Applied Linear Algebra: the decoupling principle</i> (2nd ed.), AMS, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-8218-4441-0" title="Special:BookSources/978-0-8218-4441-0"><bdi>978-0-8218-4441-0</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Applied+Linear+Algebra%3A+the+decoupling+principle&rft.edition=2nd&rft.pub=AMS&rft.date=2008&rft.isbn=978-0-8218-4441-0&rft.aulast=Sadun&rft.aufirst=Lorenzo&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALinear+algebra" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFStrang2016" class="citation cs2"><a href="/wiki/Gilbert_Strang" title="Gilbert Strang">Strang, Gilbert</a> (2016), <i>Introduction to Linear Algebra</i> (5th ed.), Wellesley-Cambridge Press, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-09802327-7-6" title="Special:BookSources/978-09802327-7-6"><bdi>978-09802327-7-6</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Introduction+to+Linear+Algebra&rft.edition=5th&rft.pub=Wellesley-Cambridge+Press&rft.date=2016&rft.isbn=978-09802327-7-6&rft.aulast=Strang&rft.aufirst=Gilbert&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALinear+algebra" class="Z3988"></span></li> <li>The Manga Guide to Linear Algebra (2012), by <a href="/wiki/Shin_Takahashi" title="Shin Takahashi">Shin Takahashi</a>, Iroha Inoue and Trend-Pro Co., Ltd., <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-59327-413-9" title="Special:BookSources/978-1-59327-413-9">978-1-59327-413-9</a></li></ul> <div class="mw-heading mw-heading3"><h3 id="Advanced_textbooks">Advanced textbooks</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Linear_algebra&action=edit&section=31" title="Edit section: Advanced textbooks"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBhatia1996" class="citation cs2">Bhatia, Rajendra (November 15, 1996), <i>Matrix Analysis</i>, <a href="/wiki/Graduate_Texts_in_Mathematics" title="Graduate Texts in Mathematics">Graduate Texts in Mathematics</a>, Springer, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-387-94846-1" title="Special:BookSources/978-0-387-94846-1"><bdi>978-0-387-94846-1</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Matrix+Analysis&rft.series=Graduate+Texts+in+Mathematics&rft.pub=Springer&rft.date=1996-11-15&rft.isbn=978-0-387-94846-1&rft.aulast=Bhatia&rft.aufirst=Rajendra&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALinear+algebra" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFDemmel1997" class="citation cs2"><a href="/wiki/James_Demmel" title="James Demmel">Demmel, James W.</a> (August 1, 1997), <i>Applied Numerical Linear Algebra</i>, SIAM, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-89871-389-3" title="Special:BookSources/978-0-89871-389-3"><bdi>978-0-89871-389-3</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Applied+Numerical+Linear+Algebra&rft.pub=SIAM&rft.date=1997-08-01&rft.isbn=978-0-89871-389-3&rft.aulast=Demmel&rft.aufirst=James+W.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALinear+algebra" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFDym2007" class="citation cs2"><a href="/wiki/Harry_Dym" title="Harry Dym">Dym, Harry</a> (2007), <i>Linear Algebra in Action</i>, AMS, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-8218-3813-6" title="Special:BookSources/978-0-8218-3813-6"><bdi>978-0-8218-3813-6</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Linear+Algebra+in+Action&rft.pub=AMS&rft.date=2007&rft.isbn=978-0-8218-3813-6&rft.aulast=Dym&rft.aufirst=Harry&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALinear+algebra" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGantmacher2005" class="citation cs2"><a href="/wiki/Felix_Gantmacher" title="Felix Gantmacher">Gantmacher, Felix R.</a> (2005), <i>Applications of the Theory of Matrices</i>, Dover Publications, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-486-44554-0" title="Special:BookSources/978-0-486-44554-0"><bdi>978-0-486-44554-0</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Applications+of+the+Theory+of+Matrices&rft.pub=Dover+Publications&rft.date=2005&rft.isbn=978-0-486-44554-0&rft.aulast=Gantmacher&rft.aufirst=Felix+R.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALinear+algebra" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGantmacher1990" class="citation cs2">Gantmacher, Felix R. (1990), <i>Matrix Theory Vol. 1</i> (2nd ed.), American Mathematical Society, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-8218-1376-8" title="Special:BookSources/978-0-8218-1376-8"><bdi>978-0-8218-1376-8</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Matrix+Theory+Vol.+1&rft.edition=2nd&rft.pub=American+Mathematical+Society&rft.date=1990&rft.isbn=978-0-8218-1376-8&rft.aulast=Gantmacher&rft.aufirst=Felix+R.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALinear+algebra" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGantmacher2000" class="citation cs2">Gantmacher, Felix R. (2000), <i>Matrix Theory Vol. 2</i> (2nd ed.), American Mathematical Society, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-8218-2664-5" title="Special:BookSources/978-0-8218-2664-5"><bdi>978-0-8218-2664-5</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Matrix+Theory+Vol.+2&rft.edition=2nd&rft.pub=American+Mathematical+Society&rft.date=2000&rft.isbn=978-0-8218-2664-5&rft.aulast=Gantmacher&rft.aufirst=Felix+R.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALinear+algebra" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGelfand1989" class="citation cs2"><a href="/wiki/Israel_Gelfand" title="Israel Gelfand">Gelfand, Israel M.</a> (1989), <i>Lectures on Linear Algebra</i>, Dover Publications, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-486-66082-0" title="Special:BookSources/978-0-486-66082-0"><bdi>978-0-486-66082-0</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Lectures+on+Linear+Algebra&rft.pub=Dover+Publications&rft.date=1989&rft.isbn=978-0-486-66082-0&rft.aulast=Gelfand&rft.aufirst=Israel+M.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALinear+algebra" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGlazmanLjubic2006" class="citation cs2">Glazman, I. M.; Ljubic, Ju. I. (2006), <i>Finite-Dimensional Linear Analysis</i>, Dover Publications, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-486-45332-3" title="Special:BookSources/978-0-486-45332-3"><bdi>978-0-486-45332-3</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Finite-Dimensional+Linear+Analysis&rft.pub=Dover+Publications&rft.date=2006&rft.isbn=978-0-486-45332-3&rft.aulast=Glazman&rft.aufirst=I.+M.&rft.au=Ljubic%2C+Ju.+I.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALinear+algebra" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGolan2007" class="citation cs2">Golan, Johnathan S. (January 2007), <i>The Linear Algebra a Beginning Graduate Student Ought to Know</i> (2nd ed.), Springer, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-4020-5494-5" title="Special:BookSources/978-1-4020-5494-5"><bdi>978-1-4020-5494-5</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Linear+Algebra+a+Beginning+Graduate+Student+Ought+to+Know&rft.edition=2nd&rft.pub=Springer&rft.date=2007-01&rft.isbn=978-1-4020-5494-5&rft.aulast=Golan&rft.aufirst=Johnathan+S.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALinear+algebra" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGolan1995" class="citation cs2">Golan, Johnathan S. (August 1995), <i>Foundations of Linear Algebra</i>, Kluwer, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-7923-3614-3" title="Special:BookSources/0-7923-3614-3"><bdi>0-7923-3614-3</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Foundations+of+Linear+Algebra&rft.pub=Kluwer&rft.date=1995-08&rft.isbn=0-7923-3614-3&rft.aulast=Golan&rft.aufirst=Johnathan+S.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALinear+algebra" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGreub1981" class="citation cs2">Greub, Werner H. (October 16, 1981), <i>Linear Algebra</i>, Graduate Texts in Mathematics (4th ed.), Springer, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-8018-5414-9" title="Special:BookSources/978-0-8018-5414-9"><bdi>978-0-8018-5414-9</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Linear+Algebra&rft.series=Graduate+Texts+in+Mathematics&rft.edition=4th&rft.pub=Springer&rft.date=1981-10-16&rft.isbn=978-0-8018-5414-9&rft.aulast=Greub&rft.aufirst=Werner+H.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALinear+algebra" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHoffmanKunze1971" class="citation cs2">Hoffman, Kenneth; <a href="/wiki/Ray_Kunze" title="Ray Kunze">Kunze, Ray</a> (1971), <i>Linear algebra</i> (2nd ed.), Englewood Cliffs, N.J.: Prentice-Hall, Inc., <a href="/wiki/MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=0276251">0276251</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Linear+algebra&rft.place=Englewood+Cliffs%2C+N.J.&rft.edition=2nd&rft.pub=Prentice-Hall%2C+Inc.&rft.date=1971&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D0276251%23id-name%3DMR&rft.aulast=Hoffman&rft.aufirst=Kenneth&rft.au=Kunze%2C+Ray&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALinear+algebra" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHalmos1993" class="citation cs2"><a href="/wiki/Paul_Halmos" title="Paul Halmos">Halmos, Paul R.</a> (August 20, 1993), <i>Finite-Dimensional Vector Spaces</i>, <a href="/wiki/Undergraduate_Texts_in_Mathematics" title="Undergraduate Texts in Mathematics">Undergraduate Texts in Mathematics</a>, Springer, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-387-90093-3" title="Special:BookSources/978-0-387-90093-3"><bdi>978-0-387-90093-3</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Finite-Dimensional+Vector+Spaces&rft.series=Undergraduate+Texts+in+Mathematics&rft.pub=Springer&rft.date=1993-08-20&rft.isbn=978-0-387-90093-3&rft.aulast=Halmos&rft.aufirst=Paul+R.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALinear+algebra" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFFriedbergInselSpence2018" class="citation cs2">Friedberg, Stephen H.; Insel, Arnold J.; Spence, Lawrence E. (September 7, 2018), <i>Linear Algebra</i> (5th ed.), Pearson, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-13-486024-4" title="Special:BookSources/978-0-13-486024-4"><bdi>978-0-13-486024-4</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Linear+Algebra&rft.edition=5th&rft.pub=Pearson&rft.date=2018-09-07&rft.isbn=978-0-13-486024-4&rft.aulast=Friedberg&rft.aufirst=Stephen+H.&rft.au=Insel%2C+Arnold+J.&rft.au=Spence%2C+Lawrence+E.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALinear+algebra" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHornJohnson1990" class="citation cs2"><a href="/wiki/Roger_Horn" title="Roger Horn">Horn, Roger A.</a>; <a href="/wiki/Charles_Royal_Johnson" title="Charles Royal Johnson">Johnson, Charles R.</a> (February 23, 1990), <i>Matrix Analysis</i>, Cambridge University Press, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-521-38632-6" title="Special:BookSources/978-0-521-38632-6"><bdi>978-0-521-38632-6</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Matrix+Analysis&rft.pub=Cambridge+University+Press&rft.date=1990-02-23&rft.isbn=978-0-521-38632-6&rft.aulast=Horn&rft.aufirst=Roger+A.&rft.au=Johnson%2C+Charles+R.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALinear+algebra" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHornJohnson1994" class="citation cs2">Horn, Roger A.; Johnson, Charles R. (June 24, 1994), <i>Topics in Matrix Analysis</i>, Cambridge University Press, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-521-46713-1" title="Special:BookSources/978-0-521-46713-1"><bdi>978-0-521-46713-1</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Topics+in+Matrix+Analysis&rft.pub=Cambridge+University+Press&rft.date=1994-06-24&rft.isbn=978-0-521-46713-1&rft.aulast=Horn&rft.aufirst=Roger+A.&rft.au=Johnson%2C+Charles+R.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALinear+algebra" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFLang2004" class="citation cs2"><a href="/wiki/Serge_Lang" title="Serge Lang">Lang, Serge</a> (March 9, 2004), <i>Linear Algebra</i>, Undergraduate Texts in Mathematics (3rd ed.), Springer, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-387-96412-6" title="Special:BookSources/978-0-387-96412-6"><bdi>978-0-387-96412-6</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Linear+Algebra&rft.series=Undergraduate+Texts+in+Mathematics&rft.edition=3rd&rft.pub=Springer&rft.date=2004-03-09&rft.isbn=978-0-387-96412-6&rft.aulast=Lang&rft.aufirst=Serge&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALinear+algebra" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMarcusMinc2010" class="citation cs2"><a href="/wiki/Marvin_Marcus" title="Marvin Marcus">Marcus, Marvin</a>; <a href="/wiki/Henryk_Minc" title="Henryk Minc">Minc, Henryk</a> (2010), <i>A Survey of Matrix Theory and Matrix Inequalities</i>, Dover Publications, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-486-67102-4" title="Special:BookSources/978-0-486-67102-4"><bdi>978-0-486-67102-4</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=A+Survey+of+Matrix+Theory+and+Matrix+Inequalities&rft.pub=Dover+Publications&rft.date=2010&rft.isbn=978-0-486-67102-4&rft.aulast=Marcus&rft.aufirst=Marvin&rft.au=Minc%2C+Henryk&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALinear+algebra" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMeyer2001" class="citation cs2">Meyer, Carl D. (February 15, 2001), <a rel="nofollow" class="external text" href="https://web.archive.org/web/20091031193126/http://matrixanalysis.com/DownloadChapters.html"><i>Matrix Analysis and Applied Linear Algebra</i></a>, Society for Industrial and Applied Mathematics (SIAM), <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-89871-454-8" title="Special:BookSources/978-0-89871-454-8"><bdi>978-0-89871-454-8</bdi></a>, archived from <a rel="nofollow" class="external text" href="http://www.matrixanalysis.com/DownloadChapters.html">the original</a> on October 31, 2009</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Matrix+Analysis+and+Applied+Linear+Algebra&rft.pub=Society+for+Industrial+and+Applied+Mathematics+%28SIAM%29&rft.date=2001-02-15&rft.isbn=978-0-89871-454-8&rft.aulast=Meyer&rft.aufirst=Carl+D.&rft_id=http%3A%2F%2Fwww.matrixanalysis.com%2FDownloadChapters.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALinear+algebra" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMirsky1990" class="citation cs2"><a href="/wiki/Leon_Mirsky" title="Leon Mirsky">Mirsky, L.</a> (1990), <i>An Introduction to Linear Algebra</i>, Dover Publications, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-486-66434-7" title="Special:BookSources/978-0-486-66434-7"><bdi>978-0-486-66434-7</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=An+Introduction+to+Linear+Algebra&rft.pub=Dover+Publications&rft.date=1990&rft.isbn=978-0-486-66434-7&rft.aulast=Mirsky&rft.aufirst=L.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALinear+algebra" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFShafarevichRemizov2012" class="citation cs2"><a href="/wiki/Igor_Shafarevich" title="Igor Shafarevich">Shafarevich, I. R.</a>; Remizov, A. O (2012), <a rel="nofollow" class="external text" href="https://www.springer.com/mathematics/algebra/book/978-3-642-30993-9"><i>Linear Algebra and Geometry</i></a>, <a href="/wiki/Springer_Science%2BBusiness_Media" title="Springer Science+Business Media">Springer</a>, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-642-30993-9" title="Special:BookSources/978-3-642-30993-9"><bdi>978-3-642-30993-9</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Linear+Algebra+and+Geometry&rft.pub=Springer&rft.date=2012&rft.isbn=978-3-642-30993-9&rft.aulast=Shafarevich&rft.aufirst=I.+R.&rft.au=Remizov%2C+A.+O&rft_id=https%3A%2F%2Fwww.springer.com%2Fmathematics%2Falgebra%2Fbook%2F978-3-642-30993-9&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALinear+algebra" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFShilov1977" class="citation cs2"><a href="/wiki/Georgiy_Shilov" title="Georgiy Shilov">Shilov, Georgi E.</a> (June 1, 1977), <i>Linear algebra</i>, Dover Publications, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-486-63518-7" title="Special:BookSources/978-0-486-63518-7"><bdi>978-0-486-63518-7</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Linear+algebra&rft.pub=Dover+Publications&rft.date=1977-06-01&rft.isbn=978-0-486-63518-7&rft.aulast=Shilov&rft.aufirst=Georgi+E.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALinear+algebra" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFShores2006" class="citation cs2">Shores, Thomas S. (December 6, 2006), <i>Applied Linear Algebra and Matrix Analysis</i>, Undergraduate Texts in Mathematics, Springer, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-387-33194-2" title="Special:BookSources/978-0-387-33194-2"><bdi>978-0-387-33194-2</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Applied+Linear+Algebra+and+Matrix+Analysis&rft.series=Undergraduate+Texts+in+Mathematics&rft.pub=Springer&rft.date=2006-12-06&rft.isbn=978-0-387-33194-2&rft.aulast=Shores&rft.aufirst=Thomas+S.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALinear+algebra" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSmith1998" class="citation cs2">Smith, Larry (May 28, 1998), <i>Linear Algebra</i>, Undergraduate Texts in Mathematics, Springer, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-387-98455-1" title="Special:BookSources/978-0-387-98455-1"><bdi>978-0-387-98455-1</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Linear+Algebra&rft.series=Undergraduate+Texts+in+Mathematics&rft.pub=Springer&rft.date=1998-05-28&rft.isbn=978-0-387-98455-1&rft.aulast=Smith&rft.aufirst=Larry&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALinear+algebra" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFTrefethenBau1997" class="citation cs2"><a href="/wiki/Lloyd_N._Trefethen" class="mw-redirect" title="Lloyd N. Trefethen">Trefethen, Lloyd N.</a>; Bau, David (1997), <i>Numerical Linear Algebra</i>, SIAM, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-898-71361-9" title="Special:BookSources/978-0-898-71361-9"><bdi>978-0-898-71361-9</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Numerical+Linear+Algebra&rft.pub=SIAM&rft.date=1997&rft.isbn=978-0-898-71361-9&rft.aulast=Trefethen&rft.aufirst=Lloyd+N.&rft.au=Bau%2C+David&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALinear+algebra" class="Z3988"></span></li></ul> <div class="mw-heading mw-heading3"><h3 id="Study_guides_and_outlines">Study guides and outlines</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Linear_algebra&action=edit&section=32" title="Edit section: Study guides and outlines"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFLeduc1996" class="citation cs2">Leduc, Steven A. (May 1, 1996), <i>Linear Algebra (Cliffs Quick Review)</i>, Cliffs Notes, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-8220-5331-6" title="Special:BookSources/978-0-8220-5331-6"><bdi>978-0-8220-5331-6</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Linear+Algebra+%28Cliffs+Quick+Review%29&rft.pub=Cliffs+Notes&rft.date=1996-05-01&rft.isbn=978-0-8220-5331-6&rft.aulast=Leduc&rft.aufirst=Steven+A.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALinear+algebra" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFLipschutzLipson2000" class="citation cs2">Lipschutz, Seymour; Lipson, Marc (December 6, 2000), <i>Schaum's Outline of Linear Algebra</i> (3rd ed.), McGraw-Hill, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-07-136200-9" title="Special:BookSources/978-0-07-136200-9"><bdi>978-0-07-136200-9</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Schaum%27s+Outline+of+Linear+Algebra&rft.edition=3rd&rft.pub=McGraw-Hill&rft.date=2000-12-06&rft.isbn=978-0-07-136200-9&rft.aulast=Lipschutz&rft.aufirst=Seymour&rft.au=Lipson%2C+Marc&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALinear+algebra" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFLipschutz1989" class="citation cs2">Lipschutz, Seymour (January 1, 1989), <i>3,000 Solved Problems in Linear Algebra</i>, McGraw–Hill, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-07-038023-3" title="Special:BookSources/978-0-07-038023-3"><bdi>978-0-07-038023-3</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=3%2C000+Solved+Problems+in+Linear+Algebra&rft.pub=McGraw%E2%80%93Hill&rft.date=1989-01-01&rft.isbn=978-0-07-038023-3&rft.aulast=Lipschutz&rft.aufirst=Seymour&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALinear+algebra" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMcMahon2005" class="citation cs2">McMahon, David (October 28, 2005), <i>Linear Algebra Demystified</i>, McGraw–Hill Professional, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-07-146579-3" title="Special:BookSources/978-0-07-146579-3"><bdi>978-0-07-146579-3</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Linear+Algebra+Demystified&rft.pub=McGraw%E2%80%93Hill+Professional&rft.date=2005-10-28&rft.isbn=978-0-07-146579-3&rft.aulast=McMahon&rft.aufirst=David&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALinear+algebra" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFZhang2009" class="citation cs2">Zhang, Fuzhen (April 7, 2009), <i>Linear Algebra: Challenging Problems for Students</i>, The Johns Hopkins University Press, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-8018-9125-0" title="Special:BookSources/978-0-8018-9125-0"><bdi>978-0-8018-9125-0</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Linear+Algebra%3A+Challenging+Problems+for+Students&rft.pub=The+Johns+Hopkins+University+Press&rft.date=2009-04-07&rft.isbn=978-0-8018-9125-0&rft.aulast=Zhang&rft.aufirst=Fuzhen&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALinear+algebra" class="Z3988"></span></li></ul> <div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Linear_algebra&action=edit&section=33" title="Edit section: External links"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1235681985">.mw-parser-output .side-box{margin:4px 0;box-sizing:border-box;border:1px solid #aaa;font-size:88%;line-height:1.25em;background-color:var(--background-color-interactive-subtle,#f8f9fa);display:flow-root}.mw-parser-output .side-box-abovebelow,.mw-parser-output .side-box-text{padding:0.25em 0.9em}.mw-parser-output .side-box-image{padding:2px 0 2px 0.9em;text-align:center}.mw-parser-output .side-box-imageright{padding:2px 0.9em 2px 0;text-align:center}@media(min-width:500px){.mw-parser-output .side-box-flex{display:flex;align-items:center}.mw-parser-output .side-box-text{flex:1;min-width:0}}@media(min-width:720px){.mw-parser-output .side-box{width:238px}.mw-parser-output .side-box-right{clear:right;float:right;margin-left:1em}.mw-parser-output .side-box-left{margin-right:1em}}</style><style data-mw-deduplicate="TemplateStyles:r1237033735">@media print{body.ns-0 .mw-parser-output .sistersitebox{display:none!important}}@media screen{html.skin-theme-clientpref-night .mw-parser-output .sistersitebox img[src*="Wiktionary-logo-en-v2.svg"]{background-color:white}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .sistersitebox img[src*="Wiktionary-logo-en-v2.svg"]{background-color:white}}</style><div class="side-box side-box-right plainlinks sistersitebox"><style data-mw-deduplicate="TemplateStyles:r1126788409">.mw-parser-output .plainlist ol,.mw-parser-output .plainlist ul{line-height:inherit;list-style:none;margin:0;padding:0}.mw-parser-output .plainlist ol li,.mw-parser-output .plainlist ul li{margin-bottom:0}</style> <div class="side-box-flex"> <div class="side-box-image"><span class="noviewer" typeof="mw:File"><a href="/wiki/File:Wikibooks-logo-en-noslogan.svg" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/d/df/Wikibooks-logo-en-noslogan.svg/40px-Wikibooks-logo-en-noslogan.svg.png" decoding="async" width="40" height="40" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/d/df/Wikibooks-logo-en-noslogan.svg/60px-Wikibooks-logo-en-noslogan.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/d/df/Wikibooks-logo-en-noslogan.svg/80px-Wikibooks-logo-en-noslogan.svg.png 2x" data-file-width="400" data-file-height="400" /></a></span></div> <div class="side-box-text plainlist">Wikibooks has a book on the topic of: <i><b><a href="https://en.wikibooks.org/wiki/Linear_Algebra" class="extiw" title="wikibooks:Linear Algebra">Linear Algebra</a></b></i></div></div> </div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1235681985"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1237033735"><div class="side-box side-box-right plainlinks sistersitebox"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1126788409"> <div class="side-box-flex"> <div class="side-box-image"><span class="noviewer" typeof="mw:File"><a href="/wiki/File:Wikiversity_logo_2017.svg" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/0/0b/Wikiversity_logo_2017.svg/40px-Wikiversity_logo_2017.svg.png" decoding="async" width="40" height="33" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/0b/Wikiversity_logo_2017.svg/60px-Wikiversity_logo_2017.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/0b/Wikiversity_logo_2017.svg/80px-Wikiversity_logo_2017.svg.png 2x" data-file-width="626" data-file-height="512" /></a></span></div> <div class="side-box-text plainlist">Wikiversity has learning resources about <i><b><a href="https://en.wikiversity.org/wiki/Linear_Algebra" class="extiw" title="v:Linear Algebra">Linear Algebra</a></b></i></div></div> </div> <div class="mw-heading mw-heading3"><h3 id="Online_Resources">Online Resources</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Linear_algebra&action=edit&section=34" title="Edit section: Online Resources"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1235681985"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1237033735"><div class="side-box side-box-right plainlinks sistersitebox"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1126788409"> <div class="side-box-flex"> <div class="side-box-image"><span class="noviewer" typeof="mw:File"><a href="/wiki/File:Commons-logo.svg" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/4/4a/Commons-logo.svg/30px-Commons-logo.svg.png" decoding="async" width="30" height="40" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/4/4a/Commons-logo.svg/45px-Commons-logo.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/4/4a/Commons-logo.svg/59px-Commons-logo.svg.png 2x" data-file-width="1024" data-file-height="1376" /></a></span></div> <div class="side-box-text plainlist">Wikimedia Commons has media related to <span style="font-weight: bold; font-style: italic;"><a href="https://commons.wikimedia.org/wiki/Category:Linear_algebra" class="extiw" title="commons:Category:Linear algebra">Linear algebra</a></span>.</div></div> </div> <ul><li><a rel="nofollow" class="external text" href="https://ocw.mit.edu/courses/mathematics/18-06-linear-algebra-spring-2010/">MIT Linear Algebra Video Lectures</a>, a series of 34 recorded lectures by Professor <a href="/wiki/Gilbert_Strang" title="Gilbert Strang">Gilbert Strang</a> (Spring 2010)</li> <li><a rel="nofollow" class="external text" href="https://www.math.technion.ac.il/iic/">International Linear Algebra Society</a></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation cs2"><a rel="nofollow" class="external text" href="https://www.encyclopediaofmath.org/index.php?title=Linear_algebra">"Linear algebra"</a>, <i><a href="/wiki/Encyclopedia_of_Mathematics" title="Encyclopedia of Mathematics">Encyclopedia of Mathematics</a></i>, <a href="/wiki/European_Mathematical_Society" title="European Mathematical Society">EMS Press</a>, 2001 [1994]</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Linear+algebra&rft.btitle=Encyclopedia+of+Mathematics&rft.pub=EMS+Press&rft.date=2001&rft_id=https%3A%2F%2Fwww.encyclopediaofmath.org%2Findex.php%3Ftitle%3DLinear_algebra&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALinear+algebra" class="Z3988"></span></li> <li><a rel="nofollow" class="external text" href="https://mathworld.wolfram.com/topics/LinearAlgebra.html">Linear Algebra</a> on <a href="/wiki/MathWorld" title="MathWorld">MathWorld</a></li> <li><a rel="nofollow" class="external text" href="http://www.economics.soton.ac.uk/staff/aldrich/matrices.htm">Matrix and Linear Algebra Terms</a> on <a rel="nofollow" class="external text" href="http://jeff560.tripod.com/mathword.html">Earliest Known Uses of Some of the Words of Mathematics</a></li> <li><a rel="nofollow" class="external text" href="http://jeff560.tripod.com/matrices.html">Earliest Uses of Symbols for Matrices and Vectors</a> on <a rel="nofollow" class="external text" href="http://jeff560.tripod.com/mathsym.html">Earliest Uses of Various Mathematical Symbols</a></li> <li><a rel="nofollow" class="external text" href="https://www.youtube.com/playlist?list=PLZHQObOWTQDPD3MizzM2xVFitgF8hE_ab">Essence of linear algebra</a>, a video presentation from <a href="/wiki/3Blue1Brown" title="3Blue1Brown">3Blue1Brown</a> of the basics of linear algebra, with emphasis on the relationship between the geometric, the matrix and the abstract points of view</li></ul> <div class="mw-heading mw-heading3"><h3 id="Online_books">Online books</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Linear_algebra&action=edit&section=35" title="Edit section: Online books"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBeezer2009" class="citation book cs1">Beezer, Robert A. (2009) [2004]. <a rel="nofollow" class="external text" href="http://linear.ups.edu"><i>A First Course in Linear Algebra</i></a>. <a href="/wiki/Gainesville,_Florida" title="Gainesville, Florida">Gainesville, Florida</a>: <a href="/wiki/University_Press_of_Florida" title="University Press of Florida">University Press of Florida</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9781616100049" title="Special:BookSources/9781616100049"><bdi>9781616100049</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=A+First+Course+in+Linear+Algebra&rft.place=Gainesville%2C+Florida&rft.pub=University+Press+of+Florida&rft.date=2009&rft.isbn=9781616100049&rft.aulast=Beezer&rft.aufirst=Robert+A.&rft_id=http%3A%2F%2Flinear.ups.edu&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALinear+algebra" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFConnell2004" class="citation book cs1">Connell, Edwin H. (2004) [1999]. <a rel="nofollow" class="external text" href="https://www.math.miami.edu/~ec/book/"><i>Elements of Abstract and Linear Algebra</i></a>. <a href="/wiki/University_of_Miami" title="University of Miami">University of Miami</a>, <a href="/wiki/Coral_Gables,_Florida" title="Coral Gables, Florida">Coral Gables, Florida</a>: Self-published.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Elements+of+Abstract+and+Linear+Algebra&rft.place=University+of+Miami%2C+Coral+Gables%2C+Florida&rft.pub=Self-published&rft.date=2004&rft.aulast=Connell&rft.aufirst=Edwin+H.&rft_id=https%3A%2F%2Fwww.math.miami.edu%2F~ec%2Fbook%2F&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALinear+algebra" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHefferon2020" class="citation book cs1"><a href="/wiki/Jim_Hefferon" title="Jim Hefferon">Hefferon, Jim</a> (2020). <a rel="nofollow" class="external text" href="https://hefferon.net/linearalgebra/"><i>Linear Algebra</i></a> (4th ed.). <a href="/wiki/Ann_Arbor,_Michigan" title="Ann Arbor, Michigan">Ann Arbor, Michigan</a>: Orthogonal Publishing. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-944325-11-4" title="Special:BookSources/978-1-944325-11-4"><bdi>978-1-944325-11-4</bdi></a>. <a href="/wiki/OCLC_(identifier)" class="mw-redirect" title="OCLC (identifier)">OCLC</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/oclc/1178900366">1178900366</a>. <a href="/wiki/OL_(identifier)" class="mw-redirect" title="OL (identifier)">OL</a> <a rel="nofollow" class="external text" href="https://openlibrary.org/books/OL30872051M">30872051M</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Linear+Algebra&rft.place=Ann+Arbor%2C+Michigan&rft.edition=4th&rft.pub=Orthogonal+Publishing&rft.date=2020&rft_id=info%3Aoclcnum%2F1178900366&rft_id=https%3A%2F%2Fopenlibrary.org%2Fbooks%2FOL30872051M%23id-name%3DOL&rft.isbn=978-1-944325-11-4&rft.aulast=Hefferon&rft.aufirst=Jim&rft_id=https%3A%2F%2Fhefferon.net%2Flinearalgebra%2F&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALinear+algebra" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMargalitRabinoff2019" class="citation book cs1"><a href="/wiki/Dan_Margalit_(mathematician)" title="Dan Margalit (mathematician)">Margalit, Dan</a>; Rabinoff, Joseph (2019). <a rel="nofollow" class="external text" href="https://textbooks.math.gatech.edu/ila/"><i>Interactive Linear Algebra</i></a>. <a href="/wiki/Georgia_Institute_of_Technology" class="mw-redirect" title="Georgia Institute of Technology">Georgia Institute of Technology</a>, <a href="/wiki/Atlanta,_Georgia" class="mw-redirect" title="Atlanta, Georgia">Atlanta, Georgia</a>: Self-published.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Interactive+Linear+Algebra&rft.place=Georgia+Institute+of+Technology%2C+Atlanta%2C+Georgia&rft.pub=Self-published&rft.date=2019&rft.aulast=Margalit&rft.aufirst=Dan&rft.au=Rabinoff%2C+Joseph&rft_id=https%3A%2F%2Ftextbooks.math.gatech.edu%2Fila%2F&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALinear+algebra" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMatthews2013" class="citation book cs1">Matthews, Keith R. (2013) [1991]. <a rel="nofollow" class="external text" href="http://www.numbertheory.org/book/"><i>Elementary Linear Algebra</i></a>. <a href="/wiki/University_of_Queensland" title="University of Queensland">University of Queensland</a>, <a href="/wiki/Brisbane,_Australia" class="mw-redirect" title="Brisbane, Australia">Brisbane, Australia</a>: Self-published.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Elementary+Linear+Algebra&rft.place=University+of+Queensland%2C+Brisbane%2C+Australia&rft.pub=Self-published&rft.date=2013&rft.aulast=Matthews&rft.aufirst=Keith+R.&rft_id=http%3A%2F%2Fwww.numbertheory.org%2Fbook%2F&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALinear+algebra" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMikaelian2020" class="citation book cs1">Mikaelian, Vahagn H. (2020) [2017]. <a rel="nofollow" class="external text" href="https://www.researchgate.net/publication/318066716"><i>Linear Algebra: Theory and Algorithms</i></a>. <a href="/wiki/Yerevan,_Armenia" class="mw-redirect" title="Yerevan, Armenia">Yerevan, Armenia</a>: Self-published – via <a href="/wiki/ResearchGate" title="ResearchGate">ResearchGate</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Linear+Algebra%3A+Theory+and+Algorithms&rft.place=Yerevan%2C+Armenia&rft.pub=Self-published&rft.date=2020&rft.aulast=Mikaelian&rft.aufirst=Vahagn+H.&rft_id=https%3A%2F%2Fwww.researchgate.net%2Fpublication%2F318066716&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALinear+algebra" class="Z3988"></span></li> <li>Sharipov, Ruslan, <i><a rel="nofollow" class="external text" href="https://arxiv.org/abs/math.HO/0405323">Course of linear algebra and multidimensional geometry</a></i></li> <li><a href="/wiki/Sergei_Treil" title="Sergei Treil">Treil, Sergei</a>, <i><a rel="nofollow" class="external text" href="https://www.math.brown.edu/~treil/papers/LADW/LADW.html">Linear Algebra Done Wrong</a></i></li></ul> <div class="navbox-styles"><style data-mw-deduplicate="TemplateStyles:r1129693374">.mw-parser-output .hlist dl,.mw-parser-output .hlist ol,.mw-parser-output .hlist ul{margin:0;padding:0}.mw-parser-output .hlist dd,.mw-parser-output .hlist dt,.mw-parser-output .hlist li{margin:0;display:inline}.mw-parser-output .hlist.inline,.mw-parser-output .hlist.inline dl,.mw-parser-output .hlist.inline ol,.mw-parser-output .hlist.inline ul,.mw-parser-output .hlist dl dl,.mw-parser-output .hlist dl ol,.mw-parser-output .hlist dl ul,.mw-parser-output .hlist ol dl,.mw-parser-output .hlist ol ol,.mw-parser-output .hlist ol ul,.mw-parser-output .hlist ul dl,.mw-parser-output .hlist ul ol,.mw-parser-output .hlist ul ul{display:inline}.mw-parser-output .hlist .mw-empty-li{display:none}.mw-parser-output .hlist dt::after{content:": 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style="font-size:114%;margin:0 4em"><a class="mw-selflink selflink">Linear algebra</a></div></th></tr><tr><td class="navbox-abovebelow" colspan="3"><div> <ul><li><a href="/wiki/Outline_of_linear_algebra" title="Outline of linear algebra">Outline</a></li> <li><a href="/wiki/Glossary_of_linear_algebra" title="Glossary of linear algebra">Glossary</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Basic concepts</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Scalar_(mathematics)" title="Scalar (mathematics)">Scalar</a></li> <li><a href="/wiki/Euclidean_vector" title="Euclidean vector">Vector</a></li> <li><a href="/wiki/Vector_space" title="Vector space">Vector space</a></li> <li><a href="/wiki/Scalar_multiplication" title="Scalar multiplication">Scalar multiplication</a></li> <li><a href="/wiki/Vector_projection" title="Vector projection">Vector projection</a></li> <li><a href="/wiki/Linear_span" title="Linear span">Linear span</a></li> <li><a href="/wiki/Linear_map" title="Linear map">Linear map</a></li> <li><a href="/wiki/Projection_(linear_algebra)" title="Projection (linear algebra)">Linear projection</a></li> <li><a href="/wiki/Linear_independence" title="Linear independence">Linear independence</a></li> <li><a href="/wiki/Linear_combination" title="Linear combination">Linear combination</a></li> <li><a href="/wiki/Multilinear_map" title="Multilinear map">Multilinear map</a></li> <li><a href="/wiki/Basis_(linear_algebra)" title="Basis (linear algebra)">Basis</a></li> <li><a href="/wiki/Change_of_basis" title="Change of basis">Change of basis</a></li> <li><a href="/wiki/Row_and_column_vectors" title="Row and column vectors">Row and column vectors</a></li> <li><a href="/wiki/Row_and_column_spaces" title="Row and column spaces">Row and column spaces</a></li> <li><a href="/wiki/Kernel_(linear_algebra)" title="Kernel (linear algebra)">Kernel</a></li> <li><a href="/wiki/Eigenvalues_and_eigenvectors" title="Eigenvalues and eigenvectors">Eigenvalues and eigenvectors</a></li> <li><a href="/wiki/Transpose" title="Transpose">Transpose</a></li> <li><a href="/wiki/System_of_linear_equations" title="System of linear equations">Linear equations</a></li></ul> </div></td><td class="noviewer navbox-image" rowspan="6" style="width:1px;padding:0 0 0 2px"><div><span typeof="mw:File"><a href="/wiki/Euclidean_space" title="Euclidean space"><img alt="Three dimensional Euclidean space" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/2f/Linear_subspaces_with_shading.svg/80px-Linear_subspaces_with_shading.svg.png" decoding="async" width="80" height="58" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/2f/Linear_subspaces_with_shading.svg/120px-Linear_subspaces_with_shading.svg.png 1.5x, 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algebra">Geometric algebra</a></li> <li><a href="/wiki/Exterior_algebra" title="Exterior algebra">Exterior algebra</a></li> <li><a href="/wiki/Bivector" title="Bivector">Bivector</a></li> <li><a href="/wiki/Multivector" title="Multivector">Multivector</a></li> <li><a href="/wiki/Tensor" title="Tensor">Tensor</a></li> <li><a href="/wiki/Outermorphism" title="Outermorphism">Outermorphism</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Vector_space" title="Vector space">Vector space</a> constructions</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Dual_space" title="Dual space">Dual</a></li> <li><a href="/wiki/Direct_sum_of_modules#Construction_for_two_vector_spaces" title="Direct sum of modules">Direct sum</a></li> <li><a href="/wiki/Function_space#In_linear_algebra" title="Function space">Function space</a></li> <li><a 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href="/wiki/Comparison_of_linear_algebra_libraries" title="Comparison of linear algebra libraries">Comparison of linear algebra libraries</a></li></ul> </div></td></tr><tr><td class="navbox-abovebelow" colspan="3"><div> <ul><li><span class="noviewer" typeof="mw:File"><span title="Category"><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/9/96/Symbol_category_class.svg/16px-Symbol_category_class.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/9/96/Symbol_category_class.svg/23px-Symbol_category_class.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/9/96/Symbol_category_class.svg/31px-Symbol_category_class.svg.png 2x" data-file-width="180" data-file-height="185" /></span></span> <a href="/wiki/Category:Linear_algebra" title="Category:Linear algebra">Category</a></li></ul> </div></td></tr></tbody></table></div> <div class="navbox-styles"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236075235"></div><div role="navigation" class="navbox" aria-labelledby="Major_mathematics_areas1051" style="padding:3px"><table class="nowraplinks hlist mw-collapsible autocollapse navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239400231"><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Areas_of_mathematics" title="Template:Areas of mathematics"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Areas_of_mathematics" title="Template talk:Areas of mathematics"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Areas_of_mathematics" title="Special:EditPage/Template:Areas of mathematics"><abbr title="Edit this template">e</abbr></a></li></ul></div><div id="Major_mathematics_areas1051" style="font-size:114%;margin:0 4em">Major <a href="/wiki/Mathematics" title="Mathematics">mathematics</a> areas</div></th></tr><tr><td class="navbox-abovebelow" colspan="2"><div> <ul><li><a href="/wiki/History_of_mathematics" title="History of mathematics">History</a> <ul><li><a href="/wiki/Timeline_of_mathematics" title="Timeline of mathematics">Timeline</a></li> <li><a href="/wiki/Future_of_mathematics" title="Future of mathematics">Future</a></li></ul></li> <li><a href="/wiki/Lists_of_mathematics_topics" title="Lists of mathematics topics">Lists</a></li> <li><a href="/wiki/Glossary_of_mathematical_symbols" title="Glossary of mathematical symbols">Glossary</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Foundations_of_mathematics" title="Foundations of mathematics">Foundations</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Category_theory" title="Category theory">Category theory</a></li> <li><a href="/wiki/Information_theory" title="Information theory">Information theory</a></li> <li><a href="/wiki/Mathematical_logic" title="Mathematical logic">Mathematical logic</a></li> <li><a href="/wiki/Philosophy_of_mathematics" title="Philosophy of mathematics">Philosophy of mathematics</a></li> <li><a href="/wiki/Set_theory" title="Set theory">Set theory</a></li> <li><a href="/wiki/Type_theory" title="Type theory">Type theory</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Algebra" title="Algebra">Algebra</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Abstract_algebra" title="Abstract algebra">Abstract</a></li> <li><a href="/wiki/Commutative_algebra" title="Commutative algebra">Commutative</a></li> <li><a href="/wiki/Elementary_algebra" title="Elementary algebra">Elementary</a></li> <li><a href="/wiki/Group_theory" title="Group theory">Group theory</a></li> <li><a class="mw-selflink selflink">Linear</a></li> <li><a href="/wiki/Multilinear_algebra" title="Multilinear algebra">Multilinear</a></li> <li><a href="/wiki/Universal_algebra" title="Universal algebra">Universal</a></li> <li><a href="/wiki/Homological_algebra" title="Homological algebra">Homological</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Mathematical_analysis" title="Mathematical analysis">Analysis</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Calculus" title="Calculus">Calculus</a></li> <li><a href="/wiki/Real_analysis" title="Real analysis">Real analysis</a></li> <li><a href="/wiki/Complex_analysis" title="Complex analysis">Complex analysis</a></li> <li><a href="/wiki/Hypercomplex_analysis" title="Hypercomplex analysis">Hypercomplex analysis</a></li> <li><a href="/wiki/Differential_equation" title="Differential equation">Differential equations</a></li> <li><a href="/wiki/Functional_analysis" title="Functional analysis">Functional analysis</a></li> <li><a href="/wiki/Harmonic_analysis" title="Harmonic analysis">Harmonic analysis</a></li> <li><a href="/wiki/Measure_(mathematics)" title="Measure (mathematics)">Measure theory</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Discrete_mathematics" title="Discrete mathematics">Discrete</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a 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<li><a href="/wiki/Finite_geometry" title="Finite geometry">Finite</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Number_theory" title="Number theory">Number theory</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Arithmetic" title="Arithmetic">Arithmetic</a></li> <li><a href="/wiki/Algebraic_number_theory" title="Algebraic number theory">Algebraic number theory</a></li> <li><a href="/wiki/Analytic_number_theory" title="Analytic number theory">Analytic number theory</a></li> <li><a href="/wiki/Diophantine_geometry" title="Diophantine geometry">Diophantine geometry</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Topology" title="Topology">Topology</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/General_topology" title="General topology">General</a></li> <li><a href="/wiki/Algebraic_topology" title="Algebraic topology">Algebraic</a></li> <li><a href="/wiki/Differential_topology" title="Differential topology">Differential</a></li> <li><a href="/wiki/Geometric_topology" title="Geometric topology">Geometric</a></li> <li><a href="/wiki/Homotopy_theory" title="Homotopy theory">Homotopy theory</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Applied_mathematics" title="Applied mathematics">Applied</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Engineering_mathematics" title="Engineering mathematics">Engineering mathematics</a></li> <li><a href="/wiki/Mathematical_and_theoretical_biology" title="Mathematical and theoretical biology">Mathematical biology</a></li> <li><a href="/wiki/Mathematical_chemistry" title="Mathematical chemistry">Mathematical chemistry</a></li> <li><a href="/wiki/Mathematical_economics" title="Mathematical economics">Mathematical economics</a></li> <li><a href="/wiki/Mathematical_finance" title="Mathematical finance">Mathematical finance</a></li> <li><a href="/wiki/Mathematical_physics" title="Mathematical physics">Mathematical physics</a></li> <li><a href="/wiki/Mathematical_psychology" title="Mathematical psychology">Mathematical psychology</a></li> <li><a href="/wiki/Mathematical_sociology" title="Mathematical sociology">Mathematical sociology</a></li> <li><a href="/wiki/Mathematical_statistics" title="Mathematical statistics">Mathematical statistics</a></li> <li><a href="/wiki/Probability_theory" title="Probability theory">Probability</a></li> <li><a href="/wiki/Statistics" title="Statistics">Statistics</a></li> <li><a href="/wiki/Systems_science" title="Systems science">Systems science</a> <ul><li><a href="/wiki/Control_theory" title="Control theory">Control 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<li><a href="/wiki/Mathematics_and_art" title="Mathematics and art">Mathematics and art</a></li> <li><a href="/wiki/Mathematics_education" title="Mathematics education">Mathematics education</a></li></ul> </div></td></tr><tr><td class="navbox-abovebelow" colspan="2"><div> <ul><li><b><span class="nowrap"><span class="noviewer" typeof="mw:File"><a href="/wiki/File:Nuvola_apps_edu_mathematics_blue-p.svg" class="mw-file-description"><img alt="icon" src="//upload.wikimedia.org/wikipedia/commons/thumb/3/3e/Nuvola_apps_edu_mathematics_blue-p.svg/16px-Nuvola_apps_edu_mathematics_blue-p.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/3e/Nuvola_apps_edu_mathematics_blue-p.svg/24px-Nuvola_apps_edu_mathematics_blue-p.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/3e/Nuvola_apps_edu_mathematics_blue-p.svg/32px-Nuvola_apps_edu_mathematics_blue-p.svg.png 2x" data-file-width="128" data-file-height="128" 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title="Wikipedia:WikiProject Mathematics">WikiProject</a></b></li></ul> </div></td></tr></tbody></table></div> <div class="navbox-styles"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236075235"><style data-mw-deduplicate="TemplateStyles:r1038841319">.mw-parser-output .tooltip-dotted{border-bottom:1px dotted;cursor:help}</style><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1038841319"></div><div role="navigation" class="navbox authority-control" aria-labelledby="Authority_control_databases_frameless&#124;text-top&#124;10px&#124;alt=Edit_this_at_Wikidata&#124;link=https&#58;//www.wikidata.org/wiki/Q82571#identifiers&#124;class=noprint&#124;Edit_this_at_Wikidata1626" style="padding:3px"><table class="nowraplinks hlist mw-collapsible autocollapse navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" 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//upload.wikimedia.org/wikipedia/en/thumb/8/8a/OOjs_UI_icon_edit-ltr-progressive.svg/20px-OOjs_UI_icon_edit-ltr-progressive.svg.png 2x" data-file-width="20" data-file-height="20" /></a></span></div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%">International</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"><ul><li><span class="uid"><a rel="nofollow" class="external text" href="http://id.worldcat.org/fast/804946/">FAST</a></span></li></ul></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">National</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"><ul><li><span class="uid"><a rel="nofollow" class="external text" href="https://d-nb.info/gnd/4035811-2">Germany</a></span></li><li><span class="uid"><a rel="nofollow" class="external text" href="https://id.loc.gov/authorities/sh85003441">United States</a></span></li><li><span class="uid"><a rel="nofollow" class="external text" href="https://catalogue.bnf.fr/ark:/12148/cb11937509n">France</a></span></li><li><span class="uid"><a rel="nofollow" class="external text" href="https://data.bnf.fr/ark:/12148/cb11937509n">BnF data</a></span></li><li><span class="uid"><a rel="nofollow" class="external text" href="https://id.ndl.go.jp/auth/ndlna/00570681">Japan</a></span></li><li><span class="uid"><span class="rt-commentedText tooltip tooltip-dotted" title="lineární algebra"><a rel="nofollow" class="external text" href="https://aleph.nkp.cz/F/?func=find-c&local_base=aut&ccl_term=ica=ph122353&CON_LNG=ENG">Czech Republic</a></span></span></li><li><span class="uid"><span class="rt-commentedText tooltip tooltip-dotted" title="Álgebra lineal"><a rel="nofollow" class="external text" href="http://catalogo.bne.es/uhtbin/authoritybrowse.cgi?action=display&authority_id=XX527736">Spain</a></span></span></li><li><span class="uid"><a rel="nofollow" class="external text" href="https://kopkatalogs.lv/F?func=direct&local_base=lnc10&doc_number=000082525&P_CON_LNG=ENG">Latvia</a></span></li><li><span class="uid"><a rel="nofollow" class="external text" href="https://www.nli.org.il/en/authorities/987007293931805171">Israel</a></span></li></ul></div></td></tr></tbody></table></div>    </div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?useformat=desktop&type=1x1&usesul3=0" alt="" width="1" height="1" style="border: none; position: absolute;"></noscript> <div class="printfooter" data-nosnippet="">Retrieved from "<a dir="ltr" href="https://en.wikipedia.org/w/index.php?title=Linear_algebra&oldid=1276158324">https://en.wikipedia.org/w/index.php?title=Linear_algebra&oldid=1276158324</a>"</div></div> <div id="catlinks" class="catlinks" data-mw="interface"><div id="mw-normal-catlinks" class="mw-normal-catlinks"><a href="/wiki/Help:Category" title="Help:Category">Categories</a>: <ul><li><a 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