Orthonormality - Wikipedia

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Available in 15 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-15" 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">15 languages</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D8%AA%D8%B9%D8%A7%D9%85%D8%AF_%D9%85%D9%86%D8%B8%D9%85" title="تعامد منظم – Arabic" lang="ar" hreflang="ar" data-title="تعامد منظم" data-language-autonym="العربية" data-language-local-name="Arabic" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Ortonormal" title="Ortonormal – Catalan" lang="ca" hreflang="ca" data-title="Ortonormal" data-language-autonym="Català" data-language-local-name="Catalan" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Ortonormalita" title="Ortonormalita – Czech" lang="cs" hreflang="cs" data-title="Ortonormalita" data-language-autonym="Čeština" data-language-local-name="Czech" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-da mw-list-item"><a href="https://da.wikipedia.org/wiki/Ortonormal" title="Ortonormal – Danish" lang="da" hreflang="da" data-title="Ortonormal" data-language-autonym="Dansk" data-language-local-name="Danish" class="interlanguage-link-target"><span>Dansk</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Ortonormal" title="Ortonormal – Spanish" lang="es" hreflang="es" data-title="Ortonormal" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E6%AD%A3%E8%A6%8F%E7%9B%B4%E4%BA%A4%E7%B3%BB" title="正規直交系 – Japanese" lang="ja" hreflang="ja" data-title="正規直交系" 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-no mw-list-item"><a href="https://no.wikipedia.org/wiki/Ortonormal" title="Ortonormal – Norwegian Bokmål" lang="nb" hreflang="nb" data-title="Ortonormal" 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-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Ortonormalno%C5%9B%C4%87" title="Ortonormalność – Polish" lang="pl" hreflang="pl" data-title="Ortonormalność" 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/Ortonormalidade" title="Ortonormalidade – Portuguese" lang="pt" hreflang="pt" data-title="Ortonormalidade" 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/Ortonormalitate" title="Ortonormalitate – Romanian" lang="ro" hreflang="ro" data-title="Ortonormalitate" data-language-autonym="Română" data-language-local-name="Romanian" class="interlanguage-link-target"><span>Română</span></a></li><li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://simple.wikipedia.org/wiki/Orthonormality" title="Orthonormality – Simple English" lang="en-simple" hreflang="en-simple" data-title="Orthonormality" 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/Ortonormalita" title="Ortonormalita – Slovak" lang="sk" hreflang="sk" data-title="Ortonormalita" data-language-autonym="Slovenčina" data-language-local-name="Slovak" class="interlanguage-link-target"><span>Slovenčina</span></a></li><li class="interlanguage-link interwiki-sl mw-list-item"><a href="https://sl.wikipedia.org/wiki/Ortonormalnost" title="Ortonormalnost – Slovenian" lang="sl" hreflang="sl" data-title="Ortonormalnost" data-language-autonym="Slovenščina" data-language-local-name="Slovenian" class="interlanguage-link-target"><span>Slovenščina</span></a></li><li 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</div> </div> </div> <div class="vector-column-end"> <div class="vector-sticky-pinned-container"> <nav class="vector-page-tools-landmark" aria-label="Page tools"> <div id="vector-page-tools-pinned-container" class="vector-pinned-container"> </div> </nav> <nav class="vector-appearance-landmark" aria-label="Appearance"> <div id="vector-appearance-pinned-container" class="vector-pinned-container"> <div id="vector-appearance" class="vector-appearance vector-pinnable-element"> <div class="vector-pinnable-header vector-appearance-pinnable-header vector-pinnable-header-pinned" data-feature-name="appearance-pinned" data-pinnable-element-id="vector-appearance" data-pinned-container-id="vector-appearance-pinned-container" data-unpinned-container-id="vector-appearance-unpinned-container" > <div class="vector-pinnable-header-label">Appearance</div> <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">Property of two or more vectors that are orthogonal and of unit length</div> <p>In <a href="/wiki/Linear_algebra" title="Linear algebra">linear algebra</a>, two <a href="/wiki/Vector_space" title="Vector space">vectors</a> in an <a href="/wiki/Inner_product_space" title="Inner product space">inner product space</a> are <b>orthonormal</b> if they are <a href="/wiki/Orthogonality" title="Orthogonality">orthogonal</a> <a href="/wiki/Unit_vector" title="Unit vector">unit vectors</a>. A unit vector means that the vector has a length of 1, which is also known as normalized. Orthogonal means that the vectors are all perpendicular to each other. A set of vectors form an <b>orthonormal set</b> if all vectors in the set are mutually orthogonal and all of unit length. An orthonormal set which forms a <a href="/wiki/Basis_(linear_algebra)" title="Basis (linear algebra)">basis</a> is called an <i><a href="/wiki/Orthonormal_basis" title="Orthonormal basis">orthonormal basis</a></i>. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Intuitive_overview">Intuitive overview</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Orthonormality&action=edit&section=1" title="Edit section: Intuitive overview"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The construction of <a href="/wiki/Orthogonality" title="Orthogonality">orthogonality</a> of vectors is motivated by a desire to extend the intuitive notion of perpendicular vectors to higher-dimensional spaces. In the <a href="/wiki/Cartesian_coordinate_system#Cartesian_coordinates_in_two_dimensions" title="Cartesian coordinate system">Cartesian plane</a>, two <a href="/wiki/Vector_(geometry)" class="mw-redirect" title="Vector (geometry)">vectors</a> are said to be <i>perpendicular</i> if the angle between them is 90° (i.e. if they form a <a href="/wiki/Right_angle" title="Right angle">right angle</a>). This definition can be formalized in Cartesian space by defining the <a href="/wiki/Dot_product" title="Dot product">dot product</a> and specifying that two vectors in the plane are orthogonal if their dot product is zero. </p><p>Similarly, the construction of the <a href="/wiki/Norm_(mathematics)" title="Norm (mathematics)">norm</a> of a vector is motivated by a desire to extend the intuitive notion of the <a href="/wiki/Norm_(mathematics)#Euclidean_norm" title="Norm (mathematics)">length</a> of a vector to higher-dimensional spaces. In Cartesian space, the <i>norm</i> of a vector is the square root of the vector dotted with itself. 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 \|\mathbf {x} \|={\sqrt {\mathbf {x} \cdot \mathbf {x} }}}"> <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">x</mi> </mrow> <mo fence="false" stretchy="false">‖</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \|\mathbf {x} \|={\sqrt {\mathbf {x} \cdot \mathbf {x} }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/78c5ff99012396a180ab607d73030e68411a60df" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:13.271ex; height:3.009ex;" alt="{\displaystyle \|\mathbf {x} \|={\sqrt {\mathbf {x} \cdot \mathbf {x} }}}"></span></dd></dl> <p>Many important results in <a href="/wiki/Linear_algebra" title="Linear algebra">linear algebra</a> deal with collections of two or more orthogonal vectors. But often, it is easier to deal with vectors of <a href="/wiki/Unit_vector" title="Unit vector">unit length</a>. That is, it often simplifies things to only consider vectors whose norm equals 1. The notion of restricting orthogonal pairs of vectors to only those of unit length is important enough to be given a special name. Two vectors which are orthogonal and of length 1 are said to be <i>orthonormal</i>. </p> <div class="mw-heading mw-heading3"><h3 id="Simple_example">Simple example</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Orthonormality&action=edit&section=2" title="Edit section: Simple example"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>What does a pair of orthonormal vectors in 2-D Euclidean space look like? </p><p>Let <b>u</b> = (x<sub>1</sub>, y<sub>1</sub>) and <b>v</b> = (x<sub>2</sub>, y<sub>2</sub>). Consider the restrictions on x<sub>1</sub>, x<sub>2</sub>, y<sub>1</sub>, y<sub>2</sub> required to make <b>u</b> and <b>v</b> form an orthonormal pair. </p> <ul><li>From the orthogonality restriction, <b>u</b> • <b>v</b> = 0.</li> <li>From the unit length restriction on <b>u</b>, ||<b>u</b>|| = 1.</li> <li>From the unit length restriction on <b>v</b>, ||<b>v</b>|| = 1.</li></ul> <p>Expanding these terms gives 3 equations: </p> <ol><li><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}x_{2}+y_{1}y_{2}=0\quad }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>=</mo> <mn>0</mn> <mspace width="1em" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{1}x_{2}+y_{1}y_{2}=0\quad }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/29c3a4a8a612e411fc05e2d09c95a63fe67a76e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:18.579ex; height:2.509ex;" alt="{\displaystyle x_{1}x_{2}+y_{1}y_{2}=0\quad }"></span></li> <li><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 {\sqrt {{x_{1}}^{2}+{y_{1}}^{2}}}=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msup> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {{x_{1}}^{2}+{y_{1}}^{2}}}=1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d36735265243284681c9819c1b127c58a14b24ab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.671ex; width:16.111ex; height:4.843ex;" alt="{\displaystyle {\sqrt {{x_{1}}^{2}+{y_{1}}^{2}}}=1}"></span></li> <li><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 {\sqrt {{x_{2}}^{2}+{y_{2}}^{2}}}=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msup> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {{x_{2}}^{2}+{y_{2}}^{2}}}=1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1551bb9238998be2d75495ae4323025b5916ec04" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.671ex; width:16.111ex; height:4.843ex;" alt="{\displaystyle {\sqrt {{x_{2}}^{2}+{y_{2}}^{2}}}=1}"></span></li></ol> <p>Converting from Cartesian to <a href="/wiki/Polar_coordinates" class="mw-redirect" title="Polar coordinates">polar coordinates</a>, and considering Equation <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 (2)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mn>2</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (2)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/43f88fdd4acbb57a291f9eb9f23ae23a1e492b30" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.972ex; height:2.843ex;" alt="{\displaystyle (2)}"></span> and Equation <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 (3)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mn>3</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (3)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a0ff45737214013a8e04d59d0de54318086be26a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.972ex; height:2.843ex;" alt="{\displaystyle (3)}"></span> immediately gives the result r<sub>1</sub> = r<sub>2</sub> = 1. In other words, requiring the vectors be of unit length restricts the vectors to lie on the <a href="/wiki/Unit_circle" title="Unit circle">unit circle</a>. </p><p>After substitution, Equation <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 (1)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (1)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a25115739469707c4758b189fe310a750092a80a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.972ex; height:2.843ex;" alt="{\displaystyle (1)}"></span> becomes <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 \cos \theta _{1}\cos \theta _{2}+\sin \theta _{1}\sin \theta _{2}=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>cos</mi> <mo>⁡</mo> <msub> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mi>cos</mi> <mo>⁡</mo> <msub> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>+</mo> <mi>sin</mi> <mo>⁡</mo> <msub> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mi>sin</mi> <mo>⁡</mo> <msub> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \cos \theta _{1}\cos \theta _{2}+\sin \theta _{1}\sin \theta _{2}=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1e4e69fe7b1e1f42e637245991d1af0f5e97797e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:29.936ex; height:2.509ex;" alt="{\displaystyle \cos \theta _{1}\cos \theta _{2}+\sin \theta _{1}\sin \theta _{2}=0}"></span>. Rearranging gives <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 \tan \theta _{1}=-\cot \theta _{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>tan</mi> <mo>⁡</mo> <msub> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mo>−</mo> <mi>cot</mi> <mo>⁡</mo> <msub> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \tan \theta _{1}=-\cot \theta _{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ccfacfa2dc4ec9a0ba8412beefe7070ca6f8e1de" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:16.816ex; height:2.509ex;" alt="{\displaystyle \tan \theta _{1}=-\cot \theta _{2}}"></span>. Using a <a href="/wiki/List_of_trigonometric_identities#Shifts_and_periodicity" title="List of trigonometric identities">trigonometric identity</a> to convert the <a href="/wiki/Cotangent" class="mw-redirect" title="Cotangent">cotangent</a> term gives </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 \tan(\theta _{1})=\tan \left(\theta _{2}+{\tfrac {\pi }{2}}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>tan</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <msub> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <mi>tan</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <msub> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mi>π</mi> <mn>2</mn> </mfrac> </mstyle> </mrow> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \tan(\theta _{1})=\tan \left(\theta _{2}+{\tfrac {\pi }{2}}\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/742564313625329ad5cd932eb03d201a36cb3f95" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:22.665ex; height:3.343ex;" alt="{\displaystyle \tan(\theta _{1})=\tan \left(\theta _{2}+{\tfrac {\pi }{2}}\right)}"></span></dd> <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 \Rightarrow \theta _{1}=\theta _{2}+{\tfrac {\pi }{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">⇒</mo> <msub> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>=</mo> <msub> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mi>π</mi> <mn>2</mn> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Rightarrow \theta _{1}=\theta _{2}+{\tfrac {\pi }{2}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d4510ce793c1c19925dd1bb534c8ca42dda2401e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:14.975ex; height:3.176ex;" alt="{\displaystyle \Rightarrow \theta _{1}=\theta _{2}+{\tfrac {\pi }{2}}}"></span></dd></dl> <p>It is clear that in the plane, orthonormal vectors are simply radii of the unit circle whose difference in angles equals 90°. </p> <div class="mw-heading mw-heading2"><h2 id="Definition">Definition</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Orthonormality&action=edit&section=3" title="Edit section: Definition"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Let <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 {\mathcal {V}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">V</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {V}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47d69f309b6deb2e5008f6130ee11e09bbabd7b6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.529ex; height:2.176ex;" alt="{\displaystyle {\mathcal {V}}}"></span> be an <a href="/wiki/Inner-product_space" class="mw-redirect" title="Inner-product space">inner-product space</a>. A set of vectors </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\{u_{1},u_{2},\ldots ,u_{n},\ldots \right\}\in {\mathcal {V}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>{</mo> <mrow> <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>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>,</mo> <mo>…</mo> </mrow> <mo>}</mo> </mrow> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">V</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left\{u_{1},u_{2},\ldots ,u_{n},\ldots \right\}\in {\mathcal {V}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b9034e221fd1224c8c7797a2f69343c7f3e71a4d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.98ex; height:2.843ex;" alt="{\displaystyle \left\{u_{1},u_{2},\ldots ,u_{n},\ldots \right\}\in {\mathcal {V}}}"></span></dd></dl> <p>is called <b>orthonormal</b> <a href="/wiki/If_and_only_if" title="If and only if">if and only if</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 \forall i,j:\langle u_{i},u_{j}\rangle =\delta _{ij}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∀</mi> <mi>i</mi> <mo>,</mo> <mi>j</mi> <mo>:</mo> <mo fence="false" stretchy="false">⟨</mo> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo fence="false" stretchy="false">⟩</mo> <mo>=</mo> <msub> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \forall i,j:\langle u_{i},u_{j}\rangle =\delta _{ij}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7ecf44b5f18b3e183c9b36d3034f56a1213631f8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:18.844ex; height:3.009ex;" alt="{\displaystyle \forall i,j:\langle u_{i},u_{j}\rangle =\delta _{ij}}"></span></dd></dl> <p>where <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 \delta _{ij}\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </msub> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \delta _{ij}\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/30ca9f0bd511de6d1e172c92dd7e8d96e7897886" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.897ex; height:3.009ex;" alt="{\displaystyle \delta _{ij}\,}"></span> is the <a href="/wiki/Kronecker_delta" title="Kronecker delta">Kronecker delta</a> and <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 }"> <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> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \langle \cdot ,\cdot \rangle }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9a50080b735975d8001c9552ac2134b49ad534c0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.137ex; height:2.843ex;" alt="{\displaystyle \langle \cdot ,\cdot \rangle }"></span> is the <a href="/wiki/Inner_product" class="mw-redirect" title="Inner product">inner product</a> defined over <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 {\mathcal {V}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">V</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {V}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47d69f309b6deb2e5008f6130ee11e09bbabd7b6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.529ex; height:2.176ex;" alt="{\displaystyle {\mathcal {V}}}"></span>. </p> <div class="mw-heading mw-heading2"><h2 id="Significance">Significance</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Orthonormality&action=edit&section=4" title="Edit section: Significance"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Orthonormal sets are not especially significant on their own. However, they display certain features that make them fundamental in exploring the notion of <a href="/wiki/Diagonalizable_matrix" title="Diagonalizable matrix">diagonalizability</a> of certain <a href="/wiki/Linear_map" title="Linear map">operators</a> on vector spaces. </p> <div class="mw-heading mw-heading3"><h3 id="Properties">Properties</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Orthonormality&action=edit&section=5" title="Edit section: Properties"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Orthonormal sets have certain very appealing properties, which make them particularly easy to work with. </p> <ul><li><b>Theorem</b>. If {<b>e</b><sub>1</sub>, <b>e</b><sub>2</sub>, ..., <b>e</b><sub><i>n</i></sub>} is an orthonormal list of vectors, then <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \forall {\textbf {a}}:=[a_{1},\cdots ,a_{n}];\ \|a_{1}{\textbf {e}}_{1}+a_{2}{\textbf {e}}_{2}+\cdots +a_{n}{\textbf {e}}_{n}\|^{2}=|a_{1}|^{2}+|a_{2}|^{2}+\cdots +|a_{n}|^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∀</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mtext mathvariant="bold">a</mtext> </mrow> </mrow> <mo>:=</mo> <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>n</mi> </mrow> </msub> <mo stretchy="false">]</mo> <mo>;</mo> <mtext> </mtext> <mo fence="false" stretchy="false">‖</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mtext mathvariant="bold">e</mtext> </mrow> </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"> <mrow class="MJX-TeXAtom-ORD"> <mtext mathvariant="bold">e</mtext> </mrow> </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>n</mi> </mrow> </msub> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mtext mathvariant="bold">e</mtext> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <msup> <mo fence="false" stretchy="false">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mo>⋯</mo> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \forall {\textbf {a}}:=[a_{1},\cdots ,a_{n}];\ \|a_{1}{\textbf {e}}_{1}+a_{2}{\textbf {e}}_{2}+\cdots +a_{n}{\textbf {e}}_{n}\|^{2}=|a_{1}|^{2}+|a_{2}|^{2}+\cdots +|a_{n}|^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4a6fa6833dee116691352686964905035e2ccb39" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:76.202ex; height:3.343ex;" alt="{\displaystyle \forall {\textbf {a}}:=[a_{1},\cdots ,a_{n}];\ \|a_{1}{\textbf {e}}_{1}+a_{2}{\textbf {e}}_{2}+\cdots +a_{n}{\textbf {e}}_{n}\|^{2}=|a_{1}|^{2}+|a_{2}|^{2}+\cdots +|a_{n}|^{2}}"></span></li> <li><b>Theorem</b>. Every orthonormal list of vectors is <a href="/wiki/Linearly_independent" class="mw-redirect" title="Linearly independent">linearly independent</a>.</li></ul> <div class="mw-heading mw-heading3"><h3 id="Existence">Existence</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Orthonormality&action=edit&section=6" title="Edit section: Existence"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><b><a href="/wiki/Gram-Schmidt_theorem" class="mw-redirect" title="Gram-Schmidt theorem">Gram-Schmidt theorem</a></b>. If {<b>v</b><sub>1</sub>, <b>v</b><sub>2</sub>,...,<b>v</b><sub>n</sub>} is a linearly independent list of vectors in an inner-product space <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 {\mathcal {V}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">V</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {V}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47d69f309b6deb2e5008f6130ee11e09bbabd7b6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.529ex; height:2.176ex;" alt="{\displaystyle {\mathcal {V}}}"></span>, then there exists an orthonormal list {<b>e</b><sub>1</sub>, <b>e</b><sub>2</sub>,...,<b>e</b><sub>n</sub>} of vectors 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 {\mathcal {V}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">V</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {V}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47d69f309b6deb2e5008f6130ee11e09bbabd7b6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.529ex; height:2.176ex;" alt="{\displaystyle {\mathcal {V}}}"></span> such that <i>span</i>(<b>e</b><sub>1</sub>, <b>e</b><sub>2</sub>,...,<b>e</b><sub>n</sub>) = <i>span</i>(<b>v</b><sub>1</sub>, <b>v</b><sub>2</sub>,...,<b>v</b><sub>n</sub>).</li></ul> <p>Proof of the Gram-Schmidt theorem is <a href="/wiki/Constructive_proof" title="Constructive proof">constructive</a>, and <a href="/wiki/Gram-Schmidt_process" class="mw-redirect" title="Gram-Schmidt process">discussed at length</a> elsewhere. The Gram-Schmidt theorem, together with the <a href="/wiki/Axiom_of_choice" title="Axiom of choice">axiom of choice</a>, guarantees that every vector space admits an orthonormal basis. This is possibly the most significant use of orthonormality, as this fact permits <a href="/wiki/Linear_map" title="Linear map">operators</a> on inner-product spaces to be discussed in terms of their action on the space's orthonormal basis vectors. What results is a deep relationship between the diagonalizability of an operator and how it acts on the orthonormal basis vectors. This relationship is characterized by the <a href="/wiki/Spectral_theorem" title="Spectral theorem">Spectral Theorem</a>. </p> <div class="mw-heading mw-heading2"><h2 id="Examples">Examples</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Orthonormality&action=edit&section=7" title="Edit section: Examples"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Standard_basis">Standard basis</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Orthonormality&action=edit&section=8" title="Edit section: Standard basis"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The <a href="/wiki/Standard_basis" title="Standard basis">standard basis</a> for the <a href="/wiki/Coordinate_space" class="mw-redirect" title="Coordinate space">coordinate space</a> <b>F</b><sup><i>n</i></sup> is </p> <dl><dd><table> <tbody><tr> <td>{<b>e</b><sub>1</sub>, <b>e</b><sub>2</sub>,...,<b>e</b><sub>n</sub>}   where </td> <td>   <b>e</b><sub>1</sub> = (1, 0, ..., 0) </td></tr> <tr> <td> </td> <td>   <b>e</b><sub>2</sub> = (0, 1, ..., 0) </td></tr> <tr> <td> </td> <td><div class="center" style="width:auto; margin-left:auto; margin-right:auto;"><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 \vdots }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>⋮</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \vdots }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f8039d9feb6596ae092e5305108722975060c083" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.647ex; height:3.676ex;" alt="{\displaystyle \vdots }"></span></div> </td></tr> <tr> <td> </td> <td>   <b>e</b><sub>n</sub> = (0, 0, ..., 1) </td></tr></tbody></table></dd></dl> <p>Any two vectors <b>e</b><sub>i</sub>, <b>e</b><sub>j</sub> where i≠j are orthogonal, and all vectors are clearly of unit length. So {<b>e</b><sub>1</sub>, <b>e</b><sub>2</sub>,...,<b>e</b><sub>n</sub>} forms an orthonormal basis. </p> <div class="mw-heading mw-heading3"><h3 id="Real-valued_functions">Real-valued functions</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Orthonormality&action=edit&section=9" title="Edit section: Real-valued functions"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>When referring to <a href="/wiki/Real_number" title="Real number">real</a>-valued <a href="/wiki/Function_(mathematics)" title="Function (mathematics)">functions</a>, usually the <a href="/wiki/Lp_space" title="Lp space">L²</a> inner product is assumed unless otherwise stated. Two functions <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 \phi (x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ϕ</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \phi (x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/546b660b2f3cfb5f34be7b3ed8371d54f5c74227" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.524ex; height:2.843ex;" alt="{\displaystyle \phi (x)}"></span> and <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 \psi (x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ψ</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \psi (x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a596a1fb4130a47f6b88c66150497338bd6cbccc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.652ex; height:2.843ex;" alt="{\displaystyle \psi (x)}"></span> are orthonormal over the <a href="/wiki/Interval_(mathematics)" title="Interval (mathematics)">interval</a> <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,b]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">[</mo> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle [a,b]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9c4b788fc5c637e26ee98b45f89a5c08c85f7935" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.555ex; height:2.843ex;" alt="{\displaystyle [a,b]}"></span> 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 (1)\quad \langle \phi (x),\psi (x)\rangle =\int _{a}^{b}\phi (x)\psi (x)dx=0,\quad {\rm {and}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mspace width="1em" /> <mo fence="false" stretchy="false">⟨</mo> <mi>ϕ</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>,</mo> <mi>ψ</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo fence="false" stretchy="false">⟩</mo> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msubsup> <mi>ϕ</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mi>ψ</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mi>d</mi> <mi>x</mi> <mo>=</mo> <mn>0</mn> <mo>,</mo> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">a</mi> <mi mathvariant="normal">n</mi> <mi mathvariant="normal">d</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (1)\quad \langle \phi (x),\psi (x)\rangle =\int _{a}^{b}\phi (x)\psi (x)dx=0,\quad {\rm {and}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4896689400f30d78553721edb75240650cfee4b7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:47.289ex; height:6.343ex;" alt="{\displaystyle (1)\quad \langle \phi (x),\psi (x)\rangle =\int _{a}^{b}\phi (x)\psi (x)dx=0,\quad {\rm {and}}}"></span></dd> <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 (2)\quad ||\phi (x)||_{2}=||\psi (x)||_{2}=\left[\int _{a}^{b}|\phi (x)|^{2}dx\right]^{\frac {1}{2}}=\left[\int _{a}^{b}|\psi (x)|^{2}dx\right]^{\frac {1}{2}}=1.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mn>2</mn> <mo stretchy="false">)</mo> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>ϕ</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>ψ</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>=</mo> <msup> <mrow> <mo>[</mo> <mrow> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>ϕ</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>d</mi> <mi>x</mi> </mrow> <mo>]</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> </msup> <mo>=</mo> <msup> <mrow> <mo>[</mo> <mrow> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>ψ</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>d</mi> <mi>x</mi> </mrow> <mo>]</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> </msup> <mo>=</mo> <mn>1.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (2)\quad ||\phi (x)||_{2}=||\psi (x)||_{2}=\left[\int _{a}^{b}|\phi (x)|^{2}dx\right]^{\frac {1}{2}}=\left[\int _{a}^{b}|\psi (x)|^{2}dx\right]^{\frac {1}{2}}=1.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0449227490994031a067a06f55e857c569b4c5ad" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:70.88ex; height:7.509ex;" alt="{\displaystyle (2)\quad ||\phi (x)||_{2}=||\psi (x)||_{2}=\left[\int _{a}^{b}|\phi (x)|^{2}dx\right]^{\frac {1}{2}}=\left[\int _{a}^{b}|\psi (x)|^{2}dx\right]^{\frac {1}{2}}=1.}"></span></dd></dl> <div class="mw-heading mw-heading3"><h3 id="Fourier_series">Fourier series</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Orthonormality&action=edit&section=10" title="Edit section: Fourier series"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The <a href="/wiki/Fourier_series" title="Fourier series">Fourier series</a> is a method of expressing a periodic function in terms of sinusoidal <a href="/wiki/Schauder_basis" title="Schauder basis">basis</a> functions. Taking <b>C</b>[−π,π] to be the space of all real-valued functions continuous on the interval [−π,π] and taking the inner product to be </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,g\rangle =\int _{-\pi }^{\pi }f(x)g(x)dx}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">⟨</mo> <mi>f</mi> <mo>,</mo> <mi>g</mi> <mo fence="false" stretchy="false">⟩</mo> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>π</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>π</mi> </mrow> </msubsup> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mi>g</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mi>d</mi> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \langle f,g\rangle =\int _{-\pi }^{\pi }f(x)g(x)dx}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/edbbeef1dbff8ef8e3807883ffda8104df4af1be" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:23.687ex; height:6.009ex;" alt="{\displaystyle \langle f,g\rangle =\int _{-\pi }^{\pi }f(x)g(x)dx}"></span></dd></dl> <p>it can be shown 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 \left\{{\frac {1}{\sqrt {2\pi }}},{\frac {\sin(x)}{\sqrt {\pi }}},{\frac {\sin(2x)}{\sqrt {\pi }}},\ldots ,{\frac {\sin(nx)}{\sqrt {\pi }}},{\frac {\cos(x)}{\sqrt {\pi }}},{\frac {\cos(2x)}{\sqrt {\pi }}},\ldots ,{\frac {\cos(nx)}{\sqrt {\pi }}}\right\},\quad n\in \mathbb {N} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>{</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msqrt> <mn>2</mn> <mi>π</mi> </msqrt> </mfrac> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <msqrt> <mi>π</mi> </msqrt> </mfrac> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <msqrt> <mi>π</mi> </msqrt> </mfrac> </mrow> <mo>,</mo> <mo>…</mo> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>n</mi> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <msqrt> <mi>π</mi> </msqrt> </mfrac> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <msqrt> <mi>π</mi> </msqrt> </mfrac> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <msqrt> <mi>π</mi> </msqrt> </mfrac> </mrow> <mo>,</mo> <mo>…</mo> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>n</mi> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <msqrt> <mi>π</mi> </msqrt> </mfrac> </mrow> </mrow> <mo>}</mo> </mrow> <mo>,</mo> <mspace width="1em" /> <mi>n</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left\{{\frac {1}{\sqrt {2\pi }}},{\frac {\sin(x)}{\sqrt {\pi }}},{\frac {\sin(2x)}{\sqrt {\pi }}},\ldots ,{\frac {\sin(nx)}{\sqrt {\pi }}},{\frac {\cos(x)}{\sqrt {\pi }}},{\frac {\cos(2x)}{\sqrt {\pi }}},\ldots ,{\frac {\cos(nx)}{\sqrt {\pi }}}\right\},\quad n\in \mathbb {N} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e71ccfcbf2b26c425d9bdbbf648d969a3fb68fe3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:79.767ex; height:6.676ex;" alt="{\displaystyle \left\{{\frac {1}{\sqrt {2\pi }}},{\frac {\sin(x)}{\sqrt {\pi }}},{\frac {\sin(2x)}{\sqrt {\pi }}},\ldots ,{\frac {\sin(nx)}{\sqrt {\pi }}},{\frac {\cos(x)}{\sqrt {\pi }}},{\frac {\cos(2x)}{\sqrt {\pi }}},\ldots ,{\frac {\cos(nx)}{\sqrt {\pi }}}\right\},\quad n\in \mathbb {N} }"></span></dd></dl> <p>forms an orthonormal set. </p><p>However, this is of little consequence, because <b>C</b>[−π,π] is infinite-dimensional, and a finite set of vectors cannot span it. But, removing the restriction that <i>n</i> be finite makes the set <a href="/wiki/Dense_subset" class="mw-redirect" title="Dense subset">dense</a> in <b>C</b>[−π,π] and therefore an orthonormal basis of <b>C</b>[−π,π]. </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=Orthonormality&action=edit&section=11" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Orthogonalization" title="Orthogonalization">Orthogonalization</a></li> <li><a href="/wiki/Orthonormal_function_system" title="Orthonormal function system">Orthonormal function system</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="Sources">Sources</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Orthonormality&action=edit&section=12" title="Edit section: Sources"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><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="CITEREFAxler1997" class="citation cs2"><a href="/wiki/Sheldon_Axler" title="Sheldon Axler">Axler, Sheldon</a> (1997), <i>Linear Algebra Done Right</i> (2nd ed.), Berlin, New York: <a href="/wiki/Springer-Verlag" class="mw-redirect" title="Springer-Verlag">Springer-Verlag</a>, p. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=ovIYVIlithQC&pg=PT106">106–110</a>, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-387-98258-8" title="Special:BookSources/978-0-387-98258-8"><bdi>978-0-387-98258-8</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+Done+Right&rft.place=Berlin%2C+New+York&rft.pages=106-110&rft.edition=2nd&rft.pub=Springer-Verlag&rft.date=1997&rft.isbn=978-0-387-98258-8&rft.aulast=Axler&rft.aufirst=Sheldon&rfr_id=info%3Asid%2Fen.wikipedia.org%3AOrthonormality" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFChen2009" class="citation cs2">Chen, Wai-Kai (2009), <i>Fundamentals of Circuits and Filters</i> (3rd ed.), <a href="/wiki/Boca_Raton,_Florida" title="Boca Raton, Florida">Boca Raton</a>: <a href="/wiki/CRC_Press" title="CRC Press">CRC Press</a>, p. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=_UVb4cxL0c0C&pg=SA6-PA62">62</a>, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-4200-5887-1" title="Special:BookSources/978-1-4200-5887-1"><bdi>978-1-4200-5887-1</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Fundamentals+of+Circuits+and+Filters&rft.place=Boca+Raton&rft.pages=62&rft.edition=3rd&rft.pub=CRC+Press&rft.date=2009&rft.isbn=978-1-4200-5887-1&rft.aulast=Chen&rft.aufirst=Wai-Kai&rfr_id=info%3Asid%2Fen.wikipedia.org%3AOrthonormality" class="Z3988"></span></li></ul>    </div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?type=1x1" 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=Orthonormality&oldid=1251419114">https://en.wikipedia.org/w/index.php?title=Orthonormality&oldid=1251419114</a>"</div></div> <div id="catlinks" class="catlinks" data-mw="interface"><div id="mw-normal-catlinks" 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Orthonormality - Wikipedia