Standardbasis – Wikipedia

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</nav> <div id="mw-content-subtitle"></div> </div> <div id="bodyContent" class="content"> <div id="mw-content-text" class="mw-body-content"> <script>function mfTempOpenSection(id){var block=document.getElementById("mf-section-"+id);block.className+=" open-block";block.previousSibling.className+=" open-block";}</script> <div class="mw-content-ltr mw-parser-output" lang="de" dir="ltr"> <section class="mf-section-0" id="mf-section-0"> <div class="hintergrundfarbe1 rahmenfarbe1 navigation-not-searchable noprint hatnote navigation-not-searchable" style="border-bottom-style: solid; border-bottom-width: 1px; font-size:95%; margin-bottom:1em; padding: 0.25em; overflow: hidden; word-break: break-word; word-wrap: break-word;" id="Vorlage_Dieser_Artikel"> <div class="noviewer noresize" style="display: table-cell; padding-bottom: 0.2em; padding-left: 0.25em; padding-right: 1em; padding-top: 0.2em; vertical-align: middle;" id="bksicon" aria-hidden="true" role="presentation"> <span typeof="mw:File"><span><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/e/ea/Disambig-dark.svg/25px-Disambig-dark.svg.png" decoding="async" width="25" height="19" class="mw-file-element" srcset="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://upload.wikimedia.org/wikipedia/commons/thumb/e/ea/Disambig-dark.svg/38px-Disambig-dark.svg.png 1.5x,https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://upload.wikimedia.org/wikipedia/commons/thumb/e/ea/Disambig-dark.svg/50px-Disambig-dark.svg.png 2x" data-file-width="444" data-file-height="340"></span></span> </div> <div style="display: table-cell; vertical-align: middle; width: 100%;"> <div role="navigation"> Dieser Artikel behandelt kanonische Basen in bestimmten Vektorräumen, für das Konzept aus der Theorie der Polynomideale siehe <a href="https://de-m-wikipedia-org.translate.goog/wiki/Gr%C3%B6bnerbasis?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Gröbnerbasis">Gröbnerbasis</a>. </div> </div> </div> <p>Als <b>Standardbasis</b>, <b>natürliche Basis</b>, <b>Einheitsbasis</b> oder <b>kanonische Basis</b> bezeichnet man im <a href="https://de-m-wikipedia-org.translate.goog/wiki/Mathematik?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Mathematik">mathematischen</a> Teilgebiet der <a href="https://de-m-wikipedia-org.translate.goog/wiki/Lineare_Algebra?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Lineare Algebra">linearen Algebra</a> eine spezielle <a href="https://de-m-wikipedia-org.translate.goog/wiki/Basis_(Vektorraum)?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Basis (Vektorraum)">Basis</a>, die in gewissen <a href="https://de-m-wikipedia-org.translate.goog/wiki/Vektorraum?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Vektorraum">Vektorräumen</a> bereits aufgrund ihrer Konstruktion unter allen möglichen Basen ausgezeichnet ist.</p> <div id="toc" class="toc" role="navigation" aria-labelledby="mw-toc-heading"> <input type="checkbox" role="button" id="toctogglecheckbox" class="toctogglecheckbox" style="display:none"> <div class="toctitle" lang="de" dir="ltr"> <h2 id="mw-toc-heading">Inhaltsverzeichnis</h2><span class="toctogglespan"><label class="toctogglelabel" for="toctogglecheckbox"></label></span> </div> <ul> <li class="toclevel-1 tocsection-1"><a href="https://de-m-wikipedia-org.translate.goog/wiki/Standardbasis?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#Basis_allgemein"><span class="tocnumber">1</span> <span class="toctext">Basis allgemein</span></a> <ul> <li class="toclevel-2 tocsection-2"><a href="https://de-m-wikipedia-org.translate.goog/wiki/Standardbasis?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#Beispiele"><span class="tocnumber">1.1</span> <span class="toctext">Beispiele</span></a></li> </ul></li> <li class="toclevel-1 tocsection-3"><a href="https://de-m-wikipedia-org.translate.goog/wiki/Standardbasis?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#Standardbasis_in_den_Standardr%C3%A4umen"><span class="tocnumber">2</span> <span class="toctext">Standardbasis in den Standardräumen</span></a> <ul> <li class="toclevel-2 tocsection-4"><a href="https://de-m-wikipedia-org.translate.goog/wiki/Standardbasis?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#Beispiel"><span class="tocnumber">2.1</span> <span class="toctext">Beispiel</span></a></li> <li class="toclevel-2 tocsection-5"><a href="https://de-m-wikipedia-org.translate.goog/wiki/Standardbasis?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#Bezeichnung"><span class="tocnumber">2.2</span> <span class="toctext">Bezeichnung</span></a></li> <li class="toclevel-2 tocsection-6"><a href="https://de-m-wikipedia-org.translate.goog/wiki/Standardbasis?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#Weitere_Eigenschaften"><span class="tocnumber">2.3</span> <span class="toctext">Weitere Eigenschaften</span></a></li> </ul></li> <li class="toclevel-1 tocsection-7"><a href="https://de-m-wikipedia-org.translate.goog/wiki/Standardbasis?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#Standardbasis_im_Matrizenraum"><span class="tocnumber">3</span> <span class="toctext">Standardbasis im Matrizenraum</span></a></li> <li class="toclevel-1 tocsection-8"><a href="https://de-m-wikipedia-org.translate.goog/wiki/Standardbasis?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#Standardbasis_in_unendlichdimensionalen_R%C3%A4umen"><span class="tocnumber">4</span> <span class="toctext">Standardbasis in unendlichdimensionalen Räumen</span></a></li> <li class="toclevel-1 tocsection-9"><a href="https://de-m-wikipedia-org.translate.goog/wiki/Standardbasis?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#Zusammenhang_mit_universellen_Eigenschaften"><span class="tocnumber">5</span> <span class="toctext">Zusammenhang mit universellen Eigenschaften</span></a></li> <li class="toclevel-1 tocsection-10"><a href="https://de-m-wikipedia-org.translate.goog/wiki/Standardbasis?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#Literatur"><span class="tocnumber">6</span> <span class="toctext">Literatur</span></a></li> </ul> </div> </section> <div class="mw-heading mw-heading2 section-heading" onclick="mfTempOpenSection(1)"> <span class="indicator mf-icon mf-icon-expand mf-icon--small"></span> <h2 id="Basis_allgemein">Basis allgemein</h2><span class="mw-editsection"> <a role="button" href="https://de-m-wikipedia-org.translate.goog/w/index.php?title=Standardbasis&action=edit&section=1&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Abschnitt bearbeiten: Basis allgemein" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> <span class="minerva-icon minerva-icon--edit"></span> <span>Bearbeiten</span> </a> </span> </div> <section class="mf-section-1 collapsible-block" id="mf-section-1"> <div class="hauptartikel" role="navigation"> <span class="hauptartikel-pfeil" title="siehe" aria-hidden="true" role="presentation">→ </span><i><span class="hauptartikel-text">Hauptartikel</span>: <a href="https://de-m-wikipedia-org.translate.goog/wiki/Basis_(Vektorraum)?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Basis (Vektorraum)">Basis (Vektorraum)</a></i> </div> <p>Allgemein ist eine Basis eines Vektorraums eine <a href="https://de-m-wikipedia-org.translate.goog/wiki/Familie_(Mathematik)?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Familie (Mathematik)">Familie</a> von Vektoren mit der Eigenschaft, dass sich jeder Vektor des Raumes eindeutig als endliche <a href="https://de-m-wikipedia-org.translate.goog/wiki/Linearkombination?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Linearkombination">Linearkombination</a> dieser darstellen lässt. Die Koeffizienten dieser Linearkombination heißen die Koordinaten des Vektors bezüglich dieser Basis. Ein Element der Basis heißt Basisvektor.</p> <p>Jeder Vektorraum hat eine Basis, im Allgemeinen sogar zahlreiche Basen, unter denen jedoch keine ausgezeichnet ist.</p> <div class="mw-heading mw-heading3"> <h3 id="Beispiele">Beispiele</h3><span class="mw-editsection"> <a role="button" href="https://de-m-wikipedia-org.translate.goog/w/index.php?title=Standardbasis&action=edit&section=2&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Abschnitt bearbeiten: Beispiele" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> <span class="minerva-icon minerva-icon--edit"></span> <span>Bearbeiten</span> </a> </span> </div> <ul> <li>Die <a href="https://de-m-wikipedia-org.translate.goog/wiki/Parallelverschiebung?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Parallelverschiebung">Parallelverschiebungen</a> der Anschauungsebene bilden einen Vektorraum (<i>siehe</i> <a href="https://de-m-wikipedia-org.translate.goog/wiki/Euklidischer_Raum?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Euklidischer Raum">Euklidischer Raum</a>) der <a href="https://de-m-wikipedia-org.translate.goog/wiki/Dimension_(Mathematik)?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Dimension (Mathematik)">Dimension</a> zwei. Es ist jedoch keine Basis ausgezeichnet. Eine mögliche Basis bestünde etwa aus der „Verschiebung um eine Einheit nach rechts“ und der „Verschiebung um eine Einheit nach oben“. Hierbei sind „Einheit“, „rechts“ und „oben“ aber Konventionen bzw. anschauungsabhängig.</li> <li>Diejenigen reellwertigen <a href="https://de-m-wikipedia-org.translate.goog/wiki/Funktion_(Mathematik)?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Funktion (Mathematik)">Funktionen</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 f\colon \mathbb {R} \to \mathbb {R} }"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> f </mi> <mo> : </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck"> R </mi> </mrow> <mo stretchy="false"> → </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck"> R </mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle f\colon \mathbb {R} \to \mathbb {R} } </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b1cacd5f7bbe1027cc75fbe2fbd9cb5e79485302" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.283ex; height:2.509ex;" alt="{\displaystyle f\colon \mathbb {R} \to \mathbb {R} }"> </noscript><span class="lazy-image-placeholder" style="width: 9.283ex;height: 2.509ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b1cacd5f7bbe1027cc75fbe2fbd9cb5e79485302" data-alt="{\displaystyle f\colon \mathbb {R} \to \mathbb {R} }" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span>, die zweimal <a href="https://de-m-wikipedia-org.translate.goog/wiki/Differenzierbar?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-redirect" title="Differenzierbar">differenzierbar</a> sind und für alle <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\in \mathbb {R} }"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> x </mi> <mo> ∈ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck"> R </mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle x\in \mathbb {R} } </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a9c6d458566aec47a7259762034790c8981aefab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.848ex; height:2.176ex;" alt="{\displaystyle x\in \mathbb {R} }"> </noscript><span class="lazy-image-placeholder" style="width: 5.848ex;height: 2.176ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a9c6d458566aec47a7259762034790c8981aefab" data-alt="{\displaystyle x\in \mathbb {R} }" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> die Gleichung <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(x)+f''(x)=0}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> f </mi> <mo stretchy="false"> ( </mo> <mi> x </mi> <mo stretchy="false"> ) </mo> <mo> + </mo> <msup> <mi> f </mi> <mo> ″ </mo> </msup> <mo stretchy="false"> ( </mo> <mi> x </mi> <mo stretchy="false"> ) </mo> <mo> = </mo> <mn> 0 </mn> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle f(x)+f''(x)=0} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/16a288f4a58c8d138f2732bf04636fe4921aa52d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.116ex; height:3.009ex;" alt="{\displaystyle f(x)+f''(x)=0}"> </noscript><span class="lazy-image-placeholder" style="width: 17.116ex;height: 3.009ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/16a288f4a58c8d138f2732bf04636fe4921aa52d" data-alt="{\displaystyle f(x)+f''(x)=0}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> erfüllen, bilden einen reellen Vektorraum <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 V}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> V </mi> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle V} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/af0f6064540e84211d0ffe4dac72098adfa52845" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.176ex;" alt="{\displaystyle V}"> </noscript><span class="lazy-image-placeholder" style="width: 1.787ex;height: 2.176ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/af0f6064540e84211d0ffe4dac72098adfa52845" data-alt="{\displaystyle V}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> der Dimension zwei. Eine mögliche Basis wird von der <a href="https://de-m-wikipedia-org.translate.goog/wiki/Sinus?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-redirect" title="Sinus">Sinus</a>- sowie der <a href="https://de-m-wikipedia-org.translate.goog/wiki/Kosinus?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-redirect" title="Kosinus">Cosinus</a>-Funktion gebildet. Diese Basis zu wählen, mag zwar naheliegen, sie ist jedoch nicht besonders vor anderen Auswahlen ausgezeichnet.</li> </ul> </section> <div class="mw-heading mw-heading2 section-heading" onclick="mfTempOpenSection(2)"> <span class="indicator mf-icon mf-icon-expand mf-icon--small"></span> <h2 id="Standardbasis_in_den_Standardräumen"><span id="Standardbasis_in_den_Standardr.C3.A4umen"></span>Standardbasis in den Standardräumen</h2><span class="mw-editsection"> <a role="button" href="https://de-m-wikipedia-org.translate.goog/w/index.php?title=Standardbasis&action=edit&section=3&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Abschnitt bearbeiten: Standardbasis in den Standardräumen" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> <span class="minerva-icon minerva-icon--edit"></span> <span>Bearbeiten</span> </a> </span> </div> <section class="mf-section-2 collapsible-block" id="mf-section-2"> <figure class="mw-default-size" typeof="mw:File/Thumb"> <a href="https://de-m-wikipedia-org.translate.goog/wiki/Datei:Unit_vectors_qtl2.svg?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-file-description"> <noscript> <img src="//upload.wikimedia.org/wikipedia/commons/thumb/8/8d/Unit_vectors_qtl2.svg/220px-Unit_vectors_qtl2.svg.png" decoding="async" width="220" height="220" class="mw-file-element" data-file-width="410" data-file-height="410"> </noscript><span class="lazy-image-placeholder" style="width: 220px;height: 220px;" data-src="//upload.wikimedia.org/wikipedia/commons/thumb/8/8d/Unit_vectors_qtl2.svg/220px-Unit_vectors_qtl2.svg.png" data-width="220" data-height="220" data-srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/8d/Unit_vectors_qtl2.svg/330px-Unit_vectors_qtl2.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/8/8d/Unit_vectors_qtl2.svg/440px-Unit_vectors_qtl2.svg.png 2x" data-class="mw-file-element"> </span></a> <figcaption> Standardbasisvektoren in der euklidischen Ebene </figcaption> </figure> <p>Die meist als erstes eingeführten Vektorräume sind die <a href="https://de-m-wikipedia-org.translate.goog/wiki/Standardraum?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-redirect" title="Standardraum">Standardräume</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 \mathbb {R} ^{n}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck"> R </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> n </mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \mathbb {R} ^{n}} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c510b63578322050121fe966f2e5770bea43308d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.897ex; height:2.343ex;" alt="{\displaystyle \mathbb {R} ^{n}}"> </noscript><span class="lazy-image-placeholder" style="width: 2.897ex;height: 2.343ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c510b63578322050121fe966f2e5770bea43308d" data-alt="{\displaystyle \mathbb {R} ^{n}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> mit <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 n\in \mathbb {N} }"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <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 n\in \mathbb {N} } </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d059936e77a2d707e9ee0a1d9575a1d693ce5d0b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.913ex; height:2.176ex;" alt="{\displaystyle n\in \mathbb {N} }"> </noscript><span class="lazy-image-placeholder" style="width: 5.913ex;height: 2.176ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d059936e77a2d707e9ee0a1d9575a1d693ce5d0b" data-alt="{\displaystyle n\in \mathbb {N} }" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span>. Elemente des <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} ^{n}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck"> R </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> n </mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \mathbb {R} ^{n}} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c510b63578322050121fe966f2e5770bea43308d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.897ex; height:2.343ex;" alt="{\displaystyle \mathbb {R} ^{n}}"> </noscript><span class="lazy-image-placeholder" style="width: 2.897ex;height: 2.343ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c510b63578322050121fe966f2e5770bea43308d" data-alt="{\displaystyle \mathbb {R} ^{n}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> sind alle <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 n}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> n </mi> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle n} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"> </noscript><span class="lazy-image-placeholder" style="width: 1.395ex;height: 1.676ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" data-alt="{\displaystyle n}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span>-<a href="https://de-m-wikipedia-org.translate.goog/wiki/Tupel?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Tupel">Tupel</a> <a href="https://de-m-wikipedia-org.translate.goog/wiki/Reelle_Zahl?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Reelle Zahl">reeller</a> Zahlen. Man kann unter allen Basen des <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} ^{n}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck"> R </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> n </mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \mathbb {R} ^{n}} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c510b63578322050121fe966f2e5770bea43308d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.897ex; height:2.343ex;" alt="{\displaystyle \mathbb {R} ^{n}}"> </noscript><span class="lazy-image-placeholder" style="width: 2.897ex;height: 2.343ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c510b63578322050121fe966f2e5770bea43308d" data-alt="{\displaystyle \mathbb {R} ^{n}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> diejenige auszeichnen, bezüglich der die Koordinaten eines Vektors genau mit seinen Tupel-Komponenten übereinstimmen. Diese Basis besteht also aus <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 e_{1},\ldots ,e_{n}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi> e </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 1 </mn> </mrow> </msub> <mo> , </mo> <mo> … </mo> <mo> , </mo> <msub> <mi> e </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> n </mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle e_{1},\ldots ,e_{n}} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c60c38b7e2450d62e9dc496b89f8e5c96c77cecf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.618ex; height:2.009ex;" alt="{\displaystyle e_{1},\ldots ,e_{n}}"> </noscript><span class="lazy-image-placeholder" style="width: 9.618ex;height: 2.009ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c60c38b7e2450d62e9dc496b89f8e5c96c77cecf" data-alt="{\displaystyle e_{1},\ldots ,e_{n}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> wobei</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{matrix}e_{1}&=&(1,0,0,\ldots ,0),\\e_{2}&=&(0,1,0,\ldots ,0),\\&\vdots &\\e_{n}&=&(0,0,0,\ldots ,1)\end{matrix}}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi> e </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 1 </mn> </mrow> </msub> </mtd> <mtd> <mo> = </mo> </mtd> <mtd> <mo stretchy="false"> ( </mo> <mn> 1 </mn> <mo> , </mo> <mn> 0 </mn> <mo> , </mo> <mn> 0 </mn> <mo> , </mo> <mo> … </mo> <mo> , </mo> <mn> 0 </mn> <mo stretchy="false"> ) </mo> <mo> , </mo> </mtd> </mtr> <mtr> <mtd> <msub> <mi> e </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msub> </mtd> <mtd> <mo> = </mo> </mtd> <mtd> <mo stretchy="false"> ( </mo> <mn> 0 </mn> <mo> , </mo> <mn> 1 </mn> <mo> , </mo> <mn> 0 </mn> <mo> , </mo> <mo> … </mo> <mo> , </mo> <mn> 0 </mn> <mo stretchy="false"> ) </mo> <mo> , </mo> </mtd> </mtr> <mtr> <mtd></mtd> <mtd> <mo> ⋮ </mo> </mtd> <mtd></mtd> </mtr> <mtr> <mtd> <msub> <mi> e </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> n </mi> </mrow> </msub> </mtd> <mtd> <mo> = </mo> </mtd> <mtd> <mo stretchy="false"> ( </mo> <mn> 0 </mn> <mo> , </mo> <mn> 0 </mn> <mo> , </mo> <mn> 0 </mn> <mo> , </mo> <mo> … </mo> <mo> , </mo> <mn> 1 </mn> <mo stretchy="false"> ) </mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle {\begin{matrix}e_{1}&=&(1,0,0,\ldots ,0),\\e_{2}&=&(0,1,0,\ldots ,0),\\&\vdots &\\e_{n}&=&(0,0,0,\ldots ,1)\end{matrix}}} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e4199b0ab88fe6a9fd275be04c7e6a0393aaa5a1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.505ex; width:23.859ex; height:14.176ex;" alt="{\displaystyle {\begin{matrix}e_{1}&=&(1,0,0,\ldots ,0),\\e_{2}&=&(0,1,0,\ldots ,0),\\&\vdots &\\e_{n}&=&(0,0,0,\ldots ,1)\end{matrix}}}"> </noscript><span class="lazy-image-placeholder" style="width: 23.859ex;height: 14.176ex;vertical-align: -6.505ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e4199b0ab88fe6a9fd275be04c7e6a0393aaa5a1" data-alt="{\displaystyle {\begin{matrix}e_{1}&=&(1,0,0,\ldots ,0),\\e_{2}&=&(0,1,0,\ldots ,0),\\&\vdots &\\e_{n}&=&(0,0,0,\ldots ,1)\end{matrix}}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> </dd> </dl> <p>und wird als die <i>Standardbasis</i> des <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} ^{n}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck"> R </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> n </mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \mathbb {R} ^{n}} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c510b63578322050121fe966f2e5770bea43308d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.897ex; height:2.343ex;" alt="{\displaystyle \mathbb {R} ^{n}}"> </noscript><span class="lazy-image-placeholder" style="width: 2.897ex;height: 2.343ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c510b63578322050121fe966f2e5770bea43308d" data-alt="{\displaystyle \mathbb {R} ^{n}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> bezeichnet.</p> <p>Dasselbe gilt für den Vektorraum <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 K^{n}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi> K </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> n </mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle K^{n}} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1d63366b3d00300e06eee81786182062b98775c5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.312ex; height:2.343ex;" alt="{\displaystyle K^{n}}"> </noscript><span class="lazy-image-placeholder" style="width: 3.312ex;height: 2.343ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1d63366b3d00300e06eee81786182062b98775c5" data-alt="{\displaystyle K^{n}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> über einem beliebigen <a href="https://de-m-wikipedia-org.translate.goog/wiki/K%C3%B6rper_(Algebra)?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Körper (Algebra)">Körper</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 K}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> K </mi> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle K} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2b76fce82a62ed5461908f0dc8f037de4e3686b0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K}"> </noscript><span class="lazy-image-placeholder" style="width: 2.066ex;height: 2.176ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2b76fce82a62ed5461908f0dc8f037de4e3686b0" data-alt="{\displaystyle K}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span>, das heißt auch hier gibt es die Standard-Basisvektoren <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 e_{1}=(1,0,\ldots ,0),\ldots ,e_{n}=(0,\ldots ,0,1)}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi> e </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 1 </mn> </mrow> </msub> <mo> = </mo> <mo stretchy="false"> ( </mo> <mn> 1 </mn> <mo> , </mo> <mn> 0 </mn> <mo> , </mo> <mo> … </mo> <mo> , </mo> <mn> 0 </mn> <mo stretchy="false"> ) </mo> <mo> , </mo> <mo> … </mo> <mo> , </mo> <msub> <mi> e </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> n </mi> </mrow> </msub> <mo> = </mo> <mo stretchy="false"> ( </mo> <mn> 0 </mn> <mo> , </mo> <mo> … </mo> <mo> , </mo> <mn> 0 </mn> <mo> , </mo> <mn> 1 </mn> <mo stretchy="false"> ) </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle e_{1}=(1,0,\ldots ,0),\ldots ,e_{n}=(0,\ldots ,0,1)} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/30c01e3f9c9b6960f167680707387d20f3d43661" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:38.832ex; height:2.843ex;" alt="{\displaystyle e_{1}=(1,0,\ldots ,0),\ldots ,e_{n}=(0,\ldots ,0,1)}"> </noscript><span class="lazy-image-placeholder" style="width: 38.832ex;height: 2.843ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/30c01e3f9c9b6960f167680707387d20f3d43661" data-alt="{\displaystyle e_{1}=(1,0,\ldots ,0),\ldots ,e_{n}=(0,\ldots ,0,1)}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span>.</p> <div class="mw-heading mw-heading3"> <h3 id="Beispiel">Beispiel</h3><span class="mw-editsection"> <a role="button" href="https://de-m-wikipedia-org.translate.goog/w/index.php?title=Standardbasis&action=edit&section=4&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Abschnitt bearbeiten: Beispiel" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> <span class="minerva-icon minerva-icon--edit"></span> <span>Bearbeiten</span> </a> </span> </div> <p>Die Standardbasis des <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} ^{2}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck"> R </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \mathbb {R} ^{2}} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e150115ab9f63023215109595b76686a1ff890fd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.732ex; height:2.676ex;" alt="{\displaystyle \mathbb {R} ^{2}}"> </noscript><span class="lazy-image-placeholder" style="width: 2.732ex;height: 2.676ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e150115ab9f63023215109595b76686a1ff890fd" data-alt="{\displaystyle \mathbb {R} ^{2}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> besteht aus <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 e_{1}=(1,0)}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi> e </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 1 </mn> </mrow> </msub> <mo> = </mo> <mo stretchy="false"> ( </mo> <mn> 1 </mn> <mo> , </mo> <mn> 0 </mn> <mo stretchy="false"> ) </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle e_{1}=(1,0)} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9c0498b5be0f2dccb99fc0685772150538b77fcf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.404ex; height:2.843ex;" alt="{\displaystyle e_{1}=(1,0)}"> </noscript><span class="lazy-image-placeholder" style="width: 10.404ex;height: 2.843ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9c0498b5be0f2dccb99fc0685772150538b77fcf" data-alt="{\displaystyle e_{1}=(1,0)}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> und <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 e_{2}=(0,1)}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi> e </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msub> <mo> = </mo> <mo stretchy="false"> ( </mo> <mn> 0 </mn> <mo> , </mo> <mn> 1 </mn> <mo stretchy="false"> ) </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle e_{2}=(0,1)} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7cac36162fa61f88374e595a45b2af71ccbb58a2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.404ex; height:2.843ex;" alt="{\displaystyle e_{2}=(0,1)}"> </noscript><span class="lazy-image-placeholder" style="width: 10.404ex;height: 2.843ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7cac36162fa61f88374e595a45b2af71ccbb58a2" data-alt="{\displaystyle e_{2}=(0,1)}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span>. Die beiden oben als Beispiel aufgeführten Vektorräume sind zwar <a href="https://de-m-wikipedia-org.translate.goog/wiki/Isomorph?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-redirect" title="Isomorph">isomorph</a> zu <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} ^{2}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck"> R </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \mathbb {R} ^{2}} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e150115ab9f63023215109595b76686a1ff890fd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.732ex; height:2.676ex;" alt="{\displaystyle \mathbb {R} ^{2}}"> </noscript><span class="lazy-image-placeholder" style="width: 2.732ex;height: 2.676ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e150115ab9f63023215109595b76686a1ff890fd" data-alt="{\displaystyle \mathbb {R} ^{2}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span>, besitzen jedoch keine Standardbasis. Infolgedessen ist auch unter den Isomorphismen zwischen diesen Räumen und <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} ^{2}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck"> R </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \mathbb {R} ^{2}} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e150115ab9f63023215109595b76686a1ff890fd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.732ex; height:2.676ex;" alt="{\displaystyle \mathbb {R} ^{2}}"> </noscript><span class="lazy-image-placeholder" style="width: 2.732ex;height: 2.676ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e150115ab9f63023215109595b76686a1ff890fd" data-alt="{\displaystyle \mathbb {R} ^{2}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> keiner ausgezeichnet.</p> <div class="mw-heading mw-heading3"> <h3 id="Bezeichnung">Bezeichnung</h3><span class="mw-editsection"> <a role="button" href="https://de-m-wikipedia-org.translate.goog/w/index.php?title=Standardbasis&action=edit&section=5&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Abschnitt bearbeiten: Bezeichnung" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> <span class="minerva-icon minerva-icon--edit"></span> <span>Bearbeiten</span> </a> </span> </div> <p>Die Bezeichnung <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 e_{1},e_{2},\ldots }"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi> e </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 1 </mn> </mrow> </msub> <mo> , </mo> <msub> <mi> e </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msub> <mo> , </mo> <mo> … </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle e_{1},e_{2},\ldots } </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/de82719660b9c03b6b5ca5301819124e08cc91fd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.067ex; height:2.009ex;" alt="{\displaystyle e_{1},e_{2},\ldots }"> </noscript><span class="lazy-image-placeholder" style="width: 9.067ex;height: 2.009ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/de82719660b9c03b6b5ca5301819124e08cc91fd" data-alt="{\displaystyle e_{1},e_{2},\ldots }" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> für die Standard-Basisvektoren ist weit verbreitet. Die drei Standard-Basisvektoren des dreidimensionalen Vektorraums <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} ^{3}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck"> R </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn> 3 </mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \mathbb {R} ^{3}} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f936ddf584f8f3dd2a0ed08917001b7a404c10b5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.732ex; height:2.676ex;" alt="{\displaystyle \mathbb {R} ^{3}}"> </noscript><span class="lazy-image-placeholder" style="width: 2.732ex;height: 2.676ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f936ddf584f8f3dd2a0ed08917001b7a404c10b5" data-alt="{\displaystyle \mathbb {R} ^{3}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> werden in den angewandten Naturwissenschaften jedoch manchmal mit <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 {i} ,\,\mathbf {j} ,\,\mathbf {k} }"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> i </mi> </mrow> <mo> , </mo> <mspace width="thinmathspace"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> j </mi> </mrow> <mo> , </mo> <mspace width="thinmathspace"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> k </mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \mathbf {i} ,\,\mathbf {j} ,\,\mathbf {k} } </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f9fb37a805bb777a383967fb24db6ca2a80e1e7d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.812ex; height:2.509ex;" alt="{\displaystyle \mathbf {i} ,\,\mathbf {j} ,\,\mathbf {k} }"> </noscript><span class="lazy-image-placeholder" style="width: 5.812ex;height: 2.509ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f9fb37a805bb777a383967fb24db6ca2a80e1e7d" data-alt="{\displaystyle \mathbf {i} ,\,\mathbf {j} ,\,\mathbf {k} }" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> bezeichnet:</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 {i} =e_{1}={\begin{pmatrix}1\\0\\0\end{pmatrix}},\quad \mathbf {j} =e_{2}={\begin{pmatrix}0\\1\\0\end{pmatrix}},\quad \mathbf {k} =e_{3}={\begin{pmatrix}0\\0\\1\end{pmatrix}}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> i </mi> </mrow> <mo> = </mo> <msub> <mi> e </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 1 </mn> </mrow> </msub> <mo> = </mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo> ( </mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn> 1 </mn> </mtd> </mtr> <mtr> <mtd> <mn> 0 </mn> </mtd> </mtr> <mtr> <mtd> <mn> 0 </mn> </mtd> </mtr> </mtable> <mo> ) </mo> </mrow> </mrow> <mo> , </mo> <mspace width="1em"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> j </mi> </mrow> <mo> = </mo> <msub> <mi> e </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msub> <mo> = </mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo> ( </mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn> 0 </mn> </mtd> </mtr> <mtr> <mtd> <mn> 1 </mn> </mtd> </mtr> <mtr> <mtd> <mn> 0 </mn> </mtd> </mtr> </mtable> <mo> ) </mo> </mrow> </mrow> <mo> , </mo> <mspace width="1em"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> k </mi> </mrow> <mo> = </mo> <msub> <mi> e </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 3 </mn> </mrow> </msub> <mo> = </mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo> ( </mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn> 0 </mn> </mtd> </mtr> <mtr> <mtd> <mn> 0 </mn> </mtd> </mtr> <mtr> <mtd> <mn> 1 </mn> </mtd> </mtr> </mtable> <mo> ) </mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \mathbf {i} =e_{1}={\begin{pmatrix}1\\0\\0\end{pmatrix}},\quad \mathbf {j} =e_{2}={\begin{pmatrix}0\\1\\0\end{pmatrix}},\quad \mathbf {k} =e_{3}={\begin{pmatrix}0\\0\\1\end{pmatrix}}} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4f67408224ea51ad6b14b0165b252f00d286782e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.005ex; width:52.629ex; height:9.176ex;" alt="{\displaystyle \mathbf {i} =e_{1}={\begin{pmatrix}1\\0\\0\end{pmatrix}},\quad \mathbf {j} =e_{2}={\begin{pmatrix}0\\1\\0\end{pmatrix}},\quad \mathbf {k} =e_{3}={\begin{pmatrix}0\\0\\1\end{pmatrix}}}"> </noscript><span class="lazy-image-placeholder" style="width: 52.629ex;height: 9.176ex;vertical-align: -4.005ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4f67408224ea51ad6b14b0165b252f00d286782e" data-alt="{\displaystyle \mathbf {i} =e_{1}={\begin{pmatrix}1\\0\\0\end{pmatrix}},\quad \mathbf {j} =e_{2}={\begin{pmatrix}0\\1\\0\end{pmatrix}},\quad \mathbf {k} =e_{3}={\begin{pmatrix}0\\0\\1\end{pmatrix}}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> </dd> </dl> <div class="mw-heading mw-heading3"> <h3 id="Weitere_Eigenschaften">Weitere Eigenschaften</h3><span class="mw-editsection"> <a role="button" href="https://de-m-wikipedia-org.translate.goog/w/index.php?title=Standardbasis&action=edit&section=6&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Abschnitt bearbeiten: Weitere Eigenschaften" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> <span class="minerva-icon minerva-icon--edit"></span> <span>Bearbeiten</span> </a> </span> </div> <p>Der <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} ^{n}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck"> R </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> n </mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \mathbb {R} ^{n}} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c510b63578322050121fe966f2e5770bea43308d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.897ex; height:2.343ex;" alt="{\displaystyle \mathbb {R} ^{n}}"> </noscript><span class="lazy-image-placeholder" style="width: 2.897ex;height: 2.343ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c510b63578322050121fe966f2e5770bea43308d" data-alt="{\displaystyle \mathbb {R} ^{n}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> hat über die Vektorraum-Eigenschaft hinaus noch weitere Eigenschaften. Auch hinsichtlich dieser erfüllen die Standard-Basisvektoren oft besondere Bedingungen. So ist die Standardbasis eine <a href="https://de-m-wikipedia-org.translate.goog/wiki/Orthonormalbasis?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Orthonormalbasis">Orthonormalbasis</a> bezüglich des <a href="https://de-m-wikipedia-org.translate.goog/wiki/Standardskalarprodukt?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Standardskalarprodukt">Standardskalarprodukts</a>.</p> </section> <div class="mw-heading mw-heading2 section-heading" onclick="mfTempOpenSection(3)"> <span class="indicator mf-icon mf-icon-expand mf-icon--small"></span> <h2 id="Standardbasis_im_Matrizenraum">Standardbasis im Matrizenraum</h2><span class="mw-editsection"> <a role="button" href="https://de-m-wikipedia-org.translate.goog/w/index.php?title=Standardbasis&action=edit&section=7&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Abschnitt bearbeiten: Standardbasis im Matrizenraum" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> <span class="minerva-icon minerva-icon--edit"></span> <span>Bearbeiten</span> </a> </span> </div> <section class="mf-section-3 collapsible-block" id="mf-section-3"> <p>Auch die Menge der <a href="https://de-m-wikipedia-org.translate.goog/wiki/Matrix_(Mathematik)?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Matrix (Mathematik)">Matrizen</a> über einem Körper <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 K^{m\times n}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi> K </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> m </mi> <mo> × </mo> <mi> n </mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle K^{m\times n}} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4a40b12a6c68466431891e4ad965fe811743d714" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.034ex; height:2.343ex;" alt="{\displaystyle K^{m\times n}}"> </noscript><span class="lazy-image-placeholder" style="width: 6.034ex;height: 2.343ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4a40b12a6c68466431891e4ad965fe811743d714" data-alt="{\displaystyle K^{m\times n}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> bildet mit der <a href="https://de-m-wikipedia-org.translate.goog/wiki/Matrizenaddition?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Matrizenaddition">Matrizenaddition</a> und der <a href="https://de-m-wikipedia-org.translate.goog/wiki/Skalarmultiplikation?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Skalarmultiplikation">Skalarmultiplikation</a> einen Vektorraum. Die Standardbasis in diesem <a href="https://de-m-wikipedia-org.translate.goog/wiki/Matrizenraum?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Matrizenraum">Matrizenraum</a> wird durch die <a href="https://de-m-wikipedia-org.translate.goog/wiki/Standardmatrix?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Standardmatrix">Standardmatrizen</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 E_{ij}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi> E </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> <mi> j </mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle E_{ij}} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/acbcb625c128efafee881204113cd9e7a8a293a0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.192ex; height:2.843ex;" alt="{\displaystyle E_{ij}}"> </noscript><span class="lazy-image-placeholder" style="width: 3.192ex;height: 2.843ex;vertical-align: -1.005ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/acbcb625c128efafee881204113cd9e7a8a293a0" data-alt="{\displaystyle E_{ij}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> gebildet, bei denen genau ein Eintrag gleich eins und alle anderen Einträge gleich null sind. Beispielsweise bilden die vier Matrizen</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 E_{11}={\begin{pmatrix}1&0\\0&0\end{pmatrix}},E_{12}={\begin{pmatrix}0&1\\0&0\end{pmatrix}},E_{21}={\begin{pmatrix}0&0\\1&0\end{pmatrix}},E_{22}={\begin{pmatrix}0&0\\0&1\end{pmatrix}}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi> E </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 11 </mn> </mrow> </msub> <mo> = </mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo> ( </mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn> 1 </mn> </mtd> <mtd> <mn> 0 </mn> </mtd> </mtr> <mtr> <mtd> <mn> 0 </mn> </mtd> <mtd> <mn> 0 </mn> </mtd> </mtr> </mtable> <mo> ) </mo> </mrow> </mrow> <mo> , </mo> <msub> <mi> E </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 12 </mn> </mrow> </msub> <mo> = </mo> <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> <mo> , </mo> <msub> <mi> E </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 21 </mn> </mrow> </msub> <mo> = </mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo> ( </mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn> 0 </mn> </mtd> <mtd> <mn> 0 </mn> </mtd> </mtr> <mtr> <mtd> <mn> 1 </mn> </mtd> <mtd> <mn> 0 </mn> </mtd> </mtr> </mtable> <mo> ) </mo> </mrow> </mrow> <mo> , </mo> <msub> <mi> E </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 22 </mn> </mrow> </msub> <mo> = </mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo> ( </mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn> 0 </mn> </mtd> <mtd> <mn> 0 </mn> </mtd> </mtr> <mtr> <mtd> <mn> 0 </mn> </mtd> <mtd> <mn> 1 </mn> </mtd> </mtr> </mtable> <mo> ) </mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle E_{11}={\begin{pmatrix}1&0\\0&0\end{pmatrix}},E_{12}={\begin{pmatrix}0&1\\0&0\end{pmatrix}},E_{21}={\begin{pmatrix}0&0\\1&0\end{pmatrix}},E_{22}={\begin{pmatrix}0&0\\0&1\end{pmatrix}}} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8ca2febdf3cbccbde808fda164c7ae01e0e0bf22" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:65.142ex; height:6.176ex;" alt="{\displaystyle E_{11}={\begin{pmatrix}1&0\\0&0\end{pmatrix}},E_{12}={\begin{pmatrix}0&1\\0&0\end{pmatrix}},E_{21}={\begin{pmatrix}0&0\\1&0\end{pmatrix}},E_{22}={\begin{pmatrix}0&0\\0&1\end{pmatrix}}}"> </noscript><span class="lazy-image-placeholder" style="width: 65.142ex;height: 6.176ex;vertical-align: -2.505ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8ca2febdf3cbccbde808fda164c7ae01e0e0bf22" data-alt="{\displaystyle E_{11}={\begin{pmatrix}1&0\\0&0\end{pmatrix}},E_{12}={\begin{pmatrix}0&1\\0&0\end{pmatrix}},E_{21}={\begin{pmatrix}0&0\\1&0\end{pmatrix}},E_{22}={\begin{pmatrix}0&0\\0&1\end{pmatrix}}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> </dd> </dl> <p>die Standardbasis des Raums der <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\times 2)}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false"> ( </mo> <mn> 2 </mn> <mo> × </mo> <mn> 2 </mn> <mo stretchy="false"> ) </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle (2\times 2)} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fd967d734835dc2bf4d3f1b10707f0052a78a650" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.975ex; height:2.843ex;" alt="{\displaystyle (2\times 2)}"> </noscript><span class="lazy-image-placeholder" style="width: 6.975ex;height: 2.843ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fd967d734835dc2bf4d3f1b10707f0052a78a650" data-alt="{\displaystyle (2\times 2)}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span>-Matrizen.</p> </section> <div class="mw-heading mw-heading2 section-heading" onclick="mfTempOpenSection(4)"> <span class="indicator mf-icon mf-icon-expand mf-icon--small"></span> <h2 id="Standardbasis_in_unendlichdimensionalen_Räumen"><span id="Standardbasis_in_unendlichdimensionalen_R.C3.A4umen"></span>Standardbasis in unendlichdimensionalen Räumen</h2><span class="mw-editsection"> <a role="button" href="https://de-m-wikipedia-org.translate.goog/w/index.php?title=Standardbasis&action=edit&section=8&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Abschnitt bearbeiten: Standardbasis in unendlichdimensionalen Räumen" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> <span class="minerva-icon minerva-icon--edit"></span> <span>Bearbeiten</span> </a> </span> </div> <section class="mf-section-4 collapsible-block" id="mf-section-4"> <p>Ist <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 K}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> K </mi> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle K} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2b76fce82a62ed5461908f0dc8f037de4e3686b0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K}"> </noscript><span class="lazy-image-placeholder" style="width: 2.066ex;height: 2.176ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2b76fce82a62ed5461908f0dc8f037de4e3686b0" data-alt="{\displaystyle K}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> ein Körper und <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}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> M </mi> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle M} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f82cade9898ced02fdd08712e5f0c0151758a0dd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}"> </noscript><span class="lazy-image-placeholder" style="width: 2.442ex;height: 2.176ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f82cade9898ced02fdd08712e5f0c0151758a0dd" data-alt="{\displaystyle M}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> eine beliebige (insb. möglicherweise unendliche) Menge, so bilden die endlichen formalen Linearkombinationen von Elementen aus <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}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> M </mi> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle M} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f82cade9898ced02fdd08712e5f0c0151758a0dd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}"> </noscript><span class="lazy-image-placeholder" style="width: 2.442ex;height: 2.176ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f82cade9898ced02fdd08712e5f0c0151758a0dd" data-alt="{\displaystyle M}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> einen Vektorraum. Dann ist <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}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> M </mi> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle M} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f82cade9898ced02fdd08712e5f0c0151758a0dd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}"> </noscript><span class="lazy-image-placeholder" style="width: 2.442ex;height: 2.176ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f82cade9898ced02fdd08712e5f0c0151758a0dd" data-alt="{\displaystyle M}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> selbst Basis dieses Vektorraumes und wird als dessen Standardbasis bezeichnet.</p> <p>Anstelle formaler Linearkombinationen betrachtet man auch alternativ den Vektorraum derjenigen Abbildungen <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\colon M\to K}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> f </mi> <mo> : </mo> <mi> M </mi> <mo stretchy="false"> → </mo> <mi> K </mi> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle f\colon M\to K} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ef155847507feab6778ac3bdc585a1f417a30707" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.435ex; height:2.509ex;" alt="{\displaystyle f\colon M\to K}"> </noscript><span class="lazy-image-placeholder" style="width: 10.435ex;height: 2.509ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ef155847507feab6778ac3bdc585a1f417a30707" data-alt="{\displaystyle f\colon M\to K}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> mit der Eigenschaft, dass <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(x)=0}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> f </mi> <mo stretchy="false"> ( </mo> <mi> x </mi> <mo stretchy="false"> ) </mo> <mo> = </mo> <mn> 0 </mn> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle f(x)=0} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cf85883d74b75fe35ca8d3f2b44802df078e4fa1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.678ex; height:2.843ex;" alt="{\displaystyle f(x)=0}"> </noscript><span class="lazy-image-placeholder" style="width: 8.678ex;height: 2.843ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cf85883d74b75fe35ca8d3f2b44802df078e4fa1" data-alt="{\displaystyle f(x)=0}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> für <a href="https://de-m-wikipedia-org.translate.goog/wiki/Fast_alle?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Fast alle">fast alle</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 x\in M}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> x </mi> <mo> ∈ </mo> <mi> M </mi> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle x\in M} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9df57d73e9532bb93a1439890bcddbc2806f5859" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.613ex; height:2.176ex;" alt="{\displaystyle x\in M}"> </noscript><span class="lazy-image-placeholder" style="width: 6.613ex;height: 2.176ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9df57d73e9532bb93a1439890bcddbc2806f5859" data-alt="{\displaystyle x\in M}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> gilt. Zu <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\in M}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> m </mi> <mo> ∈ </mo> <mi> M </mi> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle m\in M} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0c5790f9afa538086b9fea356114e77099c1a775" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.323ex; height:2.176ex;" alt="{\displaystyle m\in M}"> </noscript><span class="lazy-image-placeholder" style="width: 7.323ex;height: 2.176ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0c5790f9afa538086b9fea356114e77099c1a775" data-alt="{\displaystyle m\in M}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> sei <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 e_{m}\colon M\to K}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi> e </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> m </mi> </mrow> </msub> <mo> : </mo> <mi> M </mi> <mo stretchy="false"> → </mo> <mi> K </mi> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle e_{m}\colon M\to K} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4a4fc545fbccbf8da35793f4bf3d5e60a8a38ec6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.915ex; height:2.509ex;" alt="{\displaystyle e_{m}\colon M\to K}"> </noscript><span class="lazy-image-placeholder" style="width: 11.915ex;height: 2.509ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4a4fc545fbccbf8da35793f4bf3d5e60a8a38ec6" data-alt="{\displaystyle e_{m}\colon M\to K}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> die durch</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 e_{m}(x)={\begin{cases}1,&{\text{falls }}x=m\\0,&{\text{falls }}x\neq m\end{cases}}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi> e </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> m </mi> </mrow> </msub> <mo stretchy="false"> ( </mo> <mi> x </mi> <mo stretchy="false"> ) </mo> <mo> = </mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo> { </mo> <mtable columnalign="left left" rowspacing=".2em" columnspacing="1em" displaystyle="false"> <mtr> <mtd> <mn> 1 </mn> <mo> , </mo> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext> falls  </mtext> </mrow> <mi> x </mi> <mo> = </mo> <mi> m </mi> </mtd> </mtr> <mtr> <mtd> <mn> 0 </mn> <mo> , </mo> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext> falls  </mtext> </mrow> <mi> x </mi> <mo> ≠ </mo> <mi> m </mi> </mtd> </mtr> </mtable> <mo fence="true" stretchy="true" symmetric="true"></mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle e_{m}(x)={\begin{cases}1,&{\text{falls }}x=m\\0,&{\text{falls }}x\neq m\end{cases}}} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c7a60a6b9bc30d768656175af126a92cc89398d2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:26.756ex; height:6.176ex;" alt="{\displaystyle e_{m}(x)={\begin{cases}1,&{\text{falls }}x=m\\0,&{\text{falls }}x\neq m\end{cases}}}"> </noscript><span class="lazy-image-placeholder" style="width: 26.756ex;height: 6.176ex;vertical-align: -2.505ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c7a60a6b9bc30d768656175af126a92cc89398d2" data-alt="{\displaystyle e_{m}(x)={\begin{cases}1,&{\text{falls }}x=m\\0,&{\text{falls }}x\neq m\end{cases}}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> </dd> </dl> <p>gegebene Abbildung <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\to K}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> M </mi> <mo stretchy="false"> → </mo> <mi> K </mi> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle M\to K} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a6051ee1555b21c9e43c0031239a5873a29354df" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.122ex; height:2.176ex;" alt="{\displaystyle M\to K}"> </noscript><span class="lazy-image-placeholder" style="width: 8.122ex;height: 2.176ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a6051ee1555b21c9e43c0031239a5873a29354df" data-alt="{\displaystyle M\to K}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span>. Dann bildet die Familie <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 \{e_{m}\}_{m\in M}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false"> { </mo> <msub> <mi> e </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> m </mi> </mrow> </msub> <msub> <mo fence="false" stretchy="false"> } </mo> <mrow class="MJX-TeXAtom-ORD"> <mi> m </mi> <mo> ∈ </mo> <mi> M </mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \{e_{m}\}_{m\in M}} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e875b0657c4c24d648437be54605111d15aee4c6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.582ex; height:2.843ex;" alt="{\displaystyle \{e_{m}\}_{m\in M}}"> </noscript><span class="lazy-image-placeholder" style="width: 9.582ex;height: 2.843ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e875b0657c4c24d648437be54605111d15aee4c6" data-alt="{\displaystyle \{e_{m}\}_{m\in M}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> eine Basis des Vektorraums, die in diesem Fall ebenfalls als die Standardbasis bezeichnet wird.</p> <p>Der Vektorraum <i>aller</i> Abbildungen <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\colon M\to K}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> f </mi> <mo> : </mo> <mi> M </mi> <mo stretchy="false"> → </mo> <mi> K </mi> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle f\colon M\to K} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ef155847507feab6778ac3bdc585a1f417a30707" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.435ex; height:2.509ex;" alt="{\displaystyle f\colon M\to K}"> </noscript><span class="lazy-image-placeholder" style="width: 10.435ex;height: 2.509ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ef155847507feab6778ac3bdc585a1f417a30707" data-alt="{\displaystyle f\colon M\to K}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> besitzt hingegen, sofern <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}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> M </mi> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle M} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f82cade9898ced02fdd08712e5f0c0151758a0dd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}"> </noscript><span class="lazy-image-placeholder" style="width: 2.442ex;height: 2.176ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f82cade9898ced02fdd08712e5f0c0151758a0dd" data-alt="{\displaystyle M}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> unendlich ist, keine Standardbasis.</p> <p>Auch <a href="https://de-m-wikipedia-org.translate.goog/wiki/Polynomring?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Polynomring">Polynomringe</a> über Körpern sind Vektorräume, in denen eine Basis bereits unmittelbar aufgrund der Konstruktion ausgezeichnet ist. So sind die Elemente des Polynomringes <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} [X]}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck"> R </mi> </mrow> <mo stretchy="false"> [ </mo> <mi> X </mi> <mo stretchy="false"> ] </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \mathbb {R} [X]} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/16d740527b0b7f949b4bf9c9ce004134bb490b68" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.952ex; height:2.843ex;" alt="{\displaystyle \mathbb {R} [X]}"> </noscript><span class="lazy-image-placeholder" style="width: 4.952ex;height: 2.843ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/16d740527b0b7f949b4bf9c9ce004134bb490b68" data-alt="{\displaystyle \mathbb {R} [X]}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> definitionsgemäß die endlichen Linearkombinationen der Monome <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"> <mn> 1 </mn> <mo> , </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle 1,} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9cc5fd8163a83100c5330622e9e317fa4e872403" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.809ex; height:2.509ex;" alt="{\displaystyle 1,}"> </noscript><span class="lazy-image-placeholder" style="width: 1.809ex;height: 2.509ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9cc5fd8163a83100c5330622e9e317fa4e872403" data-alt="{\displaystyle 1,}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> <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,}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> X </mi> <mo> , </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle X,} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/09ba32eeb405f7f5f2bac1eb12987c47d2fd42df" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.627ex; height:2.509ex;" alt="{\displaystyle X,}"> </noscript><span class="lazy-image-placeholder" style="width: 2.627ex;height: 2.509ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/09ba32eeb405f7f5f2bac1eb12987c47d2fd42df" data-alt="{\displaystyle X,}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> <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^{2},}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi> X </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> <mo> , </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle X^{2},} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b43d2913fd82a09205a94045bd269603b2a727b6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.698ex; height:3.009ex;" alt="{\displaystyle X^{2},}"> </noscript><span class="lazy-image-placeholder" style="width: 3.698ex;height: 3.009ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b43d2913fd82a09205a94045bd269603b2a727b6" data-alt="{\displaystyle X^{2},}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> <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^{3}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi> X </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 3 </mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle X^{3}} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7a2439369260c9d7f7bd4fc81d3e274a00fdb7de" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.051ex; height:2.676ex;" alt="{\displaystyle X^{3}}"> </noscript><span class="lazy-image-placeholder" style="width: 3.051ex;height: 2.676ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7a2439369260c9d7f7bd4fc81d3e274a00fdb7de" data-alt="{\displaystyle X^{3}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> usw., die demnach eine Basis – die Standardbasis – von <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} [X]}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck"> R </mi> </mrow> <mo stretchy="false"> [ </mo> <mi> X </mi> <mo stretchy="false"> ] </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \mathbb {R} [X]} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/16d740527b0b7f949b4bf9c9ce004134bb490b68" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.952ex; height:2.843ex;" alt="{\displaystyle \mathbb {R} [X]}"> </noscript><span class="lazy-image-placeholder" style="width: 4.952ex;height: 2.843ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/16d740527b0b7f949b4bf9c9ce004134bb490b68" data-alt="{\displaystyle \mathbb {R} [X]}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> bilden.</p> </section> <div class="mw-heading mw-heading2 section-heading" onclick="mfTempOpenSection(5)"> <span class="indicator mf-icon mf-icon-expand mf-icon--small"></span> <h2 id="Zusammenhang_mit_universellen_Eigenschaften">Zusammenhang mit universellen Eigenschaften</h2><span class="mw-editsection"> <a role="button" href="https://de-m-wikipedia-org.translate.goog/w/index.php?title=Standardbasis&action=edit&section=9&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Abschnitt bearbeiten: Zusammenhang mit universellen Eigenschaften" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> <span class="minerva-icon minerva-icon--edit"></span> <span>Bearbeiten</span> </a> </span> </div> <section class="mf-section-5 collapsible-block" id="mf-section-5"> <p>Der Begriff <i>kanonisch</i> wird allgemein bei Konstruktionen über eine <a href="https://de-m-wikipedia-org.translate.goog/wiki/Universelle_Eigenschaft?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Universelle Eigenschaft">universelle Eigenschaft</a> verwendet. So ergibt sich auch ein Zusammenhang zwischen Standardbasen und folgender Konstruktion:</p> <p>Sei <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 K}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> K </mi> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle K} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2b76fce82a62ed5461908f0dc8f037de4e3686b0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K}"> </noscript><span class="lazy-image-placeholder" style="width: 2.066ex;height: 2.176ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2b76fce82a62ed5461908f0dc8f037de4e3686b0" data-alt="{\displaystyle K}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> ein Körper und <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}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> M </mi> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle M} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f82cade9898ced02fdd08712e5f0c0151758a0dd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}"> </noscript><span class="lazy-image-placeholder" style="width: 2.442ex;height: 2.176ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f82cade9898ced02fdd08712e5f0c0151758a0dd" data-alt="{\displaystyle M}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> eine beliebige Menge. Gesucht ist ein <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 K}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> K </mi> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle K} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2b76fce82a62ed5461908f0dc8f037de4e3686b0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K}"> </noscript><span class="lazy-image-placeholder" style="width: 2.066ex;height: 2.176ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2b76fce82a62ed5461908f0dc8f037de4e3686b0" data-alt="{\displaystyle K}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span>-Vektorraum <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 U}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> U </mi> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle U} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/458a728f53b9a0274f059cd695e067c430956025" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.783ex; height:2.176ex;" alt="{\displaystyle U}"> </noscript><span class="lazy-image-placeholder" style="width: 1.783ex;height: 2.176ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/458a728f53b9a0274f059cd695e067c430956025" data-alt="{\displaystyle U}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> zusammen mit einer Abbildung <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\colon M\to U}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> f </mi> <mo> : </mo> <mi> M </mi> <mo stretchy="false"> → </mo> <mi> U </mi> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle f\colon M\to U} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fe8bbb2f035e1b578a867e807c2f207e41a4b191" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.151ex; height:2.509ex;" alt="{\displaystyle f\colon M\to U}"> </noscript><span class="lazy-image-placeholder" style="width: 10.151ex;height: 2.509ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fe8bbb2f035e1b578a867e807c2f207e41a4b191" data-alt="{\displaystyle f\colon M\to U}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> in dessen zugrunde liegende Menge, so dass zu jedem <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 K}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> K </mi> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle K} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2b76fce82a62ed5461908f0dc8f037de4e3686b0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K}"> </noscript><span class="lazy-image-placeholder" style="width: 2.066ex;height: 2.176ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2b76fce82a62ed5461908f0dc8f037de4e3686b0" data-alt="{\displaystyle K}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span>-Vektorraum <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}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> X </mi> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle X} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}"> </noscript><span class="lazy-image-placeholder" style="width: 1.98ex;height: 2.176ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" data-alt="{\displaystyle X}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> und jeder Abbildung <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 g\colon M\to X}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> g </mi> <mo> : </mo> <mi> M </mi> <mo stretchy="false"> → </mo> <mi> X </mi> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle g\colon M\to X} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a8807ca68ffb8ad0d198f685ed8f878c76d1ae8c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.186ex; height:2.509ex;" alt="{\displaystyle g\colon M\to X}"> </noscript><span class="lazy-image-placeholder" style="width: 10.186ex;height: 2.509ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a8807ca68ffb8ad0d198f685ed8f878c76d1ae8c" data-alt="{\displaystyle g\colon M\to X}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> genau eine <a href="https://de-m-wikipedia-org.translate.goog/wiki/Lineare_Abbildung?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Lineare Abbildung">lineare Abbildung</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 h\colon U\to X}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> h </mi> <mo> : </mo> <mi> U </mi> <mo stretchy="false"> → </mo> <mi> X </mi> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle h\colon U\to X} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/44dea317a0012c58bd4842d2ffc9a04cb372b41d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:9.75ex; height:2.176ex;" alt="{\displaystyle h\colon U\to X}"> </noscript><span class="lazy-image-placeholder" style="width: 9.75ex;height: 2.176ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/44dea317a0012c58bd4842d2ffc9a04cb372b41d" data-alt="{\displaystyle h\colon U\to X}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> existiert mit <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 g=h\circ f}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> g </mi> <mo> = </mo> <mi> h </mi> <mo> ∘ </mo> <mi> f </mi> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle g=h\circ f} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6220c8d95ac75c438e6f3bf335569f45362d20f5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.027ex; height:2.509ex;" alt="{\displaystyle g=h\circ f}"> </noscript><span class="lazy-image-placeholder" style="width: 9.027ex;height: 2.509ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6220c8d95ac75c438e6f3bf335569f45362d20f5" data-alt="{\displaystyle g=h\circ f}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span>. In solch einem Paar <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 (U,f)}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false"> ( </mo> <mi> U </mi> <mo> , </mo> <mi> f </mi> <mo stretchy="false"> ) </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle (U,f)} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ec4dbdf43357db955dc953fe357a7fc4a8566d96" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.904ex; height:2.843ex;" alt="{\displaystyle (U,f)}"> </noscript><span class="lazy-image-placeholder" style="width: 5.904ex;height: 2.843ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ec4dbdf43357db955dc953fe357a7fc4a8566d96" data-alt="{\displaystyle (U,f)}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> wird dann <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}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> f </mi> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle f} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}"> </noscript><span class="lazy-image-placeholder" style="width: 1.279ex;height: 2.509ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" data-alt="{\displaystyle f}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> als <i>kanonische Abbildung</i> oder <i><a href="https://de-m-wikipedia-org.translate.goog/wiki/Universelle_L%C3%B6sung?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-redirect" title="Universelle Lösung">universelle Lösung</a></i> von <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}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> M </mi> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle M} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f82cade9898ced02fdd08712e5f0c0151758a0dd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}"> </noscript><span class="lazy-image-placeholder" style="width: 2.442ex;height: 2.176ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f82cade9898ced02fdd08712e5f0c0151758a0dd" data-alt="{\displaystyle M}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> bezüglich des <a href="https://de-m-wikipedia-org.translate.goog/wiki/Vergissfunktor?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-redirect" title="Vergissfunktor">Vergissfunktors</a>, der jedem <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 K}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> K </mi> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle K} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2b76fce82a62ed5461908f0dc8f037de4e3686b0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K}"> </noscript><span class="lazy-image-placeholder" style="width: 2.066ex;height: 2.176ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2b76fce82a62ed5461908f0dc8f037de4e3686b0" data-alt="{\displaystyle K}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span>-Vektorraum die zugrundeliegende Menge zuordnet, bezeichnet.</p> <p>Die oben angegebenen Vektorräume mit Standardbasis haben genau diese universelle Eigenschaft. Das Bild von <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}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> M </mi> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle M} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f82cade9898ced02fdd08712e5f0c0151758a0dd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}"> </noscript><span class="lazy-image-placeholder" style="width: 2.442ex;height: 2.176ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f82cade9898ced02fdd08712e5f0c0151758a0dd" data-alt="{\displaystyle M}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> unter der kanonischen Abbildung sind genau die Vektoren der kanonischen Basis bzw. die kanonische Abbildung als Familie aufgefasst <i>ist</i> die kanonische Basis.</p> <p>Daraus, dass stets eine solche universelle Lösung existiert, folgt bereits, dass eine Abbildung, die jeder Menge <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}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> M </mi> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle M} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f82cade9898ced02fdd08712e5f0c0151758a0dd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}"> </noscript><span class="lazy-image-placeholder" style="width: 2.442ex;height: 2.176ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f82cade9898ced02fdd08712e5f0c0151758a0dd" data-alt="{\displaystyle M}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> eine solche universelle Lösung <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 U}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> U </mi> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle U} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/458a728f53b9a0274f059cd695e067c430956025" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.783ex; height:2.176ex;" alt="{\displaystyle U}"> </noscript><span class="lazy-image-placeholder" style="width: 1.783ex;height: 2.176ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/458a728f53b9a0274f059cd695e067c430956025" data-alt="{\displaystyle U}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> und jedem <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 g}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> g </mi> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle g} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d3556280e66fe2c0d0140df20935a6f057381d77" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.116ex; height:2.009ex;" alt="{\displaystyle g}"> </noscript><span class="lazy-image-placeholder" style="width: 1.116ex;height: 2.009ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d3556280e66fe2c0d0140df20935a6f057381d77" data-alt="{\displaystyle g}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> ein solches <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 h}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> h </mi> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle h} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b26be3e694314bc90c3215047e4a2010c6ee184a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.339ex; height:2.176ex;" alt="{\displaystyle h}"> </noscript><span class="lazy-image-placeholder" style="width: 1.339ex;height: 2.176ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b26be3e694314bc90c3215047e4a2010c6ee184a" data-alt="{\displaystyle h}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> zuordnet, ein <a href="https://de-m-wikipedia-org.translate.goog/wiki/Funktor_(Mathematik)?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Funktor (Mathematik)">Funktor</a> ist, der <a href="https://de-m-wikipedia-org.translate.goog/wiki/Linksadjungiert?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-redirect" title="Linksadjungiert">linksadjungiert</a> zum Vergissfunktor ist. Ein solcher Funktor heißt <i><a href="https://de-m-wikipedia-org.translate.goog/w/index.php?title=Freier_Funktor&action=edit&redlink=1&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="new" title="Freier Funktor (Seite nicht vorhanden)">freier Funktor</a></i>.</p> </section> <div class="mw-heading mw-heading2 section-heading" onclick="mfTempOpenSection(6)"> <span class="indicator mf-icon mf-icon-expand mf-icon--small"></span> <h2 id="Literatur">Literatur</h2><span class="mw-editsection"> <a role="button" href="https://de-m-wikipedia-org.translate.goog/w/index.php?title=Standardbasis&action=edit&section=10&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Abschnitt bearbeiten: Literatur" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> <span class="minerva-icon minerva-icon--edit"></span> <span>Bearbeiten</span> </a> </span> </div> <section class="mf-section-6 collapsible-block" id="mf-section-6"> <ul> <li>Kowalsky und Michler: <i>Lineare Algebra</i>, Gruyter, <a href="https://de-m-wikipedia-org.translate.goog/wiki/Spezial:ISBN-Suche/9783110179637?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="internal mw-magiclink-isbn">ISBN 978-3-11-017963-7</a></li> <li>Albrecht Beutelspacher: <i>„Das ist o.B.d.A. trivial!“</i> 9. aktualisierte Auflage, Vieweg + Teubner, Braunschweig und Wiesbaden 2009, <a href="https://de-m-wikipedia-org.translate.goog/wiki/Spezial:ISBN-Suche/9783834807717?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="internal mw-magiclink-isbn">ISBN 978-3-8348-0771-7</a>, s. v. „Kanonisch“</li> </ul> </section> </div> <noscript> <img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?type=1x1&useformat=mobile" alt="" width="1" height="1" style="border: none; position: absolute;"> </noscript> <div class="printfooter" data-nosnippet=""> Abgerufen von „<a dir="ltr" href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://de.wikipedia.org/w/index.php?title%3DStandardbasis%26oldid%3D244365722">https://de.wikipedia.org/w/index.php?title=Standardbasis&oldid=244365722</a>“ </div> </div> </div> <div class="post-content" id="page-secondary-actions"> </div> </main> <footer class="mw-footer minerva-footer" role="contentinfo"><a class="last-modified-bar" href="https://de-m-wikipedia-org.translate.goog/w/index.php?title=Standardbasis&action=history&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB"> <div class="post-content last-modified-bar__content"><span class="minerva-icon minerva-icon-size-medium minerva-icon--modified-history"></span> <span class="last-modified-bar__text modified-enhancement" data-user-name="Aka" data-user-gender="male" data-timestamp="1713981422"> <span>Zuletzt bearbeitet am 24. 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