Basiswechsel (Vektorraum)

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Man bezeichnet damit den Übergang zwischen zwei verschiedenen <a href="/wiki/Basis_(Vektorraum)" title="Basis (Vektorraum)">Basen</a> eines endlichdimensionalen <a href="/wiki/Vektorraum" title="Vektorraum">Vektorraums</a> über einem <a href="/wiki/K%C3%B6rper_(Algebra)" title="Körper (Algebra)">Körper</a> <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><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}">. Dadurch ändern sich im Allgemeinen die <a href="/wiki/Koordinate" class="mw-redirect" title="Koordinate">Koordinaten</a> der <a href="/wiki/Vektor" title="Vektor">Vektoren</a> und die <a href="/wiki/Abbildungsmatrix" title="Abbildungsmatrix">Abbildungsmatrizen</a> von <a href="/wiki/Lineare_Abbildung" title="Lineare Abbildung">linearen Abbildungen</a>. Ein Basiswechsel ist somit ein Spezialfall einer <a href="/wiki/Koordinatentransformation" title="Koordinatentransformation">Koordinatentransformation</a>. Der Basiswechsel kann durch eine <a href="/wiki/Matrix_(Mathematik)" title="Matrix (Mathematik)">Matrix</a> beschrieben werden, die Basiswechselmatrix, Transformationsmatrix oder Übergangsmatrix genannt wird. Mit dieser lassen sich auch die Koordinaten bezüglich der neuen Basis ausrechnen. Stellt man die Basisvektoren der alten Basis als <a href="/wiki/Linearkombination" title="Linearkombination">Linearkombinationen</a> der Vektoren der neuen Basis dar, so bilden die Koeffizienten dieser Linearkombinationen die Einträge der Basiswechselmatrix. <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><label class="toctogglelabel" for="toctogglecheckbox"></label></div> <ul> <li class="toclevel-1 tocsection-1"><a href="#Basiswechselmatrix">1 Basiswechselmatrix</a></li> <li class="toclevel-1 tocsection-2"><a href="#Spezialfälle">2 Spezialfälle</a></li> <li class="toclevel-1 tocsection-3"><a href="#Koordinatentransformation">3 Koordinatentransformation</a></li> <li class="toclevel-1 tocsection-4"><a href="#Basiswechsel_bei_Abbildungsmatrizen">4 Basiswechsel bei Abbildungsmatrizen</a></li> <li class="toclevel-1 tocsection-5"><a href="#Beispiel">5 Beispiel</a></li> <li class="toclevel-1 tocsection-6"><a href="#Basiswechsel_mit_Hilfe_der_dualen_Basis">6 Basiswechsel mit Hilfe der dualen Basis</a> <ul> <li class="toclevel-2 tocsection-7"><a href="#Wechsel_zur_dualen_Basis">6.1 Wechsel zur dualen Basis</a></li> <li class="toclevel-2 tocsection-8"><a href="#Wechsel_zu_einer_anderen_Basis">6.2 Wechsel zu einer anderen Basis</a></li> </ul> </li> <li class="toclevel-1 tocsection-9"><a href="#Anwendungen">7 Anwendungen</a> <ul> <li class="toclevel-2 tocsection-10"><a href="#In_der_Mathematik">7.1 In der Mathematik</a></li> <li class="toclevel-2 tocsection-11"><a href="#In_der_Physik">7.2 In der Physik</a></li> </ul> </li> <li class="toclevel-1 tocsection-12"><a href="#Literatur">8 Literatur</a></li> <li class="toclevel-1 tocsection-13"><a href="#Weblinks">9 Weblinks</a></li> </ul> </div> <div class="mw-heading mw-heading2"><h2 id="Basiswechselmatrix">Basiswechselmatrix</h2>[<a href="/w/index.php?title=Basiswechsel_(Vektorraum)&veaction=edit&section=1" title="Abschnitt bearbeiten: Basiswechselmatrix" class="mw-editsection-visualeditor">Bearbeiten</a> | <a href="/w/index.php?title=Basiswechsel_(Vektorraum)&action=edit&section=1" title="Quellcode des Abschnitts bearbeiten: Basiswechselmatrix">Quelltext bearbeiten</a>]</div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/Datei:Change_of_basis_square2.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/1/19/Change_of_basis_square2.svg/220px-Change_of_basis_square2.svg.png" decoding="async" width="220" height="200" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/19/Change_of_basis_square2.svg/330px-Change_of_basis_square2.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/1/19/Change_of_basis_square2.svg/440px-Change_of_basis_square2.svg.png 2x" data-file-width="195" data-file-height="177" /></a><figcaption>Kommutatives Diagramm</figcaption></figure> Es sei <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><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}"> ein <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><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}">-dimensionaler Vektorraum über dem Körper <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><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}"> (zum Beispiel dem Körper <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/786849c765da7a84dbc3cce43e96aad58a5868dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {R} }"> der reellen Zahlen). In <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><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}"> seien zwei geordnete Basen gegeben, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B=(b_{1},\ldots ,b_{n})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> <mo>=</mo> <mo stretchy="false">(</mo> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B=(b_{1},\ldots ,b_{n})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7f2868ee9456e6538fcb2888954590ce3c5abda3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.118ex; height:2.843ex;" alt="{\displaystyle B=(b_{1},\ldots ,b_{n})}"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B'=(b_{1}',\ldots ,b_{n}')}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>B</mi> <mo>′</mo> </msup> <mo>=</mo> <mo stretchy="false">(</mo> <msubsup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mo>′</mo> </msubsup> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msubsup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> <mo>′</mo> </msubsup> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B'=(b_{1}',\ldots ,b_{n}')}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0e353326c4524f76491cb579cd0c0b1d950c053e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:16.803ex; height:3.176ex;" alt="{\displaystyle B'=(b_{1}',\ldots ,b_{n}')}">. Die Basiswechselmatrix <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T_{B'}^{B}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi>B</mi> <mo>′</mo> </msup> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T_{B'}^{B}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eb57134574715111f0c1c3dd85fd8d27a116d132" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:3.369ex; height:3.343ex;" alt="{\displaystyle T_{B'}^{B}}"> für den Basiswechsel von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47136aad860d145f75f3eed3022df827cee94d7a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle B}"> nach <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B'}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>B</mi> <mo>′</mo> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B'}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8b144e466a3757b4706daf24a45d4f1266700016" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.449ex; height:2.509ex;" alt="{\displaystyle B'}"> ist eine <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n\times n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>×</mo> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n\times n}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/59d2b4cb72e304526cf5b5887147729ea259da78" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.63ex; height:1.676ex;" alt="{\displaystyle n\times n}">-Matrix. Es handelt sich um die <a href="/wiki/Abbildungsmatrix" title="Abbildungsmatrix">Abbildungsmatrix</a> der <a href="/wiki/Identische_Abbildung" title="Identische Abbildung">Identitätsabbildung</a> auf <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><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}"> bezüglich der Basen <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47136aad860d145f75f3eed3022df827cee94d7a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle B}"> im Urbild und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B'}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>B</mi> <mo>′</mo> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B'}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8b144e466a3757b4706daf24a45d4f1266700016" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.449ex; height:2.509ex;" alt="{\displaystyle B'}"> im Bild: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T_{B'}^{B}=M_{B'}^{B}(\operatorname {id_{V}} )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi>B</mi> <mo>′</mo> </msup> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> </mrow> </msubsup> <mo>=</mo> <msubsup> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi>B</mi> <mo>′</mo> </msup> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> </mrow> </msubsup> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <mi mathvariant="normal">i</mi> <msub> <mi mathvariant="normal">d</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">V</mi> </mrow> </msub> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T_{B'}^{B}=M_{B'}^{B}(\operatorname {id_{V}} )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aacacd8609e75e7728a5a325ca602d8a2de2a36c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:15.946ex; height:3.343ex;" alt="{\displaystyle T_{B'}^{B}=M_{B'}^{B}(\operatorname {id_{V}} )}"></dd></dl> Man erhält sie, indem man die Vektoren der alten Basis <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47136aad860d145f75f3eed3022df827cee94d7a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle B}"> als Linearkombinationen der Vektoren der neuen Basis <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B{}'}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> <msup> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mo>′</mo> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B{}'}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f0aae9e8342f879f2814f464e508ecba5f182e68" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.449ex; height:2.509ex;" alt="{\displaystyle B{}'}"> darstellt: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b_{j}=a_{1j}b_{1}{}'+a_{2j}b_{2}{}'+\dots +a_{nj}b_{n}{}'=\sum _{i=1}^{n}a_{ij}b_{i}{}',\qquad j=1,\dots ,n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mi>j</mi> </mrow> </msub> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msup> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mo>′</mo> </msup> <mo>+</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>j</mi> </mrow> </msub> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msup> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mo>′</mo> </msup> <mo>+</mo> <mo>⋯</mo> <mo>+</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mi>j</mi> </mrow> </msub> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <msup> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mo>′</mo> </msup> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </msub> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msup> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mo>′</mo> </msup> <mo>,</mo> <mspace width="2em" /> <mi>j</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mo>…</mo> <mo>,</mo> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b_{j}=a_{1j}b_{1}{}'+a_{2j}b_{2}{}'+\dots +a_{nj}b_{n}{}'=\sum _{i=1}^{n}a_{ij}b_{i}{}',\qquad j=1,\dots ,n}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0cac8cef7dc54bf7956858a8efc4711f3598c498" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:63.173ex; height:6.843ex;" alt="{\displaystyle b_{j}=a_{1j}b_{1}{}'+a_{2j}b_{2}{}'+\dots +a_{nj}b_{n}{}'=\sum _{i=1}^{n}a_{ij}b_{i}{}',\qquad j=1,\dots ,n}"></dd></dl> Die Koeffizienten <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{1j},\dots ,a_{nj}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mi>j</mi> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mi>j</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{1j},\dots ,a_{nj}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0019cc95fce4ca79a5c408cafded3eec6440193d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:11.265ex; height:2.343ex;" alt="{\displaystyle a_{1j},\dots ,a_{nj}}"> bilden die <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle j}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>j</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle j}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2f461e54f5c093e92a55547b9764291390f0b5d0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.027ex; width:0.985ex; height:2.509ex;" alt="{\displaystyle j}">-te Spalte der Basiswechselmatrix <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T_{B'}^{B}={\begin{pmatrix}a_{11}&\cdots &a_{1j}&\cdots &a_{1n}\\\vdots &&\vdots &&\vdots \\a_{n1}&\cdots &a_{nj}&\cdots &a_{nn}\end{pmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi>B</mi> <mo>′</mo> </msup> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> </mrow> </msubsup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>11</mn> </mrow> </msub> </mtd> <mtd> <mo>⋯</mo> </mtd> <mtd> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mi>j</mi> </mrow> </msub> </mtd> <mtd> <mo>⋯</mo> </mtd> <mtd> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mi>n</mi> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <mo>⋮</mo> </mtd> <mtd /> <mtd> <mo>⋮</mo> </mtd> <mtd /> <mtd> <mo>⋮</mo> </mtd> </mtr> <mtr> <mtd> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mn>1</mn> </mrow> </msub> </mtd> <mtd> <mo>⋯</mo> </mtd> <mtd> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mi>j</mi> </mrow> </msub> </mtd> <mtd> <mo>⋯</mo> </mtd> <mtd> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mi>n</mi> </mrow> </msub> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T_{B'}^{B}={\begin{pmatrix}a_{11}&\cdots &a_{1j}&\cdots &a_{1n}\\\vdots &&\vdots &&\vdots \\a_{n1}&\cdots &a_{nj}&\cdots &a_{nn}\end{pmatrix}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c7dddfbb3f8a1aeab0e0eba68323cf6ba585b62c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.838ex; width:35.853ex; height:10.843ex;" alt="{\displaystyle T_{B'}^{B}={\begin{pmatrix}a_{11}&\cdots &a_{1j}&\cdots &a_{1n}\\\vdots &&\vdots &&\vdots \\a_{n1}&\cdots &a_{nj}&\cdots &a_{nn}\end{pmatrix}}}"></dd></dl> Diese Matrix ist quadratisch und <a href="/wiki/Regul%C3%A4re_Matrix" title="Reguläre Matrix">invertierbar</a> und somit ein Element der <a href="/wiki/Allgemeine_lineare_Gruppe" title="Allgemeine lineare Gruppe">allgemeinen linearen Gruppe</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {GL} \left(n,K\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">G</mi> <mi mathvariant="normal">L</mi> </mrow> <mrow> <mo>(</mo> <mrow> <mi>n</mi> <mo>,</mo> <mi>K</mi> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {GL} \left(n,K\right)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d231e7b1d10d7325eaa0cb0306cb1c7436a7b39d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.968ex; height:2.843ex;" alt="{\displaystyle \mathrm {GL} \left(n,K\right)}">. Ihre <a href="/wiki/Inverse_Matrix" title="Inverse Matrix">Inverse</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left(T_{B'}^{B}\right)^{-1}=T_{B}^{B'}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow> <mo>(</mo> <msubsup> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi>B</mi> <mo>′</mo> </msup> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> </mrow> </msubsup> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>=</mo> <msubsup> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi>B</mi> <mo>′</mo> </msup> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left(T_{B'}^{B}\right)^{-1}=T_{B}^{B'}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7e58e0fe36a2c7b41e90a14fb0d21a62ab63ee5c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:14.661ex; height:3.843ex;" alt="{\displaystyle \left(T_{B'}^{B}\right)^{-1}=T_{B}^{B'}}"> beschreibt den Basiswechsel von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B'}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>B</mi> <mo>′</mo> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B'}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8b144e466a3757b4706daf24a45d4f1266700016" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.449ex; height:2.509ex;" alt="{\displaystyle B'}"> zurück nach <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47136aad860d145f75f3eed3022df827cee94d7a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle B}">. <div class="mw-heading mw-heading2"><h2 id="Spezialfälle">Spezialfälle</h2>[<a href="/w/index.php?title=Basiswechsel_(Vektorraum)&veaction=edit&section=2" title="Abschnitt bearbeiten: Spezialfälle" class="mw-editsection-visualeditor">Bearbeiten</a> | <a href="/w/index.php?title=Basiswechsel_(Vektorraum)&action=edit&section=2" title="Quellcode des Abschnitts bearbeiten: Spezialfälle">Quelltext bearbeiten</a>]</div> Ein wichtiger Spezialfall ist der Fall <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V=K^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> <mo>=</mo> <msup> <mi>K</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V=K^{n}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3ef1bb33cc16379754d9606e92a9d5d60a094126" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.198ex; height:2.343ex;" alt="{\displaystyle V=K^{n}}">, der Vektorraum stimmt also mit dem <a href="/wiki/Koordinatenraum" title="Koordinatenraum">Koordinatenraum</a> überein. In diesem Fall sind die Basisvektoren Spaltenvektoren <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b_{1}={\begin{pmatrix}b_{11}\\\vdots \\b_{n1}\end{pmatrix}},\dots ,b_{j}={\begin{pmatrix}b_{1j}\\\vdots \\b_{nj}\end{pmatrix}},\dots ,b_{n}={\begin{pmatrix}b_{1n}\\\vdots \\b_{nn}\end{pmatrix}},\quad b_{1}'={\begin{pmatrix}b_{11}'\\\vdots \\b_{n1}'\end{pmatrix}},\dots ,b_{j}'={\begin{pmatrix}b_{1j}'\\\vdots \\b_{nj}'\end{pmatrix}},\dots ,b_{n}'={\begin{pmatrix}b_{1n}'\\\vdots \\b_{nn}'\end{pmatrix}},\quad }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>b</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> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>11</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <mo>⋮</mo> </mtd> </mtr> <mtr> <mtd> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mn>1</mn> </mrow> </msub> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mi>j</mi> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <mo>⋮</mo> </mtd> </mtr> <mtr> <mtd> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mi>j</mi> </mrow> </msub> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mi>n</mi> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <mo>⋮</mo> </mtd> </mtr> <mtr> <mtd> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mi>n</mi> </mrow> </msub> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mo>,</mo> <mspace width="1em" /> <msubsup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mo>′</mo> </msubsup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msubsup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>11</mn> </mrow> <mo>′</mo> </msubsup> </mtd> </mtr> <mtr> <mtd> <mo>⋮</mo> </mtd> </mtr> <mtr> <mtd> <msubsup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mn>1</mn> </mrow> <mo>′</mo> </msubsup> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msubsup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> <mo>′</mo> </msubsup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msubsup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mi>j</mi> </mrow> <mo>′</mo> </msubsup> </mtd> </mtr> <mtr> <mtd> <mo>⋮</mo> </mtd> </mtr> <mtr> <mtd> <msubsup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mi>j</mi> </mrow> <mo>′</mo> </msubsup> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msubsup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> <mo>′</mo> </msubsup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msubsup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mi>n</mi> </mrow> <mo>′</mo> </msubsup> </mtd> </mtr> <mtr> <mtd> <mo>⋮</mo> </mtd> </mtr> <mtr> <mtd> <msubsup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mi>n</mi> </mrow> <mo>′</mo> </msubsup> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mo>,</mo> <mspace width="1em" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b_{1}={\begin{pmatrix}b_{11}\\\vdots \\b_{n1}\end{pmatrix}},\dots ,b_{j}={\begin{pmatrix}b_{1j}\\\vdots \\b_{nj}\end{pmatrix}},\dots ,b_{n}={\begin{pmatrix}b_{1n}\\\vdots \\b_{nn}\end{pmatrix}},\quad b_{1}'={\begin{pmatrix}b_{11}'\\\vdots \\b_{n1}'\end{pmatrix}},\dots ,b_{j}'={\begin{pmatrix}b_{1j}'\\\vdots \\b_{nj}'\end{pmatrix}},\dots ,b_{n}'={\begin{pmatrix}b_{1n}'\\\vdots \\b_{nn}'\end{pmatrix}},\quad }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/90a44200c55db06003e6314019ffbba9509169d4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.171ex; width:105.157ex; height:11.509ex;" alt="{\displaystyle b_{1}={\begin{pmatrix}b_{11}\\\vdots \\b_{n1}\end{pmatrix}},\dots ,b_{j}={\begin{pmatrix}b_{1j}\\\vdots \\b_{nj}\end{pmatrix}},\dots ,b_{n}={\begin{pmatrix}b_{1n}\\\vdots \\b_{nn}\end{pmatrix}},\quad b_{1}'={\begin{pmatrix}b_{11}'\\\vdots \\b_{n1}'\end{pmatrix}},\dots ,b_{j}'={\begin{pmatrix}b_{1j}'\\\vdots \\b_{nj}'\end{pmatrix}},\dots ,b_{n}'={\begin{pmatrix}b_{1n}'\\\vdots \\b_{nn}'\end{pmatrix}},\quad }"></dd></dl> die sich zu Matrizen <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B={\begin{pmatrix}b_{11}&\dots &b_{1j}&\dots &b_{1n}\\\vdots &&\vdots &&\vdots \\b_{i1}&\dots &b_{ij}&\dots &b_{in}\\\vdots &&\vdots &&\vdots \\b_{n1}&\dots &b_{nj}&\dots &b_{nn}\end{pmatrix}}\quad {\text{ und }}\quad B'={\begin{pmatrix}b_{11}'&\dots &b_{1j}'&\dots &b_{1n}'\\\vdots &&\vdots &&\vdots \\b_{i1}'&\dots &b_{ij}'&\dots &b_{in}'\\\vdots &&\vdots &&\vdots \\b_{n1}'&\dots &b_{nj}'&\dots &b_{nn}'\end{pmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>11</mn> </mrow> </msub> </mtd> <mtd> <mo>…</mo> </mtd> <mtd> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mi>j</mi> </mrow> </msub> </mtd> <mtd> <mo>…</mo> </mtd> <mtd> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mi>n</mi> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <mo>⋮</mo> </mtd> <mtd /> <mtd> <mo>⋮</mo> </mtd> <mtd /> <mtd> <mo>⋮</mo> </mtd> </mtr> <mtr> <mtd> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mn>1</mn> </mrow> </msub> </mtd> <mtd> <mo>…</mo> </mtd> <mtd> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </msub> </mtd> <mtd> <mo>…</mo> </mtd> <mtd> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>n</mi> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <mo>⋮</mo> </mtd> <mtd /> <mtd> <mo>⋮</mo> </mtd> <mtd /> <mtd> <mo>⋮</mo> </mtd> </mtr> <mtr> <mtd> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mn>1</mn> </mrow> </msub> </mtd> <mtd> <mo>…</mo> </mtd> <mtd> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mi>j</mi> </mrow> </msub> </mtd> <mtd> <mo>…</mo> </mtd> <mtd> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mi>n</mi> </mrow> </msub> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mtext> und </mtext> </mrow> <mspace width="1em" /> <msup> <mi>B</mi> <mo>′</mo> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msubsup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>11</mn> </mrow> <mo>′</mo> </msubsup> </mtd> <mtd> <mo>…</mo> </mtd> <mtd> <msubsup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mi>j</mi> </mrow> <mo>′</mo> </msubsup> </mtd> <mtd> <mo>…</mo> </mtd> <mtd> <msubsup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mi>n</mi> </mrow> <mo>′</mo> </msubsup> </mtd> </mtr> <mtr> <mtd> <mo>⋮</mo> </mtd> <mtd /> <mtd> <mo>⋮</mo> </mtd> <mtd /> <mtd> <mo>⋮</mo> </mtd> </mtr> <mtr> <mtd> <msubsup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mn>1</mn> </mrow> <mo>′</mo> </msubsup> </mtd> <mtd> <mo>…</mo> </mtd> <mtd> <msubsup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> <mo>′</mo> </msubsup> </mtd> <mtd> <mo>…</mo> </mtd> <mtd> <msubsup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>n</mi> </mrow> <mo>′</mo> </msubsup> </mtd> </mtr> <mtr> <mtd> <mo>⋮</mo> </mtd> <mtd /> <mtd> <mo>⋮</mo> </mtd> <mtd /> <mtd> <mo>⋮</mo> </mtd> </mtr> <mtr> <mtd> <msubsup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mn>1</mn> </mrow> <mo>′</mo> </msubsup> </mtd> <mtd> <mo>…</mo> </mtd> <mtd> <msubsup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mi>j</mi> </mrow> <mo>′</mo> </msubsup> </mtd> <mtd> <mo>…</mo> </mtd> <mtd> <msubsup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mi>n</mi> </mrow> <mo>′</mo> </msubsup> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B={\begin{pmatrix}b_{11}&\dots &b_{1j}&\dots &b_{1n}\\\vdots &&\vdots &&\vdots \\b_{i1}&\dots &b_{ij}&\dots &b_{in}\\\vdots &&\vdots &&\vdots \\b_{n1}&\dots &b_{nj}&\dots &b_{nn}\end{pmatrix}}\quad {\text{ und }}\quad B'={\begin{pmatrix}b_{11}'&\dots &b_{1j}'&\dots &b_{1n}'\\\vdots &&\vdots &&\vdots \\b_{i1}'&\dots &b_{ij}'&\dots &b_{in}'\\\vdots &&\vdots &&\vdots \\b_{n1}'&\dots &b_{nj}'&\dots &b_{nn}'\end{pmatrix}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/48c284c895e72e5552054572156a04443c0db668" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -9.338ex; width:77.471ex; height:19.843ex;" alt="{\displaystyle B={\begin{pmatrix}b_{11}&\dots &b_{1j}&\dots &b_{1n}\\\vdots &&\vdots &&\vdots \\b_{i1}&\dots &b_{ij}&\dots &b_{in}\\\vdots &&\vdots &&\vdots \\b_{n1}&\dots &b_{nj}&\dots &b_{nn}\end{pmatrix}}\quad {\text{ und }}\quad B'={\begin{pmatrix}b_{11}'&\dots &b_{1j}'&\dots &b_{1n}'\\\vdots &&\vdots &&\vdots \\b_{i1}'&\dots &b_{ij}'&\dots &b_{in}'\\\vdots &&\vdots &&\vdots \\b_{n1}'&\dots &b_{nj}'&\dots &b_{nn}'\end{pmatrix}}}"></dd></dl> zusammenfassen lassen, die hier der Einfachheit halber mit den gleichen Buchstaben wie die zugehörigen Basen bezeichnet werden. Die Bedingung <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b_{j}=a_{1j}b_{1}{}'+a_{2j}b_{2}{}'+\dots +a_{nj}b_{n}{}'=\sum _{i=1}^{n}a_{ij}b_{i}{}',\qquad j=1,\dots ,n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mi>j</mi> </mrow> </msub> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msup> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mo>′</mo> </msup> <mo>+</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>j</mi> </mrow> </msub> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msup> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mo>′</mo> </msup> <mo>+</mo> <mo>⋯</mo> <mo>+</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mi>j</mi> </mrow> </msub> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <msup> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mo>′</mo> </msup> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </msub> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msup> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mo>′</mo> </msup> <mo>,</mo> <mspace width="2em" /> <mi>j</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mo>…</mo> <mo>,</mo> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b_{j}=a_{1j}b_{1}{}'+a_{2j}b_{2}{}'+\dots +a_{nj}b_{n}{}'=\sum _{i=1}^{n}a_{ij}b_{i}{}',\qquad j=1,\dots ,n}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0cac8cef7dc54bf7956858a8efc4711f3598c498" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:63.173ex; height:6.843ex;" alt="{\displaystyle b_{j}=a_{1j}b_{1}{}'+a_{2j}b_{2}{}'+\dots +a_{nj}b_{n}{}'=\sum _{i=1}^{n}a_{ij}b_{i}{}',\qquad j=1,\dots ,n}"></dd></dl> übersetzt sich dann zu <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b_{kj}=\sum _{i=1}^{n}a_{ij}b_{ki}{}'=\sum _{i=1}^{n}b_{ki}{}'a_{ij},\qquad k,j=1,\dots ,n,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mi>j</mi> </mrow> </msub> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </msub> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mi>i</mi> </mrow> </msub> <msup> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mo>′</mo> </msup> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mi>i</mi> </mrow> </msub> <msup> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mo>′</mo> </msup> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </msub> <mo>,</mo> <mspace width="2em" /> <mi>k</mi> <mo>,</mo> <mi>j</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mo>…</mo> <mo>,</mo> <mi>n</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b_{kj}=\sum _{i=1}^{n}a_{ij}b_{ki}{}'=\sum _{i=1}^{n}b_{ki}{}'a_{ij},\qquad k,j=1,\dots ,n,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/01ffbde0ec4448cd605df9d804bab7a04490c8e5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:48.899ex; height:6.843ex;" alt="{\displaystyle b_{kj}=\sum _{i=1}^{n}a_{ij}b_{ki}{}'=\sum _{i=1}^{n}b_{ki}{}'a_{ij},\qquad k,j=1,\dots ,n,}"></dd></dl> das heißt, <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B=B'\cdot T_{B'}^{B}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> <mo>=</mo> <msup> <mi>B</mi> <mo>′</mo> </msup> <mo>⋅</mo> <msubsup> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi>B</mi> <mo>′</mo> </msup> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> </mrow> </msubsup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B=B'\cdot T_{B'}^{B}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d6a97fb812dbce5c3857acd52b4c88ca5508959d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:13.006ex; height:3.343ex;" alt="{\displaystyle B=B'\cdot T_{B'}^{B}.}"></dd></dl> Die Transformationsmatrix <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T_{B'}^{B}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi>B</mi> <mo>′</mo> </msup> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T_{B'}^{B}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eb57134574715111f0c1c3dd85fd8d27a116d132" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:3.369ex; height:3.343ex;" alt="{\displaystyle T_{B'}^{B}}"> lässt sich somit durch <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T_{B'}^{B}=(B')^{-1}\cdot B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi>B</mi> <mo>′</mo> </msup> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> </mrow> </msubsup> <mo>=</mo> <mo stretchy="false">(</mo> <msup> <mi>B</mi> <mo>′</mo> </msup> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>⋅</mo> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T_{B'}^{B}=(B')^{-1}\cdot B}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c96559d73ff66d28d139fd96fceb7014deb5a53d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:16.501ex; height:3.509ex;" alt="{\displaystyle T_{B'}^{B}=(B')^{-1}\cdot B}"></dd></dl> berechnen, wobei <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (B')^{-1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msup> <mi>B</mi> <mo>′</mo> </msup> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (B')^{-1}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/18c4556a916fbd325d9349254b741a23d7a0fd61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.591ex; height:3.176ex;" alt="{\displaystyle (B')^{-1}}"> die <a href="/wiki/Inverse_Matrix" title="Inverse Matrix">inverse Matrix</a> der Matrix <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B'}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>B</mi> <mo>′</mo> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B'}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8b144e466a3757b4706daf24a45d4f1266700016" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.449ex; height:2.509ex;" alt="{\displaystyle B'}"> ist. Insbesondere gilt: Ist <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47136aad860d145f75f3eed3022df827cee94d7a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle B}"> die <a href="/wiki/Standardbasis" title="Standardbasis">Standardbasis</a>, so gilt <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T_{B'}^{B}=(B')^{-1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi>B</mi> <mo>′</mo> </msup> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> </mrow> </msubsup> <mo>=</mo> <mo stretchy="false">(</mo> <msup> <mi>B</mi> <mo>′</mo> </msup> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T_{B'}^{B}=(B')^{-1}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f17f3981d7b4e9ed3fd981cc57700590e9d4c0a0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:13.058ex; height:3.509ex;" alt="{\displaystyle T_{B'}^{B}=(B')^{-1}}">. Ist <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B'}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>B</mi> <mo>′</mo> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B'}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8b144e466a3757b4706daf24a45d4f1266700016" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.449ex; height:2.509ex;" alt="{\displaystyle B'}"> die Standardbasis, so gilt <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T_{B'}^{B}=B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi>B</mi> <mo>′</mo> </msup> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> </mrow> </msubsup> <mo>=</mo> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T_{B'}^{B}=B}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e347b15b6523004ba8d4d2e310dd1596ba176354" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:8.231ex; height:3.343ex;" alt="{\displaystyle T_{B'}^{B}=B}">. Wie im Vorangehenden wird hier die Basis <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47136aad860d145f75f3eed3022df827cee94d7a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle B}"> mit der Matrix identifiziert, die man erhält, indem man die Basisvektoren als Spaltenvektoren schreibt und diese zu einer Matrix zusammenfasst. <div class="mw-heading mw-heading2"><h2 id="Koordinatentransformation">Koordinatentransformation</h2>[<a href="/w/index.php?title=Basiswechsel_(Vektorraum)&veaction=edit&section=3" title="Abschnitt bearbeiten: Koordinatentransformation" class="mw-editsection-visualeditor">Bearbeiten</a> | <a href="/w/index.php?title=Basiswechsel_(Vektorraum)&action=edit&section=3" title="Quellcode des Abschnitts bearbeiten: Koordinatentransformation">Quelltext bearbeiten</a>]</div> Ein Vektor <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\in V}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>∈</mo> <mi>V</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\in V}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fa374e20b2db7f6b8caa71ff1865f7f84f215c9f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.958ex; height:2.176ex;" alt="{\displaystyle x\in V}"> habe bezüglich der Basis <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B=(b_{1},\dots ,b_{n})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> <mo>=</mo> <mo stretchy="false">(</mo> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B=(b_{1},\dots ,b_{n})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f48a809b22c97353ab3da3c70e576a8f7b29c7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.118ex; height:2.843ex;" alt="{\displaystyle B=(b_{1},\dots ,b_{n})}"> die Koordinaten <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{1},\dots ,x_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{1},\dots ,x_{n}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e5afdbc2d248d8fa9ba2c4f5188d946a0537e753" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.11ex; height:2.009ex;" alt="{\displaystyle x_{1},\dots ,x_{n}}">, d. h. <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x=x_{1}b_{1}+x_{2}b_{2}+\dots +x_{n}b_{n}=\sum _{i}x_{i}\,b_{i},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>=</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>+</mo> <mo>⋯</mo> <mo>+</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>=</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </munder> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mspace width="thinmathspace" /> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x=x_{1}b_{1}+x_{2}b_{2}+\dots +x_{n}b_{n}=\sum _{i}x_{i}\,b_{i},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2a7136d8575314d5395e5b27f2fc764128a472ff" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:41.109ex; height:5.509ex;" alt="{\displaystyle x=x_{1}b_{1}+x_{2}b_{2}+\dots +x_{n}b_{n}=\sum _{i}x_{i}\,b_{i},}"></dd></dl> und bezüglich der neuen Basis <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B'=(b_{1}',\dots ,b_{n}')}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>B</mi> <mo>′</mo> </msup> <mo>=</mo> <mo stretchy="false">(</mo> <msubsup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mo>′</mo> </msubsup> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msubsup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> <mo>′</mo> </msubsup> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B'=(b_{1}',\dots ,b_{n}')}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f07871e5d1a9d0ceaf9ee8103a745f5c0ac010ff" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:16.803ex; height:3.176ex;" alt="{\displaystyle B'=(b_{1}',\dots ,b_{n}')}"> die Koordinaten <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{1}',\dots ,x_{n}'}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mo>′</mo> </msubsup> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> <mo>′</mo> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{1}',\dots ,x_{n}'}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a7380746dcd6cd7b0533640f2b8716df087541c3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:10.11ex; height:2.843ex;" alt="{\displaystyle x_{1}',\dots ,x_{n}'}">, also <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x=x_{1}{}'b_{1}{}'+x_{2}{}'b_{2}{}'+\dots +x_{n}{}'b_{n}{}'=\sum _{j}x_{j}{}'\,b_{j}{}'.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>=</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msup> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mo>′</mo> </msup> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msup> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mo>′</mo> </msup> <mo>+</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msup> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mo>′</mo> </msup> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msup> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mo>′</mo> </msup> <mo>+</mo> <mo>⋯</mo> <mo>+</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <msup> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mo>′</mo> </msup> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <msup> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mo>′</mo> </msup> <mo>=</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </munder> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <msup> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mo>′</mo> </msup> <mspace width="thinmathspace" /> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <msup> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mo>′</mo> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x=x_{1}{}'b_{1}{}'+x_{2}{}'b_{2}{}'+\dots +x_{n}{}'b_{n}{}'=\sum _{j}x_{j}{}'\,b_{j}{}'.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f46c24336778b2176b29ca88c9cdd9fb781a21b0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:46.807ex; height:5.843ex;" alt="{\displaystyle x=x_{1}{}'b_{1}{}'+x_{2}{}'b_{2}{}'+\dots +x_{n}{}'b_{n}{}'=\sum _{j}x_{j}{}'\,b_{j}{}'.}"></dd></dl> Stellt man wie oben die Vektoren <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b_{j}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b_{j}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fa56eff4488494085785b7b0d6e2069bd45a3ce5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:1.907ex; height:2.843ex;" alt="{\displaystyle b_{j}}"> der alten Basis als Linearkombination der neuen Basis dar, so erhält man <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x=\sum _{j}x_{j}b_{j}=\sum _{j}x_{j}\sum _{i}a_{ij}\,b_{i}{}'=\sum _{i}\left(\sum _{j}a_{ij}\,x_{j}\right)b_{i}{}'}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>=</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </munder> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo>=</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </munder> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </munder> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </msub> <mspace width="thinmathspace" /> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msup> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mo>′</mo> </msup> <mo>=</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </munder> <mrow> <mo>(</mo> <mrow> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </munder> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </msub> <mspace width="thinmathspace" /> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msup> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mo>′</mo> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x=\sum _{j}x_{j}b_{j}=\sum _{j}x_{j}\sum _{i}a_{ij}\,b_{i}{}'=\sum _{i}\left(\sum _{j}a_{ij}\,x_{j}\right)b_{i}{}'}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dbef3cc479852d8f1376a8cf4090ba3164ec7095" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:53.568ex; height:7.676ex;" alt="{\displaystyle x=\sum _{j}x_{j}b_{j}=\sum _{j}x_{j}\sum _{i}a_{ij}\,b_{i}{}'=\sum _{i}\left(\sum _{j}a_{ij}\,x_{j}\right)b_{i}{}'}"></dd></dl> Dabei sind die <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{ij}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{ij}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ebea6cd2813c330c798921a2894b358f7b643917" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.707ex; height:2.343ex;" alt="{\displaystyle a_{ij}}"> die oben definierten Einträge der Basiswechselmatrix <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T_{B'}^{B}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi>B</mi> <mo>′</mo> </msup> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T_{B'}^{B}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eb57134574715111f0c1c3dd85fd8d27a116d132" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:3.369ex; height:3.343ex;" alt="{\displaystyle T_{B'}^{B}}">. Durch Koeffizientenvergleich erhält man <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{i}{}'=\sum _{j=1}^{n}a_{ij}\,x_{j},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msup> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mo>′</mo> </msup> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </msub> <mspace width="thinmathspace" /> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{i}{}'=\sum _{j=1}^{n}a_{ij}\,x_{j},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/73aa1fc3b50900b9b2042340ae07654136bea3ce" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:15.635ex; height:7.176ex;" alt="{\displaystyle x_{i}{}'=\sum _{j=1}^{n}a_{ij}\,x_{j},}"></dd></dl> bzw. in Matrizenschreibweise: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{pmatrix}x_{1}{}'\\\vdots \\x_{n}{}'\end{pmatrix}}={\begin{pmatrix}a_{11}&\dots &a_{1n}\\\vdots &\ddots &\vdots \\a_{n1}&\dots &a_{nn}\end{pmatrix}}{\begin{pmatrix}x_{1}\\\vdots \\x_{n}\end{pmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msup> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mo>′</mo> </msup> </mtd> </mtr> <mtr> <mtd> <mo>⋮</mo> </mtd> </mtr> <mtr> <mtd> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <msup> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mo>′</mo> </msup> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>11</mn> </mrow> </msub> </mtd> <mtd> <mo>…</mo> </mtd> <mtd> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mi>n</mi> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <mo>⋮</mo> </mtd> <mtd> <mo>⋱</mo> </mtd> <mtd> <mo>⋮</mo> </mtd> </mtr> <mtr> <mtd> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mn>1</mn> </mrow> </msub> </mtd> <mtd> <mo>…</mo> </mtd> <mtd> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mi>n</mi> </mrow> </msub> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <mo>⋮</mo> </mtd> </mtr> <mtr> <mtd> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{pmatrix}x_{1}{}'\\\vdots \\x_{n}{}'\end{pmatrix}}={\begin{pmatrix}a_{11}&\dots &a_{1n}\\\vdots &\ddots &\vdots \\a_{n1}&\dots &a_{nn}\end{pmatrix}}{\begin{pmatrix}x_{1}\\\vdots \\x_{n}\end{pmatrix}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/02e049179b6ebe25fdab77a571329f7363b89e40" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.005ex; width:37.663ex; height:11.009ex;" alt="{\displaystyle {\begin{pmatrix}x_{1}{}'\\\vdots \\x_{n}{}'\end{pmatrix}}={\begin{pmatrix}a_{11}&\dots &a_{1n}\\\vdots &\ddots &\vdots \\a_{n1}&\dots &a_{nn}\end{pmatrix}}{\begin{pmatrix}x_{1}\\\vdots \\x_{n}\end{pmatrix}}}"></dd></dl> oder kurz: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x{}'=T_{B'}^{B}\,x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <msup> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mo>′</mo> </msup> <mo>=</mo> <msubsup> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi>B</mi> <mo>′</mo> </msup> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> </mrow> </msubsup> <mspace width="thinmathspace" /> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x{}'=T_{B'}^{B}\,x}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cb654d54be9843c4b7f710895c6a70cc4b2a3e24" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:10.198ex; height:3.343ex;" alt="{\displaystyle x{}'=T_{B'}^{B}\,x}"></dd></dl> <div class="mw-heading mw-heading2"><h2 id="Basiswechsel_bei_Abbildungsmatrizen">Basiswechsel bei Abbildungsmatrizen</h2>[<a href="/w/index.php?title=Basiswechsel_(Vektorraum)&veaction=edit&section=4" title="Abschnitt bearbeiten: Basiswechsel bei Abbildungsmatrizen" class="mw-editsection-visualeditor">Bearbeiten</a> | <a href="/w/index.php?title=Basiswechsel_(Vektorraum)&action=edit&section=4" title="Quellcode des Abschnitts bearbeiten: Basiswechsel bei Abbildungsmatrizen">Quelltext bearbeiten</a>]</div> Die <a href="/wiki/Abbildungsmatrix" title="Abbildungsmatrix">Darstellungsmatrix</a> einer linearen Abbildung hängt von der Wahl der Basen im Urbild- und im Zielraum ab. Wählt man andere Basen, so erhält man möglicherweise andere Abbildungsmatrizen. <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/Datei:Change_of_basis2.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/a/ab/Change_of_basis2.svg/440px-Change_of_basis2.svg.png" decoding="async" width="440" height="183" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/ab/Change_of_basis2.svg/660px-Change_of_basis2.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/a/ab/Change_of_basis2.svg/880px-Change_of_basis2.svg.png 2x" data-file-width="425" data-file-height="177" /></a><figcaption>Kommutatives Diagramm der beteiligten Abbildungen. Mit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"> wird hier die lineare Abbildung von <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><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}}"> nach <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><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}"> bezeichnet, die <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (x_{1},\dots ,x_{n})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (x_{1},\dots ,x_{n})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/56204ada1ecf9dd5c6beddfdfd0f341cd69ff632" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.92ex; height:2.843ex;" alt="{\displaystyle (x_{1},\dots ,x_{n})}"> auf <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{1}a_{1}+\dots +x_{n}a_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <mo>⋯</mo> <mo>+</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{1}a_{1}+\dots +x_{n}a_{n}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f3487c522ae779241c34523dfda82c8b3dd1de7d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:18.068ex; height:2.343ex;" alt="{\displaystyle x_{1}a_{1}+\dots +x_{n}a_{n}}"> abbildet, etc.</figcaption></figure> Seien <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><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}"> ein <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><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}">-dimensionaler und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle W}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>W</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle W}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/54a9c4c547f4d6111f81946cad242b18298d70b7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.435ex; height:2.176ex;" alt="{\displaystyle W}"> ein <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><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0a07d98bb302f3856cbabc47b2b9016692e3f7bc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.04ex; height:1.676ex;" alt="{\displaystyle m}">-dimensionaler Vektorraum über <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><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}"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f\colon V\to W}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>:</mo> <mi>V</mi> <mo stretchy="false">→</mo> <mi>W</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f\colon V\to W}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fa58648b9aa9476995b2b41de27a9a591e631f56" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.149ex; height:2.509ex;" alt="{\displaystyle f\colon V\to W}"> eine lineare Abbildung. In <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><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}"> seien die geordneten Basen <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A=(a_{1},\dots ,a_{n})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>=</mo> <mo stretchy="false">(</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A=(a_{1},\dots ,a_{n})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3b5dd46decd1d7047cdfd4e9b228283b5bdfcfd4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.561ex; height:2.843ex;" alt="{\displaystyle A=(a_{1},\dots ,a_{n})}"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A'=(a_{1}{}',\dots ,a_{n}{}')}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>A</mi> <mo>′</mo> </msup> <mo>=</mo> <mo stretchy="false">(</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msup> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mo>′</mo> </msup> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <msup> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mo>′</mo> </msup> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A'=(a_{1}{}',\dots ,a_{n}{}')}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9c2d08ffdacf9d7b4e26d0040c0b373e03ac139e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.616ex; height:3.009ex;" alt="{\displaystyle A'=(a_{1}{}',\dots ,a_{n}{}')}"> gegeben, in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle W}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>W</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle W}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/54a9c4c547f4d6111f81946cad242b18298d70b7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.435ex; height:2.176ex;" alt="{\displaystyle W}"> die geordneten Basen <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B=(b_{1},\dots ,b_{m})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> <mo>=</mo> <mo stretchy="false">(</mo> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B=(b_{1},\dots ,b_{m})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/00c3cc61328505fc90239458c23d180deae430fa" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.574ex; height:2.843ex;" alt="{\displaystyle B=(b_{1},\dots ,b_{m})}"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B'=(b_{1}{}',\dots ,b_{m}{}')}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>B</mi> <mo>′</mo> </msup> <mo>=</mo> <mo stretchy="false">(</mo> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msup> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mo>′</mo> </msup> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> <msup> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mo>′</mo> </msup> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B'=(b_{1}{}',\dots ,b_{m}{}')}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c5cb215fcd5c3cf5dab0028803419c2084de3f26" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.629ex; height:3.009ex;" alt="{\displaystyle B'=(b_{1}{}',\dots ,b_{m}{}')}">. Dann gilt für die Darstellungsmatrizen von <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><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}"> bezüglich <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47136aad860d145f75f3eed3022df827cee94d7a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle B}"> bzw. bezüglich <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A'}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>A</mi> <mo>′</mo> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A'}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/98a12527148d6ed68adc91d9b419eb4b92d58ef6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.428ex; height:2.509ex;" alt="{\displaystyle A'}"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B'}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>B</mi> <mo>′</mo> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B'}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8b144e466a3757b4706daf24a45d4f1266700016" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.449ex; height:2.509ex;" alt="{\displaystyle B'}">: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M_{B'}^{A'}(f)=T_{B'}^{B}\cdot M_{B}^{A}(f)\cdot T_{A}^{A'}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi>B</mi> <mo>′</mo> </msup> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi>A</mi> <mo>′</mo> </msup> </mrow> </msubsup> <mo stretchy="false">(</mo> <mi>f</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msubsup> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi>B</mi> <mo>′</mo> </msup> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> </mrow> </msubsup> <mo>⋅</mo> <msubsup> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msubsup> <mo stretchy="false">(</mo> <mi>f</mi> <mo stretchy="false">)</mo> <mo>⋅</mo> <msubsup> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi>A</mi> <mo>′</mo> </msup> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M_{B'}^{A'}(f)=T_{B'}^{B}\cdot M_{B}^{A}(f)\cdot T_{A}^{A'}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f7d6ffe1ae20fcef3b53d0e0ee4aa690bff04257" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:28.176ex; height:3.676ex;" alt="{\displaystyle M_{B'}^{A'}(f)=T_{B'}^{B}\cdot M_{B}^{A}(f)\cdot T_{A}^{A'}}"></dd></dl> Man erhält diese Darstellung, indem man <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f=\operatorname {id} _{W}\circ f\circ \operatorname {id} _{V}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>=</mo> <msub> <mi>id</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>W</mi> </mrow> </msub> <mo>∘</mo> <mi>f</mi> <mo>∘</mo> <msub> <mi>id</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>V</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f=\operatorname {id} _{W}\circ f\circ \operatorname {id} _{V}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/41ab704655ab5b30df65d26e537f9555dc05a501" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:16.729ex; height:2.509ex;" alt="{\displaystyle f=\operatorname {id} _{W}\circ f\circ \operatorname {id} _{V}}"></dd></dl> schreibt. Die Abbildungsmatrix der Verkettung ist dann das <a href="/wiki/Matrizenprodukt" class="mw-redirect" title="Matrizenprodukt">Matrizenprodukt</a> der einzelnen Abbildungsmatrizen, wenn die Basen passend gewählt sind, das heißt: die Basis <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A'}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>A</mi> <mo>′</mo> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A'}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/98a12527148d6ed68adc91d9b419eb4b92d58ef6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.428ex; height:2.509ex;" alt="{\displaystyle A'}"> im Urbild von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {id} _{V}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>id</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>V</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {id} _{V}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7b7dd7ccb7d8663c3f9cb1b53d125f38703416b5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.435ex; height:2.509ex;" alt="{\displaystyle \operatorname {id} _{V}}">, die Basis <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"> im Bild von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {id} _{V}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>id</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>V</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {id} _{V}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7b7dd7ccb7d8663c3f9cb1b53d125f38703416b5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.435ex; height:2.509ex;" alt="{\displaystyle \operatorname {id} _{V}}"> und im Urbild von <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><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}">, die Basis <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47136aad860d145f75f3eed3022df827cee94d7a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle B}"> im Bild von <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><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}"> und im Urbild von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {id} _{W}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>id</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>W</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {id} _{W}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fba343c71190adf383fdaa1d6c8cbdbef73ec225" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.894ex; height:2.509ex;" alt="{\displaystyle \operatorname {id} _{W}}">, und die Basis <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B'}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>B</mi> <mo>′</mo> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B'}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8b144e466a3757b4706daf24a45d4f1266700016" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.449ex; height:2.509ex;" alt="{\displaystyle B'}"> im Bild von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {id} _{W}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>id</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>W</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {id} _{W}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fba343c71190adf383fdaa1d6c8cbdbef73ec225" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.894ex; height:2.509ex;" alt="{\displaystyle \operatorname {id} _{W}}">. Man erhält also: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M_{B'}^{A'}(f)=M_{B'}^{B}(\operatorname {id} _{W})\cdot M_{B}^{A}(f)\cdot M_{A}^{A'}(\operatorname {id} _{V})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi>B</mi> <mo>′</mo> </msup> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi>A</mi> <mo>′</mo> </msup> </mrow> </msubsup> <mo stretchy="false">(</mo> <mi>f</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msubsup> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi>B</mi> <mo>′</mo> </msup> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> </mrow> </msubsup> <mo stretchy="false">(</mo> <msub> <mi>id</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>W</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>⋅</mo> <msubsup> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msubsup> <mo stretchy="false">(</mo> <mi>f</mi> <mo stretchy="false">)</mo> <mo>⋅</mo> <msubsup> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi>A</mi> <mo>′</mo> </msup> </mrow> </msubsup> <mo stretchy="false">(</mo> <msub> <mi>id</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>V</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M_{B'}^{A'}(f)=M_{B'}^{B}(\operatorname {id} _{W})\cdot M_{B}^{A}(f)\cdot M_{A}^{A'}(\operatorname {id} _{V})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5d481c0d9e55779b983d4231d8ecb6d160f75ee1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:40.799ex; height:3.676ex;" alt="{\displaystyle M_{B'}^{A'}(f)=M_{B'}^{B}(\operatorname {id} _{W})\cdot M_{B}^{A}(f)\cdot M_{A}^{A'}(\operatorname {id} _{V})}"></dd></dl> Ein wichtiger Spezialfall ist, wenn <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f\colon V\to V}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>:</mo> <mi>V</mi> <mo stretchy="false">→</mo> <mi>V</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f\colon V\to V}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7cdad235875fd740596e229f36847d469955d96c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.501ex; height:2.509ex;" alt="{\displaystyle f\colon V\to V}"> ein <a href="/wiki/Endomorphismus" title="Endomorphismus">Endomorphismus</a> ist und im Urbild und Bild jeweils dieselbe Basis <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47136aad860d145f75f3eed3022df827cee94d7a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle B}"> bzw. <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B'}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>B</mi> <mo>′</mo> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B'}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8b144e466a3757b4706daf24a45d4f1266700016" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.449ex; height:2.509ex;" alt="{\displaystyle B'}"> benutzt wird. Dann gilt: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M_{B'}^{B'}(f)=T_{B'}^{B}\cdot M_{B}^{B}(f)\cdot T_{B}^{B'}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi>B</mi> <mo>′</mo> </msup> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi>B</mi> <mo>′</mo> </msup> </mrow> </msubsup> <mo stretchy="false">(</mo> <mi>f</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msubsup> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi>B</mi> <mo>′</mo> </msup> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> </mrow> </msubsup> <mo>⋅</mo> <msubsup> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> </mrow> </msubsup> <mo stretchy="false">(</mo> <mi>f</mi> <mo stretchy="false">)</mo> <mo>⋅</mo> <msubsup> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi>B</mi> <mo>′</mo> </msup> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M_{B'}^{B'}(f)=T_{B'}^{B}\cdot M_{B}^{B}(f)\cdot T_{B}^{B'}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d3f7edabf80cd28bb045af84c702a0b6c7bff470" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:28.221ex; height:3.509ex;" alt="{\displaystyle M_{B'}^{B'}(f)=T_{B'}^{B}\cdot M_{B}^{B}(f)\cdot T_{B}^{B'}}"></dd></dl> Setzt man <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T:=T_{B'}^{B}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> <mo>:=</mo> <msubsup> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi>B</mi> <mo>′</mo> </msup> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T:=T_{B'}^{B}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7957e52da88a7761ceeff83dc8190f20fe0691fc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:8.75ex; height:3.343ex;" alt="{\displaystyle T:=T_{B'}^{B}}">, so gilt also <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M_{B'}^{B'}(f)=T\cdot M_{B}^{B}(f)\cdot T^{-1}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi>B</mi> <mo>′</mo> </msup> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi>B</mi> <mo>′</mo> </msup> </mrow> </msubsup> <mo stretchy="false">(</mo> <mi>f</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>T</mi> <mo>⋅</mo> <msubsup> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> </mrow> </msubsup> <mo stretchy="false">(</mo> <mi>f</mi> <mo stretchy="false">)</mo> <mo>⋅</mo> <msup> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M_{B'}^{B'}(f)=T\cdot M_{B}^{B}(f)\cdot T^{-1}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7d24ee13e63f710bab15464ae2cffcffea99d80f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:27.457ex; height:3.509ex;" alt="{\displaystyle M_{B'}^{B'}(f)=T\cdot M_{B}^{B}(f)\cdot T^{-1}.}"></dd></dl> Die Abbildungsmatrizen <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M_{B'}^{B'}(f)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi>B</mi> <mo>′</mo> </msup> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi>B</mi> <mo>′</mo> </msup> </mrow> </msubsup> <mo stretchy="false">(</mo> <mi>f</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M_{B'}^{B'}(f)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d8ed9a6e3113878ea403324de45c540cebfd9661" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:7.598ex; height:3.509ex;" alt="{\displaystyle M_{B'}^{B'}(f)}"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M_{B}^{B}(f)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> </mrow> </msubsup> <mo stretchy="false">(</mo> <mi>f</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M_{B}^{B}(f)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ecac2e8b3098f605a1c5651991c8eadf3015b17c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:7.066ex; height:3.176ex;" alt="{\displaystyle M_{B}^{B}(f)}"> sind also <a href="/wiki/%C3%84hnlichkeit_(Matrix)" title="Ähnlichkeit (Matrix)">ähnlich</a>. <div class="mw-heading mw-heading2"><h2 id="Beispiel">Beispiel</h2>[<a href="/w/index.php?title=Basiswechsel_(Vektorraum)&veaction=edit&section=5" title="Abschnitt bearbeiten: Beispiel" class="mw-editsection-visualeditor">Bearbeiten</a> | <a href="/w/index.php?title=Basiswechsel_(Vektorraum)&action=edit&section=5" title="Quellcode des Abschnitts bearbeiten: Beispiel">Quelltext bearbeiten</a>]</div> Wir betrachten zwei Basen <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B=(b_{1},b_{2},b_{3})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> <mo>=</mo> <mo stretchy="false">(</mo> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B=(b_{1},b_{2},b_{3})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/be95af1343bc4075ba14bc1670719c2a696352be" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.895ex; height:2.843ex;" alt="{\displaystyle B=(b_{1},b_{2},b_{3})}"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B'=(b_{1}',b_{2}',b_{3}')}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>B</mi> <mo>′</mo> </msup> <mo>=</mo> <mo stretchy="false">(</mo> <msubsup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mo>′</mo> </msubsup> <mo>,</mo> <msubsup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> <mo>′</mo> </msubsup> <mo>,</mo> <msubsup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> <mo>′</mo> </msubsup> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B'=(b_{1}',b_{2}',b_{3}')}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7019e90084f63fb408f22ffaacc90678cce39faf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:15.58ex; height:3.176ex;" alt="{\displaystyle B'=(b_{1}',b_{2}',b_{3}')}"> des <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><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}}"> mit <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b_{1}={\begin{pmatrix}1\\0\\2\end{pmatrix}},b_{2}={\begin{pmatrix}3\\1\\0\end{pmatrix}},b_{3}={\begin{pmatrix}2\\1\\1\end{pmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>b</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>2</mn> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mo>,</mo> <msub> <mi>b</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>3</mn> </mtd> </mtr> <mtr> <mtd> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mo>,</mo> <msub> <mi>b</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>2</mn> </mtd> </mtr> <mtr> <mtd> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mn>1</mn> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b_{1}={\begin{pmatrix}1\\0\\2\end{pmatrix}},b_{2}={\begin{pmatrix}3\\1\\0\end{pmatrix}},b_{3}={\begin{pmatrix}2\\1\\1\end{pmatrix}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/50d5db642b3d4e743c49ccaec97ea592bc77acc7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.005ex; width:35.461ex; height:9.176ex;" alt="{\displaystyle b_{1}={\begin{pmatrix}1\\0\\2\end{pmatrix}},b_{2}={\begin{pmatrix}3\\1\\0\end{pmatrix}},b_{3}={\begin{pmatrix}2\\1\\1\end{pmatrix}}}"></dd></dl> und <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b_{1}'={\begin{pmatrix}1\\0\\1\end{pmatrix}},b_{2}'={\begin{pmatrix}0\\1\\1\end{pmatrix}},b_{3}'={\begin{pmatrix}1\\1\\0\end{pmatrix}}\,,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mo>′</mo> </msubsup> <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>1</mn> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mo>,</mo> <msubsup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> <mo>′</mo> </msubsup> <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>1</mn> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mo>,</mo> <msubsup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> <mo>′</mo> </msubsup> <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>1</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mspace width="thinmathspace" /> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b_{1}'={\begin{pmatrix}1\\0\\1\end{pmatrix}},b_{2}'={\begin{pmatrix}0\\1\\1\end{pmatrix}},b_{3}'={\begin{pmatrix}1\\1\\0\end{pmatrix}}\,,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c31f3eb9ec1a903cecd0b869b110ccc987ec02fc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.005ex; width:36.495ex; height:9.176ex;" alt="{\displaystyle b_{1}'={\begin{pmatrix}1\\0\\1\end{pmatrix}},b_{2}'={\begin{pmatrix}0\\1\\1\end{pmatrix}},b_{3}'={\begin{pmatrix}1\\1\\0\end{pmatrix}}\,,}"></dd></dl> wobei die Koordinatendarstellung der Vektoren die Vektoren bezüglich der <a href="/wiki/Standardbasis" title="Standardbasis">Standardbasis</a> beschreibt. Die Transformation der Koordinaten eines Vektors <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v=x_{1}b_{1}+x_{2}b_{2}+x_{3}b_{3}=x_{1}'b_{1}'+x_{2}'b_{2}'+x_{3}'b_{3}'}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>v</mi> <mo>=</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>=</mo> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mo>′</mo> </msubsup> <msubsup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mo>′</mo> </msubsup> <mo>+</mo> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> <mo>′</mo> </msubsup> <msubsup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> <mo>′</mo> </msubsup> <mo>+</mo> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> <mo>′</mo> </msubsup> <msubsup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> <mo>′</mo> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v=x_{1}b_{1}+x_{2}b_{2}+x_{3}b_{3}=x_{1}'b_{1}'+x_{2}'b_{2}'+x_{3}'b_{3}'}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5e13c7aa5b00c7e7b160d156f411be49f15c972a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:45.3ex; height:2.843ex;" alt="{\displaystyle v=x_{1}b_{1}+x_{2}b_{2}+x_{3}b_{3}=x_{1}'b_{1}'+x_{2}'b_{2}'+x_{3}'b_{3}'}"></dd></dl> ergibt sich durch die Darstellung der alten Basisvektoren <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (b_{1},b_{2},b_{3})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (b_{1},b_{2},b_{3})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/facfe95205e32ef796c45ee4702e050de002f03e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.033ex; height:2.843ex;" alt="{\displaystyle (b_{1},b_{2},b_{3})}"> bezüglich der neuen Basis <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (b_{1}',b_{2}',b_{3}')}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msubsup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mo>′</mo> </msubsup> <mo>,</mo> <msubsup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> <mo>′</mo> </msubsup> <mo>,</mo> <msubsup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> <mo>′</mo> </msubsup> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (b_{1}',b_{2}',b_{3}')}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/67744823f83d2dffd26261986f559dcc12023d91" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:10.033ex; height:3.009ex;" alt="{\displaystyle (b_{1}',b_{2}',b_{3}')}"> und deren Gewichtung mit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (x_{1},x_{2},x_{3})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (x_{1},x_{2},x_{3})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c299961b19740135135d0ef0289b16b6095048d3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.029ex; height:2.843ex;" alt="{\displaystyle (x_{1},x_{2},x_{3})}">. Um die Matrix der Basistransformation <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T_{B'}^{B}=(a_{ij})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi>B</mi> <mo>′</mo> </msup> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> </mrow> </msubsup> <mo>=</mo> <mo stretchy="false">(</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T_{B'}^{B}=(a_{ij})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/90dce48dfe26782354a35c7817b44638ee898fbb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:10.983ex; height:3.343ex;" alt="{\displaystyle T_{B'}^{B}=(a_{ij})}"> von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47136aad860d145f75f3eed3022df827cee94d7a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle B}"> nach <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B'}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>B</mi> <mo>′</mo> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B'}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8b144e466a3757b4706daf24a45d4f1266700016" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.449ex; height:2.509ex;" alt="{\displaystyle B'}"> zu berechnen, müssen wir die drei <a href="/wiki/Lineares_Gleichungssystem" title="Lineares Gleichungssystem">linearen Gleichungssysteme</a> <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{j}=a_{1j}x_{1}'+a_{2j}x_{2}'+a_{3j}x_{3}'}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mi>j</mi> </mrow> </msub> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mo>′</mo> </msubsup> <mo>+</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>j</mi> </mrow> </msub> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> <mo>′</mo> </msubsup> <mo>+</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> <mi>j</mi> </mrow> </msub> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> <mo>′</mo> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{j}=a_{1j}x_{1}'+a_{2j}x_{2}'+a_{3j}x_{3}'}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/defa37e9dfe3ae5d9c678f8eb668e939cb5320d8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:27.055ex; height:2.843ex;" alt="{\displaystyle x_{j}=a_{1j}x_{1}'+a_{2j}x_{2}'+a_{3j}x_{3}'}"></dd></dl> nach den 9 Unbekannten <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{ij}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{ij}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ebea6cd2813c330c798921a2894b358f7b643917" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.707ex; height:2.343ex;" alt="{\displaystyle a_{ij}}"> auflösen. Dies kann mit dem <a href="/wiki/Gau%C3%9F-Jordan-Algorithmus" title="Gauß-Jordan-Algorithmus">Gauß-Jordan-Algorithmus</a> für alle drei Gleichungssysteme simultan erfolgen. Dazu wird folgendes lineares Gleichungssystem aufgestellt: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left({\begin{array}{c c c | c c c}1&0&1&1&3&2\\0&1&1&0&1&1\\1&1&0&2&0&1\end{array}}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="center center center center center center" rowspacing="4pt" columnspacing="1em" columnlines="none none solid none none"> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mn>3</mn> </mtd> <mtd> <mn>2</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>2</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> </mtable> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left({\begin{array}{c c c | c c c}1&0&1&1&3&2\\0&1&1&0&1&1\\1&1&0&2&0&1\end{array}}\right)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7aab60fe811a88a587b5b862535205d60a74ffaf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.505ex; width:25.264ex; height:10.176ex;" alt="{\displaystyle \left({\begin{array}{c c c | c c c}1&0&1&1&3&2\\0&1&1&0&1&1\\1&1&0&2&0&1\end{array}}\right)}"></dd></dl> Durch Umformen mit elementaren Zeilenoperationen lässt sich die linke Seite auf die <a href="/wiki/Einheitsmatrix" title="Einheitsmatrix">Einheitsmatrix</a> bringen und auf der rechten Seite erhält man als Lösung des Systems die Transformationsmatrix <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T_{B'}^{B}={\begin{pmatrix}{\frac {3}{2}}&1&1\\{\frac {1}{2}}&-1&0\\-{\frac {1}{2}}&2&1\end{pmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi>B</mi> <mo>′</mo> </msup> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> </mrow> </msubsup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>3</mn> <mn>2</mn> </mfrac> </mrow> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> </mtd> <mtd> <mo>−</mo> <mn>1</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> </mtd> <mtd> <mn>2</mn> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T_{B'}^{B}={\begin{pmatrix}{\frac {3}{2}}&1&1\\{\frac {1}{2}}&-1&0\\-{\frac {1}{2}}&2&1\end{pmatrix}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4ee3fac1c63481f5926f089255e4986400a430cc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.012ex; margin-bottom: -0.326ex; width:23.53ex; height:11.843ex;" alt="{\displaystyle T_{B'}^{B}={\begin{pmatrix}{\frac {3}{2}}&1&1\\{\frac {1}{2}}&-1&0\\-{\frac {1}{2}}&2&1\end{pmatrix}}}">.</dd></dl> Wir betrachten den Vektor <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v=2b_{1}-b_{2}+3b_{3}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>v</mi> <mo>=</mo> <mn>2</mn> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>+</mo> <mn>3</mn> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v=2b_{1}-b_{2}+3b_{3}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/82eba19b2e76f7f11191c30586132dc22391158d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:18.387ex; height:2.509ex;" alt="{\displaystyle v=2b_{1}-b_{2}+3b_{3}}">, also den Vektor der bezüglich der Basis <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47136aad860d145f75f3eed3022df827cee94d7a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle B}"> die Koordinaten <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{pmatrix}x_{1}\\x_{2}\\x_{3}\end{pmatrix}}={\begin{pmatrix}2\\-1\\3\end{pmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>2</mn> </mtd> </mtr> <mtr> <mtd> <mo>−</mo> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mn>3</mn> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{pmatrix}x_{1}\\x_{2}\\x_{3}\end{pmatrix}}={\begin{pmatrix}2\\-1\\3\end{pmatrix}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/80ade2000efa2f653a5a3b701b749dd90538953e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.005ex; width:18.09ex; height:9.176ex;" alt="{\displaystyle {\begin{pmatrix}x_{1}\\x_{2}\\x_{3}\end{pmatrix}}={\begin{pmatrix}2\\-1\\3\end{pmatrix}}}"></dd></dl> besitzt. Um nun die Koordinaten bezüglich <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B'}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>B</mi> <mo>′</mo> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B'}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8b144e466a3757b4706daf24a45d4f1266700016" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.449ex; height:2.509ex;" alt="{\displaystyle B'}"> zu berechnen, müssen wir die Transformationsmatrix <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T_{B'}^{B}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi>B</mi> <mo>′</mo> </msup> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T_{B'}^{B}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eb57134574715111f0c1c3dd85fd8d27a116d132" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:3.369ex; height:3.343ex;" alt="{\displaystyle T_{B'}^{B}}"> mit diesem Spaltenvektor multiplizieren: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{pmatrix}x_{1}'\\x_{2}'\\x_{3}'\end{pmatrix}}={\begin{pmatrix}{\frac {3}{2}}&1&1\\{\frac {1}{2}}&-1&0\\-{\frac {1}{2}}&2&1\end{pmatrix}}{\begin{pmatrix}2\\-1\\3\end{pmatrix}}={\begin{pmatrix}5\\2\\0\end{pmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mo>′</mo> </msubsup> </mtd> </mtr> <mtr> <mtd> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> <mo>′</mo> </msubsup> </mtd> </mtr> <mtr> <mtd> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> <mo>′</mo> </msubsup> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>3</mn> <mn>2</mn> </mfrac> </mrow> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> </mtd> <mtd> <mo>−</mo> <mn>1</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> </mtd> <mtd> <mn>2</mn> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>2</mn> </mtd> </mtr> <mtr> <mtd> <mo>−</mo> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mn>3</mn> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>5</mn> </mtd> </mtr> <mtr> <mtd> <mn>2</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{pmatrix}x_{1}'\\x_{2}'\\x_{3}'\end{pmatrix}}={\begin{pmatrix}{\frac {3}{2}}&1&1\\{\frac {1}{2}}&-1&0\\-{\frac {1}{2}}&2&1\end{pmatrix}}{\begin{pmatrix}2\\-1\\3\end{pmatrix}}={\begin{pmatrix}5\\2\\0\end{pmatrix}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d5d0c841788cdbca3f497a26689c0ab9ae6d39e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.012ex; margin-bottom: -0.326ex; width:44.232ex; height:11.843ex;" alt="{\displaystyle {\begin{pmatrix}x_{1}'\\x_{2}'\\x_{3}'\end{pmatrix}}={\begin{pmatrix}{\frac {3}{2}}&1&1\\{\frac {1}{2}}&-1&0\\-{\frac {1}{2}}&2&1\end{pmatrix}}{\begin{pmatrix}2\\-1\\3\end{pmatrix}}={\begin{pmatrix}5\\2\\0\end{pmatrix}}}">.</dd></dl> Also ist <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v=5b_{1}'+2b_{2}'+0b_{3}'}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>v</mi> <mo>=</mo> <mn>5</mn> <msubsup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mo>′</mo> </msubsup> <mo>+</mo> <mn>2</mn> <msubsup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> <mo>′</mo> </msubsup> <mo>+</mo> <mn>0</mn> <msubsup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> <mo>′</mo> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v=5b_{1}'+2b_{2}'+0b_{3}'}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5b6c0992adcc11fb09d74b28953c6f94e9919893" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:19.55ex; height:2.843ex;" alt="{\displaystyle v=5b_{1}'+2b_{2}'+0b_{3}'}">. In der Tat rechnet man als Probe leicht nach, dass <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2b_{1}-b_{2}+3b_{3}=5b_{1}'+2b_{2}'+0b_{3}'}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>+</mo> <mn>3</mn> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>=</mo> <mn>5</mn> <msubsup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mo>′</mo> </msubsup> <mo>+</mo> <mn>2</mn> <msubsup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> <mo>′</mo> </msubsup> <mo>+</mo> <mn>0</mn> <msubsup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> <mo>′</mo> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2b_{1}-b_{2}+3b_{3}=5b_{1}'+2b_{2}'+0b_{3}'}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/38d38e8dfc3c3ea3c93e0a07ca2b6488ab1d6f9a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:32.583ex; height:2.843ex;" alt="{\displaystyle 2b_{1}-b_{2}+3b_{3}=5b_{1}'+2b_{2}'+0b_{3}'}"></dd></dl> gilt. <div class="mw-heading mw-heading2"><h2 id="Basiswechsel_mit_Hilfe_der_dualen_Basis">Basiswechsel mit Hilfe der dualen Basis</h2>[<a href="/w/index.php?title=Basiswechsel_(Vektorraum)&veaction=edit&section=6" title="Abschnitt bearbeiten: Basiswechsel mit Hilfe der dualen Basis" class="mw-editsection-visualeditor">Bearbeiten</a> | <a href="/w/index.php?title=Basiswechsel_(Vektorraum)&action=edit&section=6" title="Quellcode des Abschnitts bearbeiten: Basiswechsel mit Hilfe der dualen Basis">Quelltext bearbeiten</a>]</div> Im wichtigen und anschaulichen Spezialfall des <a href="/wiki/Euklidischer_Vektorraum" class="mw-redirect" title="Euklidischer Vektorraum">euklidischen Vektorraums</a> (V, ·) kann der Basiswechsel elegant mit der <a href="/wiki/Duale_Basis" title="Duale Basis">dualen Basis</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ({\vec {b}}^{1},\ldots ,{\vec {b}}^{n})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>b</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msup> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>b</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ({\vec {b}}^{1},\ldots ,{\vec {b}}^{n})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/57643d5c1c804a50d7d4a750cef29b8e4c6b8e2e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.448ex; height:3.843ex;" alt="{\displaystyle ({\vec {b}}^{1},\ldots ,{\vec {b}}^{n})}"> einer Basis <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ({\vec {b}}_{1},\ldots ,{\vec {b}}_{n})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>b</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>b</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ({\vec {b}}_{1},\ldots ,{\vec {b}}_{n})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f260ce39585cf8fbf2f2f1c11ae32444306f9fd9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.448ex; height:3.343ex;" alt="{\displaystyle ({\vec {b}}_{1},\ldots ,{\vec {b}}_{n})}"> durchgeführt werden. Für die Basisvektoren gilt dann <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {b}}_{i}\cdot {\vec {b}}^{j}=\delta _{i}^{j}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>b</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>⋅</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>b</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msup> <mo>=</mo> <msubsup> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msubsup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {b}}_{i}\cdot {\vec {b}}^{j}=\delta _{i}^{j}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5665b28f06ebe9fdc8a810a6019518448bd9856e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:11.285ex; height:4.009ex;" alt="{\displaystyle {\vec {b}}_{i}\cdot {\vec {b}}^{j}=\delta _{i}^{j}.}"></dd></dl> mit dem <a href="/wiki/Kronecker-Delta" title="Kronecker-Delta">Kronecker-Delta</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \delta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>δ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \delta }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c5321cfa797202b3e1f8620663ff43c4660ea03a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.049ex; height:2.343ex;" alt="{\displaystyle \delta }">. Skalare Multiplikation eines Vektors <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {v}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>v</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {v}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/85820588abd7333ef4d0c56539cb31c20e730753" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.175ex; height:2.343ex;" alt="{\displaystyle {\vec {v}}}"> mit den Basisvektoren <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {b}}^{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>b</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {b}}^{i}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1f29f4c2e0e44e24a2984f299c73fc13adc32ab5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.894ex; height:3.343ex;" alt="{\displaystyle {\vec {b}}^{i}}">, Multiplikation dieser Skalarprodukte mit den Basisvektoren <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {b}}_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>b</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {b}}_{i}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8b8d268dde2fbd9df47a558bb7723895939fb933" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.894ex; height:3.176ex;" alt="{\displaystyle {\vec {b}}_{i}}"> und Addition aller Gleichungen ergibt einen Vektor <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {w}}:=({\vec {b}}^{i}\cdot {\vec {v}}){\vec {b}}_{i}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>w</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>:=</mo> <mo stretchy="false">(</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>b</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msup> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>v</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo stretchy="false">)</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>b</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {w}}:=({\vec {b}}^{i}\cdot {\vec {v}}){\vec {b}}_{i}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ad5836c2cfa0c240cbc877d81f2c58c109cb7736" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.507ex; height:3.843ex;" alt="{\displaystyle {\vec {w}}:=({\vec {b}}^{i}\cdot {\vec {v}}){\vec {b}}_{i}.}"> Hier wie im Folgenden ist die <a href="/wiki/Einsteinsche_Summenkonvention" title="Einsteinsche Summenkonvention">Einsteinsche Summenkonvention</a> anzuwenden, der zufolge über in einem Produkt doppelt vorkommende Indizes, im vorhergehenden Satz beispielsweise nur <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>i</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/add78d8608ad86e54951b8c8bd6c8d8416533d20" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.802ex; height:2.176ex;" alt="{\displaystyle i}">, von eins bis <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><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}"> zu summieren ist. Skalare Multiplikation von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {w}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>w</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {w}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8b6c48cdaecf8d81481ea21b1d0c046bf34b68ec" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.664ex; height:2.343ex;" alt="{\displaystyle {\vec {w}}}"> mit irgendeinem Basisvektor <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {b}}^{k}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>b</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {b}}^{k}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/26e03e92f2f2f5535e30879f8e9fd10ec9c7cfbf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.183ex; height:3.509ex;" alt="{\displaystyle {\vec {b}}^{k}}"> ergibt wegen <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {w}}\cdot {\vec {b}}^{k}=({\vec {b}}^{i}\cdot {\vec {v}}){\vec {b}}_{i}\cdot {\vec {b}}^{k}=({\vec {b}}^{i}\cdot {\vec {v}})\delta _{i}^{k}={\vec {b}}^{k}\cdot {\vec {v}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>w</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>⋅</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>b</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> <mo>=</mo> <mo stretchy="false">(</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>b</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msup> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>v</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo stretchy="false">)</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>b</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>⋅</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>b</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> <mo>=</mo> <mo stretchy="false">(</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>b</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msup> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>v</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo stretchy="false">)</mo> <msubsup> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msubsup> <mo>=</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>b</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>v</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {w}}\cdot {\vec {b}}^{k}=({\vec {b}}^{i}\cdot {\vec {v}}){\vec {b}}_{i}\cdot {\vec {b}}^{k}=({\vec {b}}^{i}\cdot {\vec {v}})\delta _{i}^{k}={\vec {b}}^{k}\cdot {\vec {v}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a6ed4c49b2b30efc21212aa0006da3388303867d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:40.871ex; height:4.176ex;" alt="{\displaystyle {\vec {w}}\cdot {\vec {b}}^{k}=({\vec {b}}^{i}\cdot {\vec {v}}){\vec {b}}_{i}\cdot {\vec {b}}^{k}=({\vec {b}}^{i}\cdot {\vec {v}})\delta _{i}^{k}={\vec {b}}^{k}\cdot {\vec {v}}}"></dd></dl> dasselbe Ergebnis wie die skalare Multiplikation von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {v}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>v</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {v}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/85820588abd7333ef4d0c56539cb31c20e730753" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.175ex; height:2.343ex;" alt="{\displaystyle {\vec {v}}}"> mit diesem Basisvektor, weswegen die beiden Vektoren identisch sind: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {v}}=({\vec {b}}^{i}\cdot {\vec {v}}){\vec {b}}_{i}=:v^{i}{\vec {b}}_{i}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>v</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>=</mo> <mo stretchy="false">(</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>b</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msup> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>v</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo stretchy="false">)</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>b</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>=:</mo> <msup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msup> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>b</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {v}}=({\vec {b}}^{i}\cdot {\vec {v}}){\vec {b}}_{i}=:v^{i}{\vec {b}}_{i}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b1c6999b27b62c7c55bc1d5ccd3df80f7a713b3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.938ex; height:3.843ex;" alt="{\displaystyle {\vec {v}}=({\vec {b}}^{i}\cdot {\vec {v}}){\vec {b}}_{i}=:v^{i}{\vec {b}}_{i}.}"></dd></dl> Analog zeigt sich: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {v}}=({\vec {b}}_{i}\cdot {\vec {v}}){\vec {b}}^{i}=:v_{i}{\vec {b}}^{i}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>v</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>=</mo> <mo stretchy="false">(</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>b</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>v</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo stretchy="false">)</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>b</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msup> <mo>=:</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>b</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {v}}=({\vec {b}}_{i}\cdot {\vec {v}}){\vec {b}}^{i}=:v_{i}{\vec {b}}^{i}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/866a555043a589228407b37454185bd3a52bfd53" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.938ex; height:3.843ex;" alt="{\displaystyle {\vec {v}}=({\vec {b}}_{i}\cdot {\vec {v}}){\vec {b}}^{i}=:v_{i}{\vec {b}}^{i}.}"></dd></dl> Dieser Zusammenhang zwischen den Basisvektoren und einem Vektor, seinen Komponenten und Koordinaten, gilt für jeden Vektor im gegebenen Vektorraum. <div class="mw-heading mw-heading3"><h3 id="Wechsel_zur_dualen_Basis">Wechsel zur dualen Basis</h3>[<a href="/w/index.php?title=Basiswechsel_(Vektorraum)&veaction=edit&section=7" title="Abschnitt bearbeiten: Wechsel zur dualen Basis" class="mw-editsection-visualeditor">Bearbeiten</a> | <a href="/w/index.php?title=Basiswechsel_(Vektorraum)&action=edit&section=7" title="Quellcode des Abschnitts bearbeiten: Wechsel zur dualen Basis">Quelltext bearbeiten</a>]</div> Skalare Multiplikation beider Gleichungen mit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {b}}^{k}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>b</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {b}}^{k}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/26e03e92f2f2f5535e30879f8e9fd10ec9c7cfbf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.183ex; height:3.509ex;" alt="{\displaystyle {\vec {b}}^{k}}"> liefert <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v^{i}{\vec {b}}_{i}\cdot {\vec {b}}^{k}=v_{i}{\vec {b}}^{i}\cdot {\vec {b}}^{k}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msup> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>b</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>⋅</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>b</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> <mo>=</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>b</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msup> <mo>⋅</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>b</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v^{i}{\vec {b}}_{i}\cdot {\vec {b}}^{k}=v_{i}{\vec {b}}^{i}\cdot {\vec {b}}^{k}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5854dde6a8c143e7350cb019af7b779afea98658" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:18.464ex; height:3.843ex;" alt="{\displaystyle v^{i}{\vec {b}}_{i}\cdot {\vec {b}}^{k}=v_{i}{\vec {b}}^{i}\cdot {\vec {b}}^{k}}"> oder <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v^{k}=b^{ki}v_{i}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> <mo>=</mo> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mi>i</mi> </mrow> </msup> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v^{k}=b^{ki}v_{i}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1981e389aa562d2c7d620315fd5d8eb5d8810cda" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.543ex; height:3.009ex;" alt="{\displaystyle v^{k}=b^{ki}v_{i}.}"></dd></dl> Die Umkehroperation mit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {b}}_{k}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>b</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {b}}_{k}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7b92656c7a8794ce34ee3e686d1cc6501aa67efa" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.183ex; height:3.176ex;" alt="{\displaystyle {\vec {b}}_{k}}"> ist <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v_{k}=v^{i}{\vec {b}}_{i}\cdot {\vec {b}}_{k}=b_{ki}v^{i}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo>=</mo> <msup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msup> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>b</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>⋅</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>b</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mi>i</mi> </mrow> </msub> <msup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v_{k}=v^{i}{\vec {b}}_{i}\cdot {\vec {b}}_{k}=b_{ki}v^{i}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eda09e27d499b0ff9c7874d504085b2f47a6aca3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:21.324ex; height:3.176ex;" alt="{\displaystyle v_{k}=v^{i}{\vec {b}}_{i}\cdot {\vec {b}}_{k}=b_{ki}v^{i}.}"></dd></dl> Für die oben benutzten Skalarprodukte <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b_{ij}:={\vec {b}}_{i}\cdot {\vec {b}}_{j}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </msub> <mo>:=</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>b</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>⋅</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>b</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b_{ij}:={\vec {b}}_{i}\cdot {\vec {b}}_{j}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/37f2faf7bbf19c22751518e1bd3af4d7d6a60a65" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:11.796ex; height:3.509ex;" alt="{\displaystyle b_{ij}:={\vec {b}}_{i}\cdot {\vec {b}}_{j}}"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b^{kl}:={\vec {b}}^{k}\cdot {\vec {b}}^{l}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mi>l</mi> </mrow> </msup> <mo>:=</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>b</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> <mo>⋅</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>b</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>l</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b^{kl}:={\vec {b}}^{k}\cdot {\vec {b}}^{l}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6fd5726bde8439a6ebce52098041abbebc5fccac" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:12ex; height:3.509ex;" alt="{\displaystyle b^{kl}:={\vec {b}}^{k}\cdot {\vec {b}}^{l}}"> gilt: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b_{ik}b^{kj}=b^{jk}b_{ki}=({\vec {b}}^{j}\cdot {\vec {b}}^{k})({\vec {b}}_{k}\cdot {\vec {b}}_{i})=[({\vec {b}}^{j}\cdot {\vec {b}}^{k}){\vec {b}}_{k}]\cdot {\vec {b}}_{i}={\vec {b}}^{j}\cdot {\vec {b}}_{i}=\delta _{i}^{j}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>k</mi> </mrow> </msub> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mi>j</mi> </mrow> </msup> <mo>=</mo> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mi>k</mi> </mrow> </msup> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mi>i</mi> </mrow> </msub> <mo>=</mo> <mo stretchy="false">(</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>b</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msup> <mo>⋅</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>b</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>b</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo>⋅</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>b</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">[</mo> <mo stretchy="false">(</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>b</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msup> <mo>⋅</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>b</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> <mo stretchy="false">)</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>b</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo stretchy="false">]</mo> <mo>⋅</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>b</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>=</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>b</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msup> <mo>⋅</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>b</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>=</mo> <msubsup> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msubsup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b_{ik}b^{kj}=b^{jk}b_{ki}=({\vec {b}}^{j}\cdot {\vec {b}}^{k})({\vec {b}}_{k}\cdot {\vec {b}}_{i})=[({\vec {b}}^{j}\cdot {\vec {b}}^{k}){\vec {b}}_{k}]\cdot {\vec {b}}_{i}={\vec {b}}^{j}\cdot {\vec {b}}_{i}=\delta _{i}^{j}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a815ee39140c3225dfbedb0eef1b4cd7ca2bd05a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:64.477ex; height:4.176ex;" alt="{\displaystyle b_{ik}b^{kj}=b^{jk}b_{ki}=({\vec {b}}^{j}\cdot {\vec {b}}^{k})({\vec {b}}_{k}\cdot {\vec {b}}_{i})=[({\vec {b}}^{j}\cdot {\vec {b}}^{k}){\vec {b}}_{k}]\cdot {\vec {b}}_{i}={\vec {b}}^{j}\cdot {\vec {b}}_{i}=\delta _{i}^{j}.}"></dd></dl> <div class="mw-heading mw-heading3"><h3 id="Wechsel_zu_einer_anderen_Basis">Wechsel zu einer anderen Basis</h3>[<a href="/w/index.php?title=Basiswechsel_(Vektorraum)&veaction=edit&section=8" title="Abschnitt bearbeiten: Wechsel zu einer anderen Basis" class="mw-editsection-visualeditor">Bearbeiten</a> | <a href="/w/index.php?title=Basiswechsel_(Vektorraum)&action=edit&section=8" title="Quellcode des Abschnitts bearbeiten: Wechsel zu einer anderen Basis">Quelltext bearbeiten</a>]</div> Gegeben sei ein Vektor <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {v}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>v</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {v}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/85820588abd7333ef4d0c56539cb31c20e730753" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.175ex; height:2.343ex;" alt="{\displaystyle {\vec {v}}}">, der von einer Basis <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ({\vec {a}}_{1},\ldots ,{\vec {a}}_{n})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>a</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>a</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ({\vec {a}}_{1},\ldots ,{\vec {a}}_{n})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8783917e26ded4cfc81c4d0493e14ed66911d4af" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.72ex; height:2.843ex;" alt="{\displaystyle ({\vec {a}}_{1},\ldots ,{\vec {a}}_{n})}"> zur Basis <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ({\vec {b}}_{1},\ldots ,{\vec {b}}_{n})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>b</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>b</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ({\vec {b}}_{1},\ldots ,{\vec {b}}_{n})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f260ce39585cf8fbf2f2f1c11ae32444306f9fd9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.448ex; height:3.343ex;" alt="{\displaystyle ({\vec {b}}_{1},\ldots ,{\vec {b}}_{n})}"> wechseln soll. Das gelingt, indem jeder Basisvektor gemäß <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {a}}_{j}=({\vec {b}}^{i}\cdot {\vec {a}}_{j}){\vec {b}}_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>a</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo>=</mo> <mo stretchy="false">(</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>b</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msup> <mo>⋅</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>a</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo stretchy="false">)</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>b</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {a}}_{j}=({\vec {b}}^{i}\cdot {\vec {a}}_{j}){\vec {b}}_{i}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3b43b88852d132941bec00445aa1eeda305e7b71" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:14.653ex; height:4.009ex;" alt="{\displaystyle {\vec {a}}_{j}=({\vec {b}}^{i}\cdot {\vec {a}}_{j}){\vec {b}}_{i}}"> durch die neue Basis ausgedrückt wird: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {v}}=x_{j}{\vec {a}}_{j}=x_{j}({\vec {b}}^{i}\cdot {\vec {a}}_{j}){\vec {b}}_{i}=x_{i}^{\prime }{\vec {b}}_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>v</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>=</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>a</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo stretchy="false">(</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>b</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msup> <mo>⋅</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>a</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo stretchy="false">)</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>b</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>=</mo> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-variant" mathvariant="normal">′</mi> </mrow> </msubsup> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>b</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {v}}=x_{j}{\vec {a}}_{j}=x_{j}({\vec {b}}^{i}\cdot {\vec {a}}_{j}){\vec {b}}_{i}=x_{i}^{\prime }{\vec {b}}_{i}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3e405eb0fd9c6dd46bfcf0c1b2e58b1a5fbc6c00" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:30.527ex; height:4.009ex;" alt="{\displaystyle {\vec {v}}=x_{j}{\vec {a}}_{j}=x_{j}({\vec {b}}^{i}\cdot {\vec {a}}_{j}){\vec {b}}_{i}=x_{i}^{\prime }{\vec {b}}_{i}}"> mit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{i}^{\prime }:=({\vec {b}}^{i}\cdot {\vec {a}}_{j})x_{j}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-variant" mathvariant="normal">′</mi> </mrow> </msubsup> <mo>:=</mo> <mo stretchy="false">(</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>b</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msup> <mo>⋅</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>a</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo stretchy="false">)</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{i}^{\prime }:=({\vec {b}}^{i}\cdot {\vec {a}}_{j})x_{j}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5bee74ba1308e32ceae22083a59d5f45153ec10" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:16.282ex; height:4.009ex;" alt="{\displaystyle x_{i}^{\prime }:=({\vec {b}}^{i}\cdot {\vec {a}}_{j})x_{j}.}"></dd></dl> Die Umkehrung davon ist <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{i}:=({\vec {b}}_{i}\cdot {\vec {a}}^{j})x_{j}^{\prime }.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>:=</mo> <mo stretchy="false">(</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>b</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>⋅</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>a</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msup> <mo stretchy="false">)</mo> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-variant" mathvariant="normal">′</mi> </mrow> </msubsup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{i}:=({\vec {b}}_{i}\cdot {\vec {a}}^{j})x_{j}^{\prime }.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3c3ad6913a8f27bbd72c86e3ba88f0a249996f50" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:16.282ex; height:3.843ex;" alt="{\displaystyle x_{i}:=({\vec {b}}_{i}\cdot {\vec {a}}^{j})x_{j}^{\prime }.}"> Der Basiswechsel bei Tensoren zweiter Stufe wird analog durchgeführt: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {M} :=M_{ij}{\vec {a}}_{i}\otimes {\vec {b}}_{j}:=M_{ij}[({\vec {c}}^{k}\cdot {\vec {a}}_{i}){\vec {c}}_{k}]\otimes [({\vec {d}}^{l}\cdot {\vec {b}}_{j}){\vec {d}}_{l}]:=M_{kl}^{\prime }{\vec {c}}_{k}\otimes {\vec {d}}_{l}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">M</mi> </mrow> <mo>:=</mo> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </msub> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>a</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>⊗</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>b</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo>:=</mo> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </msub> <mo stretchy="false">[</mo> <mo stretchy="false">(</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>c</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> <mo>⋅</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>a</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">)</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>c</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo stretchy="false">]</mo> <mo>⊗</mo> <mo stretchy="false">[</mo> <mo stretchy="false">(</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>d</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>l</mi> </mrow> </msup> <mo>⋅</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>b</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo stretchy="false">)</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>d</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>l</mi> </mrow> </msub> <mo stretchy="false">]</mo> <mo>:=</mo> <msubsup> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mi>l</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-variant" mathvariant="normal">′</mi> </mrow> </msubsup> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>c</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo>⊗</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>d</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>l</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {M} :=M_{ij}{\vec {a}}_{i}\otimes {\vec {b}}_{j}:=M_{ij}[({\vec {c}}^{k}\cdot {\vec {a}}_{i}){\vec {c}}_{k}]\otimes [({\vec {d}}^{l}\cdot {\vec {b}}_{j}){\vec {d}}_{l}]:=M_{kl}^{\prime }{\vec {c}}_{k}\otimes {\vec {d}}_{l}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/10badc3f0706b8fb6e32a1a7d01844d6fdab7267" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:64.965ex; height:4.176ex;" alt="{\displaystyle \mathbf {M} :=M_{ij}{\vec {a}}_{i}\otimes {\vec {b}}_{j}:=M_{ij}[({\vec {c}}^{k}\cdot {\vec {a}}_{i}){\vec {c}}_{k}]\otimes [({\vec {d}}^{l}\cdot {\vec {b}}_{j}){\vec {d}}_{l}]:=M_{kl}^{\prime }{\vec {c}}_{k}\otimes {\vec {d}}_{l}}"> mit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M_{kl}^{\prime }=({\vec {c}}^{k}\cdot {\vec {a}}_{i})M_{ij}({\vec {d}}^{l}\cdot {\vec {b}}_{j})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mi>l</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-variant" mathvariant="normal">′</mi> </mrow> </msubsup> <mo>=</mo> <mo stretchy="false">(</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>c</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> <mo>⋅</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>a</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">)</mo> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </msub> <mo stretchy="false">(</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>d</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>l</mi> </mrow> </msup> <mo>⋅</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>b</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M_{kl}^{\prime }=({\vec {c}}^{k}\cdot {\vec {a}}_{i})M_{ij}({\vec {d}}^{l}\cdot {\vec {b}}_{j})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6e2c1816e426b07fbac8a0b0c61f30221a441554" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:26.254ex; height:4.176ex;" alt="{\displaystyle M_{kl}^{\prime }=({\vec {c}}^{k}\cdot {\vec {a}}_{i})M_{ij}({\vec {d}}^{l}\cdot {\vec {b}}_{j})}"></dd></dl> was sich ohne weiteres auf Tensoren höherer Stufe verallgemeinern lässt. Das Rechenzeichen „<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \otimes }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>⊗</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \otimes }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/de29098f5a34ee296a505681a0d5e875070f2aea" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:1.808ex; height:2.176ex;" alt="{\displaystyle \otimes }">“ bildet das <a href="/wiki/Dyadisches_Produkt" title="Dyadisches Produkt">dyadische Produkt</a>. Der Zusammenhang zwischen den Koordinaten <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{i}^{\prime }:=({\vec {b}}^{i}\cdot {\vec {a}}_{j})x_{j}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-variant" mathvariant="normal">′</mi> </mrow> </msubsup> <mo>:=</mo> <mo stretchy="false">(</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>b</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msup> <mo>⋅</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>a</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo stretchy="false">)</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{i}^{\prime }:=({\vec {b}}^{i}\cdot {\vec {a}}_{j})x_{j}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8ad6c62a0e8e42528956be1b8faa045504c0f73b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:15.636ex; height:4.009ex;" alt="{\displaystyle x_{i}^{\prime }:=({\vec {b}}^{i}\cdot {\vec {a}}_{j})x_{j}}"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M_{kl}^{\prime }=({\vec {c}}^{k}\cdot {\vec {a}}_{i})M_{ij}({\vec {d}}^{l}\cdot {\vec {b}}_{j})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mi>l</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-variant" mathvariant="normal">′</mi> </mrow> </msubsup> <mo>=</mo> <mo stretchy="false">(</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>c</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> <mo>⋅</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>a</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">)</mo> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </msub> <mo stretchy="false">(</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>d</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>l</mi> </mrow> </msup> <mo>⋅</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>b</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M_{kl}^{\prime }=({\vec {c}}^{k}\cdot {\vec {a}}_{i})M_{ij}({\vec {d}}^{l}\cdot {\vec {b}}_{j})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6e2c1816e426b07fbac8a0b0c61f30221a441554" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:26.254ex; height:4.176ex;" alt="{\displaystyle M_{kl}^{\prime }=({\vec {c}}^{k}\cdot {\vec {a}}_{i})M_{ij}({\vec {d}}^{l}\cdot {\vec {b}}_{j})}"></dd></dl> kann kompakt mit Basiswechselmatrizen <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T_{Q}^{P}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>Q</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>P</mi> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T_{Q}^{P}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3030c9546b3643759d65a7bce2f103112b79bbf0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:3.186ex; height:3.509ex;" alt="{\displaystyle T_{Q}^{P}}"> mit den Komponenten <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (T_{Q}^{P})_{ij}={\vec {q}}^{i}\cdot {\vec {p}}_{j}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msubsup> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>Q</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>P</mi> </mrow> </msubsup> <msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </msub> <mo>=</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>q</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msup> <mo>⋅</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>p</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (T_{Q}^{P})_{ij}={\vec {q}}^{i}\cdot {\vec {p}}_{j}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b5d68bb30210e1038d44551828aa87a183a538b7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:15.594ex; height:3.843ex;" alt="{\displaystyle (T_{Q}^{P})_{ij}={\vec {q}}^{i}\cdot {\vec {p}}_{j}}"> bei einem Basiswechsel von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ({\vec {p}}_{1},\ldots ,{\vec {p}}_{n})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>p</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>p</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ({\vec {p}}_{1},\ldots ,{\vec {p}}_{n})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b5b026a98ed225f2f62e8eca5ebe72bbd09306a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.911ex; height:2.843ex;" alt="{\displaystyle ({\vec {p}}_{1},\ldots ,{\vec {p}}_{n})}"> nach <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ({\vec {q}}_{1},\ldots ,{\vec {q}}_{n})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>q</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>q</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ({\vec {q}}_{1},\ldots ,{\vec {q}}_{n})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/99d9421bfcbcc3929e7495a93b023ce33dd1a72e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.877ex; height:2.843ex;" alt="{\displaystyle ({\vec {q}}_{1},\ldots ,{\vec {q}}_{n})}"> und ihren dualen Partnern dargestellt werden. Die <a href="/wiki/Inverse_Matrix" title="Inverse Matrix">Inverse</a> der Basiswechselmatrix hat, wie oben angedeutet, die Komponenten <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (T_{Q}^{P})_{ij}^{-1}={\vec {p}}^{i}\cdot {\vec {q}}_{j},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msubsup> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>Q</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>P</mi> </mrow> </msubsup> <msubsup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msubsup> <mo>=</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>p</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msup> <mo>⋅</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>q</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (T_{Q}^{P})_{ij}^{-1}={\vec {p}}^{i}\cdot {\vec {q}}_{j},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/65b0f0598a3ff91d41991e9aeeacfecd70b7e5fd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:17.096ex; height:3.843ex;" alt="{\displaystyle (T_{Q}^{P})_{ij}^{-1}={\vec {p}}^{i}\cdot {\vec {q}}_{j},}"> denn bei der Matrizenmultiplikation ergibt sich für Komponenten <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ij}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>i</mi> <mi>j</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ij}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/53fcc7b57da64979c370eb150eb5a61a625a08e8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.761ex; height:2.509ex;" alt="{\displaystyle ij}">: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [T_{Q}^{P}\cdot (T_{Q}^{P})^{-1}]_{ij}=({\vec {q}}^{i}\cdot {\vec {p}}_{k})({\vec {p}}^{k}\cdot {\vec {q}}_{j})=[({\vec {q}}^{i}\cdot {\vec {p}}_{k}){\vec {p}}^{k}]\cdot {\vec {q}}_{j}={\vec {q}}^{i}\cdot {\vec {q}}_{j}=\delta _{j}^{i}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">[</mo> <msubsup> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>Q</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>P</mi> </mrow> </msubsup> <mo>⋅</mo> <mo stretchy="false">(</mo> <msubsup> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>Q</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>P</mi> </mrow> </msubsup> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <msub> <mo stretchy="false">]</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </msub> <mo>=</mo> <mo stretchy="false">(</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>q</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msup> <mo>⋅</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>p</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>p</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> <mo>⋅</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>q</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">[</mo> <mo stretchy="false">(</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>q</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msup> <mo>⋅</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>p</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo stretchy="false">)</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>p</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> <mo stretchy="false">]</mo> <mo>⋅</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>q</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo>=</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>q</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msup> <mo>⋅</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>q</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo>=</mo> <msubsup> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msubsup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle [T_{Q}^{P}\cdot (T_{Q}^{P})^{-1}]_{ij}=({\vec {q}}^{i}\cdot {\vec {p}}_{k})({\vec {p}}^{k}\cdot {\vec {q}}_{j})=[({\vec {q}}^{i}\cdot {\vec {p}}_{k}){\vec {p}}^{k}]\cdot {\vec {q}}_{j}={\vec {q}}^{i}\cdot {\vec {q}}_{j}=\delta _{j}^{i}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b7b992074529b432635274ebffaae3a6f9cd5cab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:67.7ex; height:3.843ex;" alt="{\displaystyle [T_{Q}^{P}\cdot (T_{Q}^{P})^{-1}]_{ij}=({\vec {q}}^{i}\cdot {\vec {p}}_{k})({\vec {p}}^{k}\cdot {\vec {q}}_{j})=[({\vec {q}}^{i}\cdot {\vec {p}}_{k}){\vec {p}}^{k}]\cdot {\vec {q}}_{j}={\vec {q}}^{i}\cdot {\vec {q}}_{j}=\delta _{j}^{i}.}"></dd></dl> <div class="mw-heading mw-heading2"><h2 id="Anwendungen">Anwendungen</h2>[<a href="/w/index.php?title=Basiswechsel_(Vektorraum)&veaction=edit&section=9" title="Abschnitt bearbeiten: Anwendungen" class="mw-editsection-visualeditor">Bearbeiten</a> | <a href="/w/index.php?title=Basiswechsel_(Vektorraum)&action=edit&section=9" title="Quellcode des Abschnitts bearbeiten: Anwendungen">Quelltext bearbeiten</a>]</div> Basiswechselmatrizen besitzen vielfältige Anwendungsmöglichkeiten in der Mathematik und Physik. <div class="mw-heading mw-heading3"><h3 id="In_der_Mathematik">In der Mathematik</h3>[<a href="/w/index.php?title=Basiswechsel_(Vektorraum)&veaction=edit&section=10" title="Abschnitt bearbeiten: In der Mathematik" class="mw-editsection-visualeditor">Bearbeiten</a> | <a href="/w/index.php?title=Basiswechsel_(Vektorraum)&action=edit&section=10" title="Quellcode des Abschnitts bearbeiten: In der Mathematik">Quelltext bearbeiten</a>]</div> Eine Anwendung von Basiswechselmatrizen in der Mathematik ist die Veränderung der Gestalt der Abbildungsmatrix einer linearen Abbildung, um die Rechnung zu vereinfachen. Möchte man zum Beispiel die Potenz <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A^{p}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A^{p}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/531f6b3fa540ff52e0b045dabf6d14d0be92247c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.802ex; height:2.343ex;" alt="{\displaystyle A^{p}}"> einer <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n\times n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>×</mo> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n\times n}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/59d2b4cb72e304526cf5b5887147729ea259da78" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.63ex; height:1.676ex;" alt="{\displaystyle n\times n}">-Matrix <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"> mit einem Exponenten <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p>1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mo>></mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p>1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0f127e7a5f2449ddf3edb8164c2d2439120641f9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:5.52ex; height:2.509ex;" alt="{\displaystyle p>1}"> berechnen, so ist die Zahl der benötigten Matrizenmultiplikationen von der Größenordnung <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle O(\log p)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>O</mi> <mo stretchy="false">(</mo> <mi>log</mi> <mo>⁡</mo> <mi>p</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle O(\log p)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1dd864c9c3116369675faa60061e985e0ec11760" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.111ex; height:2.843ex;" alt="{\displaystyle O(\log p)}">. Ist <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"> diagonalisierbar, so existieren eine <a href="/wiki/Diagonalmatrix" title="Diagonalmatrix">Diagonalmatrix</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>D</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle D}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f34a0c600395e5d4345287e21fb26efd386990e6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.924ex; height:2.176ex;" alt="{\displaystyle D}"> und eine Basiswechselmatrix <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T\in Gl\left(n,K\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> <mo>∈</mo> <mi>G</mi> <mi>l</mi> <mrow> <mo>(</mo> <mrow> <mi>n</mi> <mo>,</mo> <mi>K</mi> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T\in Gl\left(n,K\right)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/82e001847c95dd0d9243c2092e7c77b0fb8073ab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.688ex; height:2.843ex;" alt="{\displaystyle T\in Gl\left(n,K\right)}">, sodass <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A=T\cdot D\cdot T^{-1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>=</mo> <mi>T</mi> <mo>⋅</mo> <mi>D</mi> <mo>⋅</mo> <msup> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A=T\cdot D\cdot T^{-1}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1183ba4192dd415f611f822c0043be867bb81b6d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:15.813ex; height:2.676ex;" alt="{\displaystyle A=T\cdot D\cdot T^{-1}}"> und somit <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A^{p}=\left(T\cdot D\cdot T^{-1}\right)^{p}=T\cdot D^{p}\cdot T^{-1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msup> <mo>=</mo> <msup> <mrow> <mo>(</mo> <mrow> <mi>T</mi> <mo>⋅</mo> <mi>D</mi> <mo>⋅</mo> <msup> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msup> <mo>=</mo> <mi>T</mi> <mo>⋅</mo> <msup> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msup> <mo>⋅</mo> <msup> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A^{p}=\left(T\cdot D\cdot T^{-1}\right)^{p}=T\cdot D^{p}\cdot T^{-1}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4d55e52e2d64a9c650b0ea4f90e50dadf6bba4cb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:35.19ex; height:3.343ex;" alt="{\displaystyle A^{p}=\left(T\cdot D\cdot T^{-1}\right)^{p}=T\cdot D^{p}\cdot T^{-1}}"></dd></dl> Die Zahl der für die Berechnung der rechten Seite benötigten Multiplikationen ist nur von der Größenordnung: <ul><li><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n\log p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mi>log</mi> <mo>⁡</mo> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n\log p}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d5d700d07dc95463cc60f63bf18347d8be425217" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.31ex; height:2.509ex;" alt="{\displaystyle n\log p}"> zur Berechnung von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D^{p}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle D^{p}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/db4ed76952aa125fb95c69be6c96b92ab852bcac" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.983ex; height:2.343ex;" alt="{\displaystyle D^{p}}">,</li> <li><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n^{2}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ac9810bbdafe4a6a8061338db0f74e25b7952620" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.449ex; height:2.676ex;" alt="{\displaystyle n^{2}}"> zur Berechnung des Produkts <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D^{p}\cdot T^{-1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msup> <mo>⋅</mo> <msup> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle D^{p}\cdot T^{-1}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/97137edcb2b0c6a6d3356f8727d091ac338feccf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.715ex; height:2.676ex;" alt="{\displaystyle D^{p}\cdot T^{-1}}"></li> <li>sowie einer Matrixmultiplikation für das Produkt <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T\cdot (D^{p}T^{-1})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> <mo>⋅</mo> <mo stretchy="false">(</mo> <msup> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msup> <msup> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T\cdot (D^{p}T^{-1})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/347e86fbd0bb7140cae4c54e597200f6eb8bd523" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.161ex; height:3.176ex;" alt="{\displaystyle T\cdot (D^{p}T^{-1})}"></li></ul> Da die <a href="/wiki/Matrixmultiplikation#Algorithmen_mit_besserer_Komplexität" class="mw-redirect" title="Matrixmultiplikation">Matrixmultiplikation</a> von der Größenordnung <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle O\left(n^{2{,}3727}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>O</mi> <mrow> <mo>(</mo> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2,372</mn> <mn>7</mn> </mrow> </msup> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle O\left(n^{2{,}3727}\right)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fc7a313b1cf6d14657ae84c69200d9b500e9798f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:10.484ex; height:3.343ex;" alt="{\displaystyle O\left(n^{2{,}3727}\right)}"> ist, erhalten wir eine Komplexität von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle O\left(n^{2{,}3727}+n\cdot \log(p)\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>O</mi> <mrow> <mo>(</mo> <mrow> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2,372</mn> <mn>7</mn> </mrow> </msup> <mo>+</mo> <mi>n</mi> <mo>⋅</mo> <mi>log</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>p</mi> <mo stretchy="false">)</mo> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle O\left(n^{2{,}3727}+n\cdot \log(p)\right)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dafcc6fcc678d0dc2aa27986835c46afc8670014" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:22.349ex; height:3.343ex;" alt="{\displaystyle O\left(n^{2{,}3727}+n\cdot \log(p)\right)}"> anstelle von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle O\left(n^{2{,}3727}\cdot \log(p)\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>O</mi> <mrow> <mo>(</mo> <mrow> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2,372</mn> <mn>7</mn> </mrow> </msup> <mo>⋅</mo> <mi>log</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>p</mi> <mo stretchy="false">)</mo> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle O\left(n^{2{,}3727}\cdot \log(p)\right)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/82c57454097f40fbfa33a09bd1f5a168234e3059" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:18.114ex; height:3.343ex;" alt="{\displaystyle O\left(n^{2{,}3727}\cdot \log(p)\right)}">. <div class="mw-heading mw-heading3"><h3 id="In_der_Physik">In der Physik</h3>[<a href="/w/index.php?title=Basiswechsel_(Vektorraum)&veaction=edit&section=11" title="Abschnitt bearbeiten: In der Physik" class="mw-editsection-visualeditor">Bearbeiten</a> | <a href="/w/index.php?title=Basiswechsel_(Vektorraum)&action=edit&section=11" title="Quellcode des Abschnitts bearbeiten: In der Physik">Quelltext bearbeiten</a>]</div> Eine Anwendung von Basiswechselmatrizen in der Physik findet bspw. in der <a href="/wiki/%C3%84hnlichkeitstheorie" title="Ähnlichkeitstheorie">Ähnlichkeitstheorie</a> statt, um <a href="/wiki/Dimensionslose_Kennzahl" title="Dimensionslose Kennzahl">dimensionslose Kennzahlen</a> zu ermitteln. Hierbei werden durch einen Basiswechsel einer physikalischen Größe neue Basisdimensionen zugeordnet. Die dimensionslosen Kennzahlen stellen dann genau das Verhältnis der physikalischen Größe zu seiner Dimensionsvorschrift dar. <div class="mw-heading mw-heading2"><h2 id="Literatur">Literatur</h2>[<a href="/w/index.php?title=Basiswechsel_(Vektorraum)&veaction=edit&section=12" title="Abschnitt bearbeiten: Literatur" class="mw-editsection-visualeditor">Bearbeiten</a> | <a href="/w/index.php?title=Basiswechsel_(Vektorraum)&action=edit&section=12" title="Quellcode des Abschnitts bearbeiten: Literatur">Quelltext bearbeiten</a>]</div> <ul><li><a href="/wiki/Peter_Knabner" title="Peter Knabner">Peter Knabner</a>, <a href="/wiki/Wolf_Barth_(Mathematiker)" title="Wolf Barth (Mathematiker)">Wolf Barth</a>: <cite style="font-style:italic">Lineare Algebra. Grundlagen und Anwendungen</cite>. Springer Spektrum, Berlin/Heidelberg 2013, <a href="/wiki/Spezial:ISBN-Suche/9783642321856" class="internal mw-magiclink-isbn">ISBN 978-3-642-32185-6</a>.<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rfr_id=info:sid/de.wikipedia.org:Basiswechsel+%28Vektorraum%29&rft.au=Peter+Knabner%2C+Wolf+Barth&rft.btitle=Lineare+Algebra.+Grundlagen+und+Anwendungen&rft.date=2013&rft.genre=book&rft.isbn=9783642321856&rft.place=Berlin%2FHeidelberg&rft.pub=Springer+Spektrum" style="display:none"> </li> <li>Gerd Fischer: <cite style="font-style:italic">Lineare Algebra: Eine Einführung für Studienanfänger</cite>. 13. Auflage. Vieweg, 2002, <a href="/wiki/Spezial:ISBN-Suche/3528972173" class="internal mw-magiclink-isbn">ISBN 3-528-97217-3</a>.<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rfr_id=info:sid/de.wikipedia.org:Basiswechsel+%28Vektorraum%29&rft.au=Gerd+Fischer&rft.btitle=Lineare+Algebra%3A+Eine+Einf%C3%BChrung+f%C3%BCr+Studienanf%C3%A4nger&rft.date=2002&rft.edition=13&rft.genre=book&rft.isbn=3528972173&rft.pub=Vieweg" style="display:none"> </li></ul> <div class="mw-heading mw-heading2"><h2 id="Weblinks">Weblinks</h2>[<a href="/w/index.php?title=Basiswechsel_(Vektorraum)&veaction=edit&section=13" title="Abschnitt bearbeiten: Weblinks" class="mw-editsection-visualeditor">Bearbeiten</a> | <a href="/w/index.php?title=Basiswechsel_(Vektorraum)&action=edit&section=13" title="Quellcode des Abschnitts bearbeiten: Weblinks">Quelltext bearbeiten</a>]</div> <div class="sisterproject" style="margin:0.1em 0 0 0;"><div class="noviewer" style="display:inline-block; line-height:10px; min-width:1.6em; text-align:center;" aria-hidden="true" role="presentation"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/0/0b/Wikiversity_logo_2017.svg/20px-Wikiversity_logo_2017.svg.png" decoding="async" width="20" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/0b/Wikiversity_logo_2017.svg/29px-Wikiversity_logo_2017.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/0b/Wikiversity_logo_2017.svg/39px-Wikiversity_logo_2017.svg.png 2x" data-file-width="626" data-file-height="512" /></div><a href="https://de.wikiversity.org/wiki/Kurs:Lineare_Algebra_(Osnabr%C3%BCck_2015-2016)/Teil_I/Vorlesung_9" class="extiw" title="v:Kurs:Lineare Algebra (Osnabrück 2015-2016)/Teil I/Vorlesung 9">Wikiversity: Vorlesung zu Basiswechsel</a> – Kursmaterialien</div> <ul><li><a rel="nofollow" class="external text" href="https://mathepedia.de/Basiswechsel.html">Seite mit ausführlichen Beispielen</a></li></ul> <div class="hintergrundfarbe1 rahmenfarbe1 navigation-not-searchable normdaten-typ-s" style="border-style: solid; border-width: 1px; clear: left; margin-bottom:1em; margin-top:1em; padding: 0.25em; overflow: hidden; word-break: break-word; word-wrap: break-word;" id="normdaten"> <div style="display: table-cell; vertical-align: middle; width: 100%;"> <div> Normdaten (Sachbegriff): <a href="/wiki/Gemeinsame_Normdatei" title="Gemeinsame Normdatei">GND</a>: <a rel="nofollow" class="external text" href="https://d-nb.info/gnd/4144107-2">4144107-2</a> (<a rel="nofollow" class="external text" href="https://lobid.org/gnd/4144107-2">lobid</a>, <a rel="nofollow" class="external text" href="https://swb.bsz-bw.de/DB=2.104/SET=1/TTL=1/CMD?retrace=0&trm_old=&ACT=SRCHA&IKT=2999&SRT=RLV&TRM=4144107-2">OGND</a>, <a rel="nofollow" class="external text" href="https://prometheus.lmu.de/gnd/4144107-2">AKS</a>) </div> </div></div></div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?useformat=desktop&type=1x1&usesul3=0" alt="" width="1" height="1" style="border: none; position: absolute;"></noscript> <div class="printfooter" data-nosnippet="">Abgerufen von „<a dir="ltr" href="https://de.wikipedia.org/w/index.php?title=Basiswechsel_(Vektorraum)&oldid=221424393">https://de.wikipedia.org/w/index.php?title=Basiswechsel_(Vektorraum)&oldid=221424393</a>“</div></div> <div id="catlinks" class="catlinks" data-mw="interface"><div id="mw-normal-catlinks" class="mw-normal-catlinks"><a href="/wiki/Wikipedia:Kategorien" title="Wikipedia:Kategorien">Kategorie</a>: <ul><li><a href="/wiki/Kategorie:Lineare_Algebra" title="Kategorie:Lineare Algebra">Lineare Algebra</a></li></ul></div></div> </div> </div> <div id="mw-navigation"> <h2>Navigationsmenü</h2> <div id="mw-head"> <nav id="p-personal" class="mw-portlet mw-portlet-personal vector-user-menu-legacy vector-menu" aria-labelledby="p-personal-label" > <h3 id="p-personal-label" class="vector-menu-heading " > Meine Werkzeuge </h3> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="pt-anonuserpage" class="mw-list-item">Nicht angemeldet</li><li id="pt-anontalk" class="mw-list-item"><a href="/wiki/Spezial:Meine_Diskussionsseite" title="Diskussion über Änderungen von dieser IP-Adresse [n]" accesskey="n">Diskussionsseite</a></li><li id="pt-anoncontribs" class="mw-list-item"><a href="/wiki/Spezial:Meine_Beitr%C3%A4ge" title="Eine Liste der Bearbeitungen, die von dieser IP-Adresse gemacht wurden [y]" accesskey="y">Beiträge</a></li><li id="pt-createaccount" class="mw-list-item"><a href="/w/index.php?title=Spezial:Benutzerkonto_anlegen&returnto=Basiswechsel+%28Vektorraum%29" title="Wir ermutigen dich dazu, ein Benutzerkonto zu erstellen und dich anzumelden. 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interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D8%AA%D8%BA%D9%8A%D9%8A%D8%B1_%D8%A7%D9%84%D9%82%D8%A7%D8%B9%D8%AF%D8%A9_(%D8%AC%D8%A8%D8%B1_%D8%AE%D8%B7%D9%8A)" title="تغيير القاعدة (جبر خطي) – Arabisch" lang="ar" hreflang="ar" data-title="تغيير القاعدة (جبر خطي)" data-language-autonym="العربية" data-language-local-name="Arabisch" class="interlanguage-link-target">العربية</a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Canvi_de_base" title="Canvi de base – Katalanisch" lang="ca" hreflang="ca" data-title="Canvi de base" data-language-autonym="Català" data-language-local-name="Katalanisch" class="interlanguage-link-target">Català</a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Matice_p%C5%99echodu" title="Matice přechodu – Tschechisch" lang="cs" hreflang="cs" data-title="Matice přechodu" data-language-autonym="Čeština" data-language-local-name="Tschechisch" class="interlanguage-link-target">Čeština</a></li><li class="interlanguage-link interwiki-en mw-list-item"><a href="https://en.wikipedia.org/wiki/Change_of_basis" title="Change of basis – Englisch" lang="en" hreflang="en" data-title="Change of basis" data-language-autonym="English" data-language-local-name="Englisch" class="interlanguage-link-target">English</a></li><li class="interlanguage-link interwiki-eo mw-list-item"><a href="https://eo.wikipedia.org/wiki/%C5%9Can%C4%9Do_de_bazo" title="Ŝanĝo de bazo – Esperanto" lang="eo" hreflang="eo" data-title="Ŝanĝo de bazo" data-language-autonym="Esperanto" data-language-local-name="Esperanto" class="interlanguage-link-target">Esperanto</a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Cambio_de_base" title="Cambio de base – Spanisch" lang="es" hreflang="es" data-title="Cambio de base" data-language-autonym="Español" data-language-local-name="Spanisch" class="interlanguage-link-target">Español</a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/Kannanvaihto" title="Kannanvaihto – Finnisch" lang="fi" hreflang="fi" data-title="Kannanvaihto" data-language-autonym="Suomi" data-language-local-name="Finnisch" class="interlanguage-link-target">Suomi</a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Changement_de_base_(alg%C3%A8bre_lin%C3%A9aire)" title="Changement de base (algèbre linéaire) – Französisch" lang="fr" hreflang="fr" data-title="Changement de base (algèbre linéaire)" data-language-autonym="Français" data-language-local-name="Französisch" class="interlanguage-link-target">Français</a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%9E%D7%98%D7%A8%D7%99%D7%A6%D7%AA_%D7%9E%D7%A2%D7%91%D7%A8" title="מטריצת מעבר – Hebräisch" lang="he" hreflang="he" data-title="מטריצת מעבר" data-language-autonym="עברית" data-language-local-name="Hebräisch" class="interlanguage-link-target">עברית</a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/B%C3%A1zistranszform%C3%A1ci%C3%B3" title="Bázistranszformáció – Ungarisch" lang="hu" hreflang="hu" data-title="Bázistranszformáció" data-language-autonym="Magyar" data-language-local-name="Ungarisch" class="interlanguage-link-target">Magyar</a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Perubahan_basis" title="Perubahan basis – Indonesisch" lang="id" hreflang="id" data-title="Perubahan basis" data-language-autonym="Bahasa Indonesia" data-language-local-name="Indonesisch" class="interlanguage-link-target">Bahasa Indonesia</a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Matrice_di_cambiamento_di_base" title="Matrice di cambiamento di base – Italienisch" lang="it" hreflang="it" data-title="Matrice di cambiamento di base" data-language-autonym="Italiano" data-language-local-name="Italienisch" class="interlanguage-link-target">Italiano</a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E5%9F%BA%E5%BA%95%E5%A4%89%E6%8F%9B" title="基底変換 – Japanisch" lang="ja" hreflang="ja" data-title="基底変換" data-language-autonym="日本語" data-language-local-name="Japanisch" class="interlanguage-link-target">日本語</a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Basistransformatie" title="Basistransformatie – Niederländisch" lang="nl" hreflang="nl" data-title="Basistransformatie" data-language-autonym="Nederlands" data-language-local-name="Niederländisch" class="interlanguage-link-target">Nederlands</a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Mudan%C3%A7a_de_base" title="Mudança de base – Portugiesisch" lang="pt" hreflang="pt" data-title="Mudança de base" data-language-autonym="Português" data-language-local-name="Portugiesisch" class="interlanguage-link-target">Português</a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Basbyte" title="Basbyte – Schwedisch" lang="sv" hreflang="sv" data-title="Basbyte" data-language-autonym="Svenska" data-language-local-name="Schwedisch" class="interlanguage-link-target">Svenska</a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%97%D0%BC%D1%96%D0%BD%D0%B0_%D0%B1%D0%B0%D0%B7%D0%B8%D1%81%D1%83" title="Зміна базису – Ukrainisch" lang="uk" hreflang="uk" data-title="Зміна базису" data-language-autonym="Українська" data-language-local-name="Ukrainisch" class="interlanguage-link-target">Українська</a></li><li class="interlanguage-link interwiki-ur mw-list-item"><a href="https://ur.wikipedia.org/wiki/%D8%AA%D8%A8%D8%AF%DB%8C%D9%84%DB%8C_%D8%A7%D8%B2_%D8%A8%D9%86%DB%8C%D8%A7%D8%AF_%D8%B3%D9%85%D8%AA%DB%8C%DB%81" title="تبدیلی از بنیاد سمتیہ – Urdu" lang="ur" hreflang="ur" data-title="تبدیلی از بنیاد سمتیہ" data-language-autonym="اردو" data-language-local-name="Urdu" class="interlanguage-link-target">اردو</a></li><li class="interlanguage-link interwiki-vi mw-list-item"><a href="https://vi.wikipedia.org/wiki/Ph%C3%A9p_chuy%E1%BB%83n_c%C6%A1_s%E1%BB%9F" title="Phép chuyển cơ sở – Vietnamesisch" lang="vi" hreflang="vi" data-title="Phép chuyển cơ sở" data-language-autonym="Tiếng Việt" data-language-local-name="Vietnamesisch" class="interlanguage-link-target">Tiếng Việt</a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/%E5%9F%BA%E5%8F%98%E6%9B%B4" title="基变更 – Chinesisch" lang="zh" hreflang="zh" data-title="基变更" data-language-autonym="中文" data-language-local-name="Chinesisch" 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Basiswechsel (Vektorraum) – Wikipedia