Lineare Unabhängigkeit

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href="//de.wikipedia.org/w/api.php?action=rsd"> <link rel="canonical" href="https://de.wikipedia.org/wiki/Lineare_Unabh%C3%A4ngigkeit"> <link rel="license" href="https://creativecommons.org/licenses/by-sa/4.0/deed.de"> <link rel="alternate" type="application/atom+xml" title="Atom-Feed für „Wikipedia“" href="/w/index.php?title=Spezial:Letzte_%C3%84nderungen&feed=atom"> <link rel="dns-prefetch" href="//meta.wikimedia.org" /> <link rel="dns-prefetch" href="//login.wikimedia.org"> </head> <body class="skin-vector-legacy mediawiki ltr sitedir-ltr mw-hide-empty-elt ns-0 ns-subject mw-editable page-Lineare_Unabhängigkeit rootpage-Lineare_Unabhängigkeit skin-vector action-view"><div id="mw-page-base" class="noprint"></div> <div id="mw-head-base" class="noprint"></div> <div id="content" class="mw-body" role="main"> <a id="top"></a> <div id="siteNotice"></div> <div class="mw-indicators"> </div> <h1 id="firstHeading" class="firstHeading mw-first-heading">Lineare Unabhängigkeit</h1> <div id="bodyContent" class="vector-body"> <div id="siteSub" class="noprint">aus Wikipedia, der freien Enzyklopädie</div> <div id="contentSub"><div id="mw-content-subtitle">(Weitergeleitet von <a href="/w/index.php?title=Linear_unabh%C3%A4ngig&redirect=no" class="mw-redirect" title="Linear unabhängig">Linear unabhängig</a>)</div></div> <div id="contentSub2"></div> <div id="jump-to-nav"></div> <a class="mw-jump-link" href="#mw-head">Zur Navigation springen</a> <a class="mw-jump-link" href="#searchInput">Zur Suche springen</a> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="de" dir="ltr"><figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/Datei:Vec-indep.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/b/bc/Vec-indep.png/220px-Vec-indep.png" decoding="async" width="220" height="220" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/b/bc/Vec-indep.png/330px-Vec-indep.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/b/bc/Vec-indep.png/440px-Vec-indep.png 2x" data-file-width="2000" data-file-height="2000" /></a><figcaption>Linear unabhängige Vektoren in ℝ3</figcaption></figure> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/Datei:Vec-dep.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/a/ab/Vec-dep.png/220px-Vec-dep.png" decoding="async" width="220" height="110" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/ab/Vec-dep.png/330px-Vec-dep.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/a/ab/Vec-dep.png/440px-Vec-dep.png 2x" data-file-width="4000" data-file-height="2000" /></a><figcaption>Linear abhängige Vektoren in einer Ebene in ℝ3</figcaption></figure> In der <a href="/wiki/Lineare_Algebra" title="Lineare Algebra">linearen Algebra</a> wird eine <a href="/wiki/Familie_(Mathematik)" title="Familie (Mathematik)">Familie</a> von <a href="/wiki/Vektor" title="Vektor">Vektoren</a> eines <a href="/wiki/Vektorraum" title="Vektorraum">Vektorraums</a> linear unabhängig genannt, wenn sich der <a href="/wiki/Nullvektor" title="Nullvektor">Nullvektor</a> nur durch eine <a href="/wiki/Linearkombination" title="Linearkombination">Linearkombination</a> der Vektoren erzeugen lässt, in der alle Koeffizienten der Kombination auf den Wert null gesetzt werden. Äquivalent dazu ist, dass sich keiner der Vektoren als Linearkombination der anderen Vektoren der Familie darstellen lässt. Andernfalls heißen sie linear abhängig. In diesem Fall lässt sich mindestens einer der Vektoren (aber nicht notwendigerweise jeder) als Linearkombination der anderen darstellen. Zum Beispiel sind im dreidimensionalen <a href="/wiki/Euklidischer_Raum" title="Euklidischer Raum">euklidischen Raum</a> <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}}"> die Vektoren <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (1,0,0)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mn>1</mn> <mo>,</mo> <mn>0</mn> <mo>,</mo> <mn>0</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (1,0,0)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8ea6e8ece35797224448db97fa0ea17544a7f756" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.365ex; height:2.843ex;" alt="{\displaystyle (1,0,0)}">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (0,1,0)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo>,</mo> <mn>0</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (0,1,0)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3e28bfe8a44d06939011e3ba2159e894c0f22f23" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.365ex; height:2.843ex;" alt="{\displaystyle (0,1,0)}"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (0,0,1)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (0,0,1)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/25b35543ef5f96df2e9aea6cced9bdd57dab7e1a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.365ex; height:2.843ex;" alt="{\displaystyle (0,0,1)}"> linear unabhängig. Die Vektoren <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (2,{-1},1)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mn>2</mn> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> <mo>,</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (2,{-1},1)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4eea05be8ceed0ced0a7f76f8db71f6148b296e8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.173ex; height:2.843ex;" alt="{\displaystyle (2,{-1},1)}">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (1,0,1)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mn>1</mn> <mo>,</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (1,0,1)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e62ee87058de1e6f24731a117229cf7020b14ab6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.365ex; height:2.843ex;" alt="{\displaystyle (1,0,1)}"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (3,{-1},2)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mn>3</mn> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> <mo>,</mo> <mn>2</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (3,{-1},2)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9633215e6513fef3ff259b0e86355f21f0658821" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.173ex; height:2.843ex;" alt="{\displaystyle (3,{-1},2)}"> sind hingegen linear abhängig, denn der dritte Vektor ist die Summe der beiden ersten, d. h. die Differenz von der Summe der ersten beiden und dem dritten ist der Nullvektor. Die Vektoren <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (1,2,{-3})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>3</mn> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (1,2,{-3})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1d0e3269991d8e0d7609e79238176b0798cd715e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.173ex; height:2.843ex;" alt="{\displaystyle (1,2,{-3})}">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ({-2},{-4},6)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>2</mn> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>4</mn> </mrow> <mo>,</mo> <mn>6</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ({-2},{-4},6)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5553af01866d9be3e74f19d7ff4f40d3f9ec5d31" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.981ex; height:2.843ex;" alt="{\displaystyle ({-2},{-4},6)}"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (1,1,1)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mn>1</mn> <mo>,</mo> <mn>1</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (1,1,1)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/efeac0c817a90342bab7878eb40cb33dd6facbbc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.365ex; height:2.843ex;" alt="{\displaystyle (1,1,1)}"> sind wegen <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2\cdot (1,2,{-3})+({-2},{-4},6)=(0,0,0)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <mo>⋅</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>3</mn> </mrow> <mo stretchy="false">)</mo> <mo>+</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>2</mn> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>4</mn> </mrow> <mo>,</mo> <mn>6</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mn>0</mn> <mo>,</mo> <mn>0</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2\cdot (1,2,{-3})+({-2},{-4},6)=(0,0,0)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7492ebfc83c067004cdf3769695c1847259ab1b8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:36.298ex; height:2.843ex;" alt="{\displaystyle 2\cdot (1,2,{-3})+({-2},{-4},6)=(0,0,0)}"> ebenfalls linear abhängig; jedoch ist hier der dritte Vektor nicht als Linearkombination der beiden anderen darstellbar. <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="#Definition">1 Definition</a></li> <li class="toclevel-1 tocsection-2"><a href="#Andere_Charakterisierungen_und_einfache_Eigenschaften">2 Andere Charakterisierungen und einfache Eigenschaften</a></li> <li class="toclevel-1 tocsection-3"><a href="#Ermittlung_mittels_Determinante">3 Ermittlung mittels Determinante</a></li> <li class="toclevel-1 tocsection-4"><a href="#Basis_eines_Vektorraums">4 Basis eines Vektorraums</a></li> <li class="toclevel-1 tocsection-5"><a href="#Beispiele">5 Beispiele</a> <ul> <li class="toclevel-2 tocsection-6"><a href="#Einzelner_Vektor">5.1 Einzelner Vektor</a></li> <li class="toclevel-2 tocsection-7"><a href="#Vektoren_in_der_Ebene">5.2 Vektoren in der Ebene</a></li> <li class="toclevel-2 tocsection-8"><a href="#Standardbasis_im_n-dimensionalen_Raum">5.3 Standardbasis im n-dimensionalen Raum</a></li> <li class="toclevel-2 tocsection-9"><a href="#Funktionen_als_Vektoren">5.4 Funktionen als Vektoren</a></li> <li class="toclevel-2 tocsection-10"><a href="#Reihen">5.5 Reihen</a></li> <li class="toclevel-2 tocsection-11"><a href="#Zeilen_und_Spalten_einer_Matrix">5.6 Zeilen und Spalten einer Matrix</a></li> <li class="toclevel-2 tocsection-12"><a href="#Rationale_Unabhängigkeit">5.7 Rationale Unabhängigkeit</a></li> </ul> </li> <li class="toclevel-1 tocsection-13"><a href="#Verallgemeinerungen">6 Verallgemeinerungen</a></li> <li class="toclevel-1 tocsection-14"><a href="#Literatur">7 Literatur</a></li> <li class="toclevel-1 tocsection-15"><a href="#Einzelnachweise">8 Einzelnachweise</a></li> </ul> </div> <div class="mw-heading mw-heading2"><h2 id="Definition">Definition</h2>[<a href="/w/index.php?title=Lineare_Unabh%C3%A4ngigkeit&veaction=edit&section=1" title="Abschnitt bearbeiten: Definition" class="mw-editsection-visualeditor">Bearbeiten</a> | <a href="/w/index.php?title=Lineare_Unabh%C3%A4ngigkeit&action=edit&section=1" title="Quellcode des Abschnitts bearbeiten: Definition">Quelltext bearbeiten</a>]</div> Ist <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 Vektorraum 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}">, so heißen die Vektoren <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {v}}_{1},{\vec {v}}_{2},...,{\vec {v}}_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>v</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>v</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>,</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>v</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {v}}_{1},{\vec {v}}_{2},...,{\vec {v}}_{n}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f74ad6bb6601fa30f508e9ed418377720fb0e3b2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:13.057ex; height:2.676ex;" alt="{\displaystyle {\vec {v}}_{1},{\vec {v}}_{2},...,{\vec {v}}_{n}}"> aus <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}"> linear unabhängig, wenn die einzig mögliche Darstellung des Nullvektors als Linearkombination <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{1}{\vec {v}}_{1}+a_{2}{\vec {v}}_{2}+\dotsb +a_{n}{\vec {v}}_{n}={\vec {0}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>v</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>v</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>+</mo> <mo>⋯</mo> <mo>+</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>v</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mn>0</mn> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{1}{\vec {v}}_{1}+a_{2}{\vec {v}}_{2}+\dotsb +a_{n}{\vec {v}}_{n}={\vec {0}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/56fef3024163172619a683409d5746adc25dd58e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:29.375ex; height:3.176ex;" alt="{\displaystyle a_{1}{\vec {v}}_{1}+a_{2}{\vec {v}}_{2}+\dotsb +a_{n}{\vec {v}}_{n}={\vec {0}}}"></dd></dl> mit Koeffizienten <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{1},a_{2},\dots ,a_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{1},a_{2},\dots ,a_{n}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ccc2b55ae65455992fe3ab08c90c55a0cc2e7709" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:13.229ex; height:2.009ex;" alt="{\displaystyle a_{1},a_{2},\dots ,a_{n}}"> aus dem Grundkörper <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}"> diejenige ist, bei der alle Koeffizienten <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{i}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0bc77764b2e74e64a63341054fa90f3e07db275f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.029ex; height:2.009ex;" alt="{\displaystyle a_{i}}"> gleich null sind. Lässt sich dagegen der Nullvektor auch nichttrivial (mit Koeffizienten ungleich null) erzeugen, dann sind die Vektoren linear abhängig.<a href="#cite_note-1">[1]</a><a href="#cite_note-:0-2">[2]</a><a href="#cite_note-:1-3">[3]</a> Ist <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/535ea7fc4134a31cbe2251d9d3511374bc41be9f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.172ex; height:2.176ex;" alt="{\displaystyle I}"> eine beliebige <a href="/wiki/Indexmenge_(Mathematik)" title="Indexmenge (Mathematik)">Indexmenge</a>, so heißt eine Familie <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ({\vec {v}}_{i})_{i\in I}}"> <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>v</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>∈</mo> <mi>I</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ({\vec {v}}_{i})_{i\in I}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b1bc7770abda464218610cd58c159914333d0678" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.509ex; height:2.843ex;" alt="{\displaystyle ({\vec {v}}_{i})_{i\in I}}"> von Vektoren aus <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}"> linear unabhängig, falls jede endliche Teilfamilie linear unabhängig ist.<a href="#cite_note-:0-2">[2]</a><a href="#cite_note-:1-3">[3]</a> Die Familie <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ({\vec {v}}_{i})_{i\in I}}"> <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>v</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>∈</mo> <mi>I</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ({\vec {v}}_{i})_{i\in I}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b1bc7770abda464218610cd58c159914333d0678" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.509ex; height:2.843ex;" alt="{\displaystyle ({\vec {v}}_{i})_{i\in I}}"> ist also genau dann linear abhängig, wenn es eine endliche Teilmenge <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle J\subseteq I}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>J</mi> <mo>⊆</mo> <mi>I</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle J\subseteq I}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/76ac10d0192d1da551ee96115a645f1faa4465de" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.742ex; height:2.343ex;" alt="{\displaystyle J\subseteq I}"> gibt, sowie Koeffizienten <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (a_{j})_{j\in J}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mo>∈</mo> <mi>J</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (a_{j})_{j\in J}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8ad12de65c9c825750836768373da16909975125" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:6.995ex; height:3.009ex;" alt="{\displaystyle (a_{j})_{j\in J}}">, von denen mindestens einer ungleich 0 ist, so dass <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sum _{j\in J}a_{j}{\vec {v}}_{j}={\vec {0}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mo>∈</mo> <mi>J</mi> </mrow> </munder> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>v</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mn>0</mn> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum _{j\in J}a_{j}{\vec {v}}_{j}={\vec {0}}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2a7af4029b0a017605a9afce93cecb8255c64160" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:12.874ex; height:5.843ex;" alt="{\displaystyle \sum _{j\in J}a_{j}{\vec {v}}_{j}={\vec {0}}.}"></dd></dl> Der Nullvektor <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {0}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mn>0</mn> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {0}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e76498919cf387316fc79d04120c59a8d430ef36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.843ex;" alt="{\displaystyle {\vec {0}}}"> ist ein Element des Vektorraumes <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}">. Im Gegensatz dazu ist 0 ein Element des Körpers <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}">. Der Begriff wird auch für Teilmengen eines Vektorraums verwendet: Eine Teilmenge <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S\subseteq V}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> <mo>⊆</mo> <mi>V</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S\subseteq V}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/659806b81175edf69ba2fe8050d7aaf773c29bc0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.385ex; height:2.343ex;" alt="{\displaystyle S\subseteq V}"> eines Vektorraums <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}"> heißt linear unabhängig, wenn jede Linearkombination von paarweise verschiedenen Vektoren aus <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4611d85173cd3b508e67077d4a1252c9c05abca2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.499ex; height:2.176ex;" alt="{\displaystyle S}"> nur dann den Nullvektor darstellen kann, wenn alle Koeffizienten in dieser Linearkombination den Wert null haben. Man beachte folgenden Unterschied: Ist etwa <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ({\vec {v}}_{1},{\vec {v}}_{2})}"> <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>v</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>v</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ({\vec {v}}_{1},{\vec {v}}_{2})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aaddf2ebfb1ff9337c56a2a7843769a8d2e057de" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.302ex; height:2.843ex;" alt="{\displaystyle ({\vec {v}}_{1},{\vec {v}}_{2})}"> eine linear unabhängige Familie, so ist <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ({\vec {v}}_{1},{\vec {v}}_{1},{\vec {v}}_{2})}"> <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>v</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>v</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>v</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ({\vec {v}}_{1},{\vec {v}}_{1},{\vec {v}}_{2})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/78469bb1bd794385e8a0b594c1270cea20e7948c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.566ex; height:2.843ex;" alt="{\displaystyle ({\vec {v}}_{1},{\vec {v}}_{1},{\vec {v}}_{2})}"> offenbar eine linear abhängige Familie. Die Menge <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{{\vec {v}}_{1},{\vec {v}}_{1},{\vec {v}}_{2}\}=\{{\vec {v}}_{1},{\vec {v}}_{2}\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>v</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>v</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>v</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo fence="false" stretchy="false">}</mo> <mo>=</mo> <mo fence="false" stretchy="false">{</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>v</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>v</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{{\vec {v}}_{1},{\vec {v}}_{1},{\vec {v}}_{2}\}=\{{\vec {v}}_{1},{\vec {v}}_{2}\}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9f72e1ed69ddc2ae020e19627078d5378e84815d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.998ex; height:2.843ex;" alt="{\displaystyle \{{\vec {v}}_{1},{\vec {v}}_{1},{\vec {v}}_{2}\}=\{{\vec {v}}_{1},{\vec {v}}_{2}\}}"> ist dann aber linear unabhängig. <div class="mw-heading mw-heading2"><h2 id="Andere_Charakterisierungen_und_einfache_Eigenschaften">Andere Charakterisierungen und einfache Eigenschaften</h2>[<a href="/w/index.php?title=Lineare_Unabh%C3%A4ngigkeit&veaction=edit&section=2" title="Abschnitt bearbeiten: Andere Charakterisierungen und einfache Eigenschaften" class="mw-editsection-visualeditor">Bearbeiten</a> | <a href="/w/index.php?title=Lineare_Unabh%C3%A4ngigkeit&action=edit&section=2" title="Quellcode des Abschnitts bearbeiten: Andere Charakterisierungen und einfache Eigenschaften">Quelltext bearbeiten</a>]</div> <ul><li>Die Familie <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (v_{i})_{i\in I}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>∈</mo> <mi>I</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (v_{i})_{i\in I}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b8a41d171710e2f4ea57c79d3ff95ec971e2bed4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.461ex; height:2.843ex;" alt="{\displaystyle (v_{i})_{i\in I}}"> von Elementen eines <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}">-Vektorraums <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}"> ist genau dann linear unabhängig, wenn die lineare Abbildung <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m\colon K^{(I)}\to V,(s_{i})_{i\in I}\mapsto \sum _{i:s_{i}\neq 0}s_{i}\cdot v_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> <mo>:</mo> <msup> <mi>K</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>I</mi> <mo stretchy="false">)</mo> </mrow> </msup> <mo stretchy="false">→</mo> <mi>V</mi> <mo>,</mo> <mo stretchy="false">(</mo> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>∈</mo> <mi>I</mi> </mrow> </msub> <mo stretchy="false">↦</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>:</mo> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>≠</mo> <mn>0</mn> </mrow> </munder> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>⋅</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m\colon K^{(I)}\to V,(s_{i})_{i\in I}\mapsto \sum _{i:s_{i}\neq 0}s_{i}\cdot v_{i}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/049885b763ceac351a8821d0308187ad8dfb8f79" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.505ex; width:34.387ex; height:6.009ex;" alt="{\displaystyle m\colon K^{(I)}\to V,(s_{i})_{i\in I}\mapsto \sum _{i:s_{i}\neq 0}s_{i}\cdot v_{i}}"> den <a href="/wiki/Lineare_Abbildung#Bild_und_Kern" title="Lineare Abbildung">Kern</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{0\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <mn>0</mn> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{0\}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0ff0df9ef65c0572eb676580ce1c02b8ec40f694" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.487ex; height:2.843ex;" alt="{\displaystyle \{0\}}"> hat.</li></ul> <ul><li>Die Vektoren <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {v}}_{1},\ldots ,{\vec {v}}_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>v</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>v</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {v}}_{1},\ldots ,{\vec {v}}_{n}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2ece5619855a9c8c86ae53771d9ceba8d19e71a8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.802ex; height:2.676ex;" alt="{\displaystyle {\vec {v}}_{1},\ldots ,{\vec {v}}_{n}}"> sind genau dann linear unabhängig, wenn sich keiner von ihnen als Linearkombination der anderen darstellen lässt. Diese Aussage gilt nicht im allgemeineren Kontext von <a href="/wiki/Modul_(Mathematik)" title="Modul (Mathematik)">Moduln</a> über <a href="/wiki/Ring_(Algebra)" title="Ring (Algebra)">Ringen</a>.</li></ul> <ul><li>Eine Variante dieser Aussage ist das Abhängigkeitslemma: Sind <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {v}}_{1},\ldots ,{\vec {v}}_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>v</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>v</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {v}}_{1},\ldots ,{\vec {v}}_{n}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2ece5619855a9c8c86ae53771d9ceba8d19e71a8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.802ex; height:2.676ex;" alt="{\displaystyle {\vec {v}}_{1},\ldots ,{\vec {v}}_{n}}"> linear unabhängig und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {v}}_{1},\ldots ,{\vec {v}}_{n},{\vec {w}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>v</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>v</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>,</mo> <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 {v}}_{1},\ldots ,{\vec {v}}_{n},{\vec {w}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/80f1408eff35a20690b400461dc4d01ebe579e87" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.5ex; height:2.676ex;" alt="{\displaystyle {\vec {v}}_{1},\ldots ,{\vec {v}}_{n},{\vec {w}}}"> linear abhängig, so lässt sich <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}}}"> als Linearkombination von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {v}}_{1},\ldots ,{\vec {v}}_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>v</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>v</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {v}}_{1},\ldots ,{\vec {v}}_{n}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2ece5619855a9c8c86ae53771d9ceba8d19e71a8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.802ex; height:2.676ex;" alt="{\displaystyle {\vec {v}}_{1},\ldots ,{\vec {v}}_{n}}"> schreiben.</li></ul> <ul><li>Ist eine Familie von Vektoren linear unabhängig, so ist jede Teilfamilie dieser Familie ebenfalls linear unabhängig. Ist eine Familie hingegen linear abhängig, so ist jede Familie, die diese abhängige Familie beinhaltet, ebenso linear abhängig.</li></ul> <ul><li><a href="/wiki/Elementarmatrix" title="Elementarmatrix">Elementare Umformungen</a> der Vektoren verändern die lineare Abhängigkeit oder die lineare Unabhängigkeit nicht.</li></ul> <ul><li>Ist einer der <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {v}}_{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>v</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}}_{i}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4c1e99e843fffeae2c302f24e96edb76b2cbf915" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.975ex; height:2.676ex;" alt="{\displaystyle {\vec {v}}_{i}}"> der Nullvektor (hier: Sei <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {v}}_{j}={\vec {0}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>v</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mn>0</mn> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {v}}_{j}={\vec {0}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1643e92e3ed7b300e2e45ff2df6f974d66e5e3b9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:6.346ex; height:3.509ex;" alt="{\displaystyle {\vec {v}}_{j}={\vec {0}}}">), so sind diese linear abhängig – der Nullvektor kann erzeugt werden, indem alle <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{i}=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{i}=0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d1d2e283d81c74e9c6742dcbbcad8c622ef0c3c9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.29ex; height:2.509ex;" alt="{\displaystyle a_{i}=0}"> gesetzt werden mit Ausnahme von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{j}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{j}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d0096fb78d6843c9fb67a840dc796b61ad93eec2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.14ex; height:2.343ex;" alt="{\displaystyle a_{j}}">, welches als Koeffizient des Nullvektors <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {v}}_{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>v</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 {\vec {v}}_{j}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ef89ae6be2895528c01d3ea80918b19aa470826e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.085ex; height:3.009ex;" alt="{\displaystyle {\vec {v}}_{j}}"> beliebig (also insbesondere auch ungleich null) sein darf.</li></ul> <ul><li>In einem <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/e85ff03cbe0c7341af6b982e47e9f90d235c66ab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.216ex; height:2.176ex;" alt="{\displaystyle d}">-<a href="/wiki/Dimension_(Mathematik)" title="Dimension (Mathematik)">dimensionalen</a> Raum ist eine Familie aus mehr als <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/e85ff03cbe0c7341af6b982e47e9f90d235c66ab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.216ex; height:2.176ex;" alt="{\displaystyle d}"> Vektoren immer linear abhängig (siehe <a href="/wiki/Schranken-Lemma" title="Schranken-Lemma">Schranken-Lemma</a>).</li></ul> <div class="mw-heading mw-heading2"><h2 id="Ermittlung_mittels_Determinante">Ermittlung mittels Determinante</h2>[<a href="/w/index.php?title=Lineare_Unabh%C3%A4ngigkeit&veaction=edit&section=3" title="Abschnitt bearbeiten: Ermittlung mittels Determinante" class="mw-editsection-visualeditor">Bearbeiten</a> | <a href="/w/index.php?title=Lineare_Unabh%C3%A4ngigkeit&action=edit&section=3" title="Quellcode des Abschnitts bearbeiten: Ermittlung mittels Determinante">Quelltext bearbeiten</a>]</div> Hat man <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}"> Vektoren eines <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}">-dimensionalen Vektorraums als Zeilen- oder Spaltenvektoren bzgl. einer festen Basis gegeben, so kann man deren lineare Unabhängigkeit dadurch prüfen, dass man diese <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}"> Zeilen- bzw. Spaltenvektoren zu 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 zusammenfasst und dann deren <a href="/wiki/Determinante_(Mathematik)" class="mw-redirect" title="Determinante (Mathematik)">Determinante</a> ausrechnet. Die Vektoren sind genau dann linear unabhängig, wenn die Determinante ungleich 0 ist. <div class="mw-heading mw-heading2"><h2 id="Basis_eines_Vektorraums">Basis eines Vektorraums</h2>[<a href="/w/index.php?title=Lineare_Unabh%C3%A4ngigkeit&veaction=edit&section=4" title="Abschnitt bearbeiten: Basis eines Vektorraums" class="mw-editsection-visualeditor">Bearbeiten</a> | <a href="/w/index.php?title=Lineare_Unabh%C3%A4ngigkeit&action=edit&section=4" title="Quellcode des Abschnitts bearbeiten: Basis eines Vektorraums">Quelltext bearbeiten</a>]</div> <div class="hauptartikel" role="navigation">→ Hauptartikel: <a href="/wiki/Basis_(Vektorraum)" title="Basis (Vektorraum)">Basis (Vektorraum)</a></div> Eine wichtige Rolle spielt das Konzept der linear unabhängigen Vektoren bei der Definition beziehungsweise beim Umgang mit Vektorraumbasen. Eine Basis eines Vektorraums ist ein linear unabhängiges <a href="/wiki/Erzeugendensystem" title="Erzeugendensystem">Erzeugendensystem</a>. Basen erlauben es, insbesondere bei endlichdimensionalen Vektorräumen mit Koordinaten zu rechnen. <div class="mw-heading mw-heading2"><h2 id="Beispiele">Beispiele</h2>[<a href="/w/index.php?title=Lineare_Unabh%C3%A4ngigkeit&veaction=edit&section=5" title="Abschnitt bearbeiten: Beispiele" class="mw-editsection-visualeditor">Bearbeiten</a> | <a href="/w/index.php?title=Lineare_Unabh%C3%A4ngigkeit&action=edit&section=5" title="Quellcode des Abschnitts bearbeiten: Beispiele">Quelltext bearbeiten</a>]</div> <figure class="mw-default-size mw-halign-right" typeof="mw:File"><a href="/wiki/Datei:Vectores_independientes.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/2/27/Vectores_independientes.png" decoding="async" width="405" height="122" class="mw-file-element" data-file-width="405" data-file-height="122" /></a><figcaption></figcaption></figure> <ul><li><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {u}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>u</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {u}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/89c41e9cf70c5e5b56e2128a136985a75f90ba43" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:2.343ex;" alt="{\displaystyle {\vec {u}}}"> und <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}}}"> sind linear unabhängig und definieren die <a href="/wiki/Ebene_(Mathematik)" title="Ebene (Mathematik)">Ebene</a> P.</li> <li><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {u}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>u</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {u}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/89c41e9cf70c5e5b56e2128a136985a75f90ba43" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:2.343ex;" alt="{\displaystyle {\vec {u}}}">, <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}}}"> und <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}}}"> sind linear abhängig, weil sie in derselben Ebene liegen.</li> <li><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {u}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>u</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {u}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/89c41e9cf70c5e5b56e2128a136985a75f90ba43" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:2.343ex;" alt="{\displaystyle {\vec {u}}}"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {j}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>j</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {j}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8ce1ed1de8493f7cc7d856ca5427cf311b1597f1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.094ex; height:3.176ex;" alt="{\displaystyle {\vec {j}}}"> sind linear abhängig, da sie parallel zueinander verlaufen.</li> <li><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {u}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>u</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {u}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/89c41e9cf70c5e5b56e2128a136985a75f90ba43" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:2.343ex;" alt="{\displaystyle {\vec {u}}}">, <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}}}"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {k}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>k</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {k}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5ccd4b98d198d6538010ae815ee1199baabd3493" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.843ex;" alt="{\displaystyle {\vec {k}}}"> sind linear unabhängig, da <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {u}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>u</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {u}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/89c41e9cf70c5e5b56e2128a136985a75f90ba43" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:2.343ex;" alt="{\displaystyle {\vec {u}}}"> und <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}}}"> voneinander unabhängig sind und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {k}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>k</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {k}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5ccd4b98d198d6538010ae815ee1199baabd3493" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.843ex;" alt="{\displaystyle {\vec {k}}}"> sich nicht als lineare Kombination der beiden darstellen lässt bzw. weil sie nicht auf einer gemeinsamen Ebene liegen. Die drei Vektoren definieren einen drei-dimensionalen Raum.</li> <li>Die Vektoren <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {0}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mn>0</mn> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {0}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e76498919cf387316fc79d04120c59a8d430ef36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.843ex;" alt="{\displaystyle {\vec {0}}}"> (<a href="/wiki/Nullvektor" title="Nullvektor">Nullvektor</a>) und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {k}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>k</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {k}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5ccd4b98d198d6538010ae815ee1199baabd3493" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.843ex;" alt="{\displaystyle {\vec {k}}}"> sind linear abhängig, da  <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {0}}=0\cdot {\vec {k}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mn>0</mn> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>=</mo> <mn>0</mn> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>k</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {0}}=0\cdot {\vec {k}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/61a6acb7ed0c3861c4c533ec45c61db9442e688f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.314ex; height:2.843ex;" alt="{\displaystyle {\vec {0}}=0\cdot {\vec {k}}}"></li></ul> <div class="mw-heading mw-heading3"><h3 id="Einzelner_Vektor">Einzelner Vektor</h3>[<a href="/w/index.php?title=Lineare_Unabh%C3%A4ngigkeit&veaction=edit&section=6" title="Abschnitt bearbeiten: Einzelner Vektor" class="mw-editsection-visualeditor">Bearbeiten</a> | <a href="/w/index.php?title=Lineare_Unabh%C3%A4ngigkeit&action=edit&section=6" title="Quellcode des Abschnitts bearbeiten: Einzelner Vektor">Quelltext bearbeiten</a>]</div> Der 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}}}"> sei ein Element des Vektorraums <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}"> 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}">. Dann ist der einzelne 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}}}"> für sich genau dann linear unabhängig, wenn er nicht der Nullvektor ist. Denn aus der Definition des <a href="/wiki/Vektorraum" title="Vektorraum">Vektorraums</a> folgt, dass wenn <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a\,{\vec {v}}={\vec {0}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>v</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mn>0</mn> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a\,{\vec {v}}={\vec {0}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d3fbc8fc85fdf14b6b2dbe0d95a99d99c2c86240" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.053ex; height:2.843ex;" alt="{\displaystyle a\,{\vec {v}}={\vec {0}}}"> mit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a\in K}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>∈</mo> <mi>K</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a\in K}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f97d838bfcfb39f7a33ffe31cd1c2a989b8ca3f6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.136ex; height:2.176ex;" alt="{\displaystyle a\in K}">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {v}}\in 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> <mo>∈</mo> <mi>V</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {v}}\in V}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f9945ab48b1b66ef19c197dc63553ba0cc12e043" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.803ex; height:2.343ex;" alt="{\displaystyle {\vec {v}}\in V}"></dd></dl> nur <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a=0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/90d476e5e765a5d77bbcff32e4584579207ec7d8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.491ex; height:2.176ex;" alt="{\displaystyle a=0}"> oder <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {v}}={\vec {0}}}"> <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> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mn>0</mn> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {v}}={\vec {0}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/add3dc0454eda873e35f853cf997e664514f3786" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.436ex; height:2.843ex;" alt="{\displaystyle {\vec {v}}={\vec {0}}}"> sein kann. <div class="mw-heading mw-heading3"><h3 id="Vektoren_in_der_Ebene">Vektoren in der Ebene</h3>[<a href="/w/index.php?title=Lineare_Unabh%C3%A4ngigkeit&veaction=edit&section=7" title="Abschnitt bearbeiten: Vektoren in der Ebene" class="mw-editsection-visualeditor">Bearbeiten</a> | <a href="/w/index.php?title=Lineare_Unabh%C3%A4ngigkeit&action=edit&section=7" title="Quellcode des Abschnitts bearbeiten: Vektoren in der Ebene">Quelltext bearbeiten</a>]</div> Die Vektoren <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {u}}={\begin{pmatrix}1\\1\end{pmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>u</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <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> </mtable> <mo>)</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {u}}={\begin{pmatrix}1\\1\end{pmatrix}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/08b71cbd127ed9d41811f585a84d5a9b93e90bac" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:9.763ex; height:6.176ex;" alt="{\displaystyle {\vec {u}}={\begin{pmatrix}1\\1\end{pmatrix}}}"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {v}}={\begin{pmatrix}-3\\2\end{pmatrix}}}"> <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> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mo>−</mo> <mn>3</mn> </mtd> </mtr> <mtr> <mtd> <mn>2</mn> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {v}}={\begin{pmatrix}-3\\2\end{pmatrix}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d0bf71b488a2cd10ed325a6a96afc2becb4ff95c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:11.417ex; height:6.176ex;" alt="{\displaystyle {\vec {v}}={\begin{pmatrix}-3\\2\end{pmatrix}}}"> sind in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{2}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e150115ab9f63023215109595b76686a1ff890fd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.732ex; height:2.676ex;" alt="{\displaystyle \mathbb {R} ^{2}}"> linear unabhängig. Beweis: Für <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a,b\in \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a,b\in \mathbb {R} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fe0bf0e2324bac843fab916c46d1b3349017a616" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.78ex; height:2.509ex;" alt="{\displaystyle a,b\in \mathbb {R} }"> gelte <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a\,{\vec {u}}+b\,{\vec {v}}={\vec {0}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>u</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>+</mo> <mi>b</mi> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>v</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mn>0</mn> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a\,{\vec {u}}+b\,{\vec {v}}={\vec {0}},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5ad517321735d09a64d762f36f008282e398664d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:13.255ex; height:3.176ex;" alt="{\displaystyle a\,{\vec {u}}+b\,{\vec {v}}={\vec {0}},}"></dd></dl> d. h. <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a\,{\begin{pmatrix}1\\1\end{pmatrix}}+b\,{\begin{pmatrix}-3\\2\end{pmatrix}}={\begin{pmatrix}0\\0\end{pmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mspace width="thinmathspace" /> <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> </mtable> <mo>)</mo> </mrow> </mrow> <mo>+</mo> <mi>b</mi> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mo>−</mo> <mn>3</mn> </mtd> </mtr> <mtr> <mtd> <mn>2</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>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a\,{\begin{pmatrix}1\\1\end{pmatrix}}+b\,{\begin{pmatrix}-3\\2\end{pmatrix}}={\begin{pmatrix}0\\0\end{pmatrix}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d21516b86e5c54996b597e2b794c13d85261c56a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:26.754ex; height:6.176ex;" alt="{\displaystyle a\,{\begin{pmatrix}1\\1\end{pmatrix}}+b\,{\begin{pmatrix}-3\\2\end{pmatrix}}={\begin{pmatrix}0\\0\end{pmatrix}}}"> </dd></dl> Dann gilt <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{pmatrix}a-3b\\a+2b\end{pmatrix}}={\begin{pmatrix}0\\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> <mi>a</mi> <mo>−</mo> <mn>3</mn> <mi>b</mi> </mtd> </mtr> <mtr> <mtd> <mi>a</mi> <mo>+</mo> <mn>2</mn> <mi>b</mi> </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>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{pmatrix}a-3b\\a+2b\end{pmatrix}}={\begin{pmatrix}0\\0\end{pmatrix}},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/15f264f811c1bb41e64fc775875a3bef8cda9114" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:19.483ex; height:6.176ex;" alt="{\displaystyle {\begin{pmatrix}a-3b\\a+2b\end{pmatrix}}={\begin{pmatrix}0\\0\end{pmatrix}},}"></dd></dl> also <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a-3b=0\ \wedge \ a+2b=0.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>−</mo> <mn>3</mn> <mi>b</mi> <mo>=</mo> <mn>0</mn> <mtext> </mtext> <mo>∧</mo> <mtext> </mtext> <mi>a</mi> <mo>+</mo> <mn>2</mn> <mi>b</mi> <mo>=</mo> <mn>0.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a-3b=0\ \wedge \ a+2b=0.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/41850417eaf336367c5da8ee42cd32288e1987ed" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:25.373ex; height:2.343ex;" alt="{\displaystyle a-3b=0\ \wedge \ a+2b=0.}"></dd></dl> Dieses Gleichungssystem ist nur für die Lösung <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a=0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/90d476e5e765a5d77bbcff32e4584579207ec7d8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.491ex; height:2.176ex;" alt="{\displaystyle a=0}">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>b</mi> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b=0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/19206e7d4dab695ccb34c502eff0741e98dbdfc2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.258ex; height:2.176ex;" alt="{\displaystyle b=0}"> (die sogenannte triviale Lösung) erfüllt; d. h. <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {u}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>u</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {u}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/89c41e9cf70c5e5b56e2128a136985a75f90ba43" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:2.343ex;" alt="{\displaystyle {\vec {u}}}"> und <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}}}"> sind linear unabhängig. <div class="mw-heading mw-heading3"><h3 id="Standardbasis_im_n-dimensionalen_Raum">Standardbasis im n-dimensionalen Raum</h3>[<a href="/w/index.php?title=Lineare_Unabh%C3%A4ngigkeit&veaction=edit&section=8" title="Abschnitt bearbeiten: Standardbasis im n-dimensionalen Raum" class="mw-editsection-visualeditor">Bearbeiten</a> | <a href="/w/index.php?title=Lineare_Unabh%C3%A4ngigkeit&action=edit&section=8" title="Quellcode des Abschnitts bearbeiten: Standardbasis im n-dimensionalen Raum">Quelltext bearbeiten</a>]</div> Die kanonischen Einheitsvektoren <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{matrix}{\vec {e}}_{1}&=&(1,0,0,\dots ,0),\\{\vec {e}}_{2}&=&(0,1,0,\dots ,0),\\&\vdots &\\{\vec {e}}_{n}&=&(0,0,0,\dots ,1)\end{matrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>e</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mtd> <mtd> <mo>=</mo> </mtd> <mtd> <mo stretchy="false">(</mo> <mn>1</mn> <mo>,</mo> <mn>0</mn> <mo>,</mo> <mn>0</mn> <mo>,</mo> <mo>…</mo> <mo>,</mo> <mn>0</mn> <mo stretchy="false">)</mo> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>e</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mtd> <mtd> <mo>=</mo> </mtd> <mtd> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo>,</mo> <mn>0</mn> <mo>,</mo> <mo>…</mo> <mo>,</mo> <mn>0</mn> <mo stretchy="false">)</mo> <mo>,</mo> </mtd> </mtr> <mtr> <mtd /> <mtd> <mo>⋮</mo> </mtd> <mtd /> </mtr> <mtr> <mtd> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>e</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mtd> <mtd> <mo>=</mo> </mtd> <mtd> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mn>0</mn> <mo>,</mo> <mn>0</mn> <mo>,</mo> <mo>…</mo> <mo>,</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{matrix}{\vec {e}}_{1}&=&(1,0,0,\dots ,0),\\{\vec {e}}_{2}&=&(0,1,0,\dots ,0),\\&\vdots &\\{\vec {e}}_{n}&=&(0,0,0,\dots ,1)\end{matrix}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b2fa36bc098e7cc58fadc9ac6d9d84069ecc53fc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.505ex; width:23.998ex; height:14.176ex;" alt="{\displaystyle {\begin{matrix}{\vec {e}}_{1}&=&(1,0,0,\dots ,0),\\{\vec {e}}_{2}&=&(0,1,0,\dots ,0),\\&\vdots &\\{\vec {e}}_{n}&=&(0,0,0,\dots ,1)\end{matrix}}}"></dd></dl> sind im Vektorraum <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{n}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c510b63578322050121fe966f2e5770bea43308d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.897ex; height:2.343ex;" alt="{\displaystyle \mathbb {R} ^{n}}"> linear unabhängig. Beweis:    Für <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{1},a_{2},\dots ,a_{n}\in \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{1},a_{2},\dots ,a_{n}\in \mathbb {R} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5c70a5ab74bffafcd0e2eaa49ac3b6012121314b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:17.747ex; height:2.509ex;" alt="{\displaystyle a_{1},a_{2},\dots ,a_{n}\in \mathbb {R} }"> gelte <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{1}\,{\vec {e}}_{1}+a_{2}\,{\vec {e}}_{2}+\dotsb +a_{n}\,{\vec {e}}_{n}={\vec {0}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mspace width="thinmathspace" /> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>e</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mspace width="thinmathspace" /> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>e</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>+</mo> <mo>⋯</mo> <mo>+</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mspace width="thinmathspace" /> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>e</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mn>0</mn> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{1}\,{\vec {e}}_{1}+a_{2}\,{\vec {e}}_{2}+\dotsb +a_{n}\,{\vec {e}}_{n}={\vec {0}}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/07f441635031e8fde88002a2bde79c7751162da2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:31.326ex; height:3.176ex;" alt="{\displaystyle a_{1}\,{\vec {e}}_{1}+a_{2}\,{\vec {e}}_{2}+\dotsb +a_{n}\,{\vec {e}}_{n}={\vec {0}}.}"></dd></dl> Dann gilt aber auch <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{1}\,{\vec {e}}_{1}+a_{2}\,{\vec {e}}_{2}+\dots +a_{n}\,{\vec {e}}_{n}=(a_{1},a_{2},\ \dots ,a_{n})={\vec {0}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mspace width="thinmathspace" /> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>e</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mspace width="thinmathspace" /> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>e</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>+</mo> <mo>⋯</mo> <mo>+</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mspace width="thinmathspace" /> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>e</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>=</mo> <mo stretchy="false">(</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <mtext> </mtext> <mo>…</mo> <mo>,</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mn>0</mn> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{1}\,{\vec {e}}_{1}+a_{2}\,{\vec {e}}_{2}+\dots +a_{n}\,{\vec {e}}_{n}=(a_{1},a_{2},\ \dots ,a_{n})={\vec {0}},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2f0ea0c71514360e39fa1fdff0c9727cdf3b9f62" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:50.43ex; height:3.343ex;" alt="{\displaystyle a_{1}\,{\vec {e}}_{1}+a_{2}\,{\vec {e}}_{2}+\dots +a_{n}\,{\vec {e}}_{n}=(a_{1},a_{2},\ \dots ,a_{n})={\vec {0}},}"></dd></dl> und daraus folgt, dass <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{i}=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{i}=0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d1d2e283d81c74e9c6742dcbbcad8c622ef0c3c9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.29ex; height:2.509ex;" alt="{\displaystyle a_{i}=0}"> für alle <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i\in \{1,2,\dots ,n\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>i</mi> <mo>∈</mo> <mo fence="false" stretchy="false">{</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mo>…</mo> <mo>,</mo> <mi>n</mi> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i\in \{1,2,\dots ,n\}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d0ca4f29c266187e3eb4950224e67a106c693c50" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.9ex; height:2.843ex;" alt="{\displaystyle i\in \{1,2,\dots ,n\}}">. <div class="mw-heading mw-heading3"><h3 id="Funktionen_als_Vektoren">Funktionen als Vektoren</h3>[<a href="/w/index.php?title=Lineare_Unabh%C3%A4ngigkeit&veaction=edit&section=9" title="Abschnitt bearbeiten: Funktionen als Vektoren" class="mw-editsection-visualeditor">Bearbeiten</a> | <a href="/w/index.php?title=Lineare_Unabh%C3%A4ngigkeit&action=edit&section=9" title="Quellcode des Abschnitts bearbeiten: Funktionen als Vektoren">Quelltext bearbeiten</a>]</div> 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}"> der Vektorraum aller Funktionen <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f\colon \mathbb {R} \to \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>:</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mo stretchy="false">→</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f\colon \mathbb {R} \to \mathbb {R} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b1cacd5f7bbe1027cc75fbe2fbd9cb5e79485302" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.283ex; height:2.509ex;" alt="{\displaystyle f\colon \mathbb {R} \to \mathbb {R} }">. Die beiden Funktionen <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {e} ^{t}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">e</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {e} ^{t}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f1df06a51ce791640b5a424aaddaddfb22d4398d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.858ex; height:2.509ex;" alt="{\displaystyle \mathrm {e} ^{t}}"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {e} ^{2t}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">e</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>t</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {e} ^{2t}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6f0070b52dc5c58d36450ca8a355da9c61626565" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.68ex; height:2.676ex;" alt="{\displaystyle \mathrm {e} ^{2t}}"> 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}"> sind linear unabhängig. Beweis: Es seien <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a,b\in \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a,b\in \mathbb {R} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fe0bf0e2324bac843fab916c46d1b3349017a616" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.78ex; height:2.509ex;" alt="{\displaystyle a,b\in \mathbb {R} }"> und es gelte <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a\,\mathrm {e} ^{t}+b\,\mathrm {e} ^{2t}=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mspace width="thinmathspace" /> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">e</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msup> <mo>+</mo> <mi>b</mi> <mspace width="thinmathspace" /> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">e</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>t</mi> </mrow> </msup> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a\,\mathrm {e} ^{t}+b\,\mathrm {e} ^{2t}=0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9c7d6ba16dfd53e6edf07fae7d3029fd17b6c4ca" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:14.642ex; height:2.843ex;" alt="{\displaystyle a\,\mathrm {e} ^{t}+b\,\mathrm {e} ^{2t}=0}"></dd></dl> für alle <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t\in \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>t</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t\in \mathbb {R} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/592bced0c39b10fc90e74c6a66223abfbfb029de" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.358ex; height:2.176ex;" alt="{\displaystyle t\in \mathbb {R} }">. Leitet man diese Gleichung nach <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/65658b7b223af9e1acc877d848888ecdb4466560" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.84ex; height:2.009ex;" alt="{\displaystyle t}"> ab, dann erhält man eine zweite Gleichung <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a\,\mathrm {e} ^{t}+2b\,\mathrm {e} ^{2t}=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mspace width="thinmathspace" /> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">e</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msup> <mo>+</mo> <mn>2</mn> <mi>b</mi> <mspace width="thinmathspace" /> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">e</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>t</mi> </mrow> </msup> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a\,\mathrm {e} ^{t}+2b\,\mathrm {e} ^{2t}=0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0e1827621c841ed41d5c737a4b94fad44c224343" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:15.804ex; height:2.843ex;" alt="{\displaystyle a\,\mathrm {e} ^{t}+2b\,\mathrm {e} ^{2t}=0}">.</dd></dl> Indem man von der zweiten Gleichung die erste subtrahiert, erhält man <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b\,\mathrm {e} ^{2t}=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>b</mi> <mspace width="thinmathspace" /> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">e</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>t</mi> </mrow> </msup> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b\,\mathrm {e} ^{2t}=0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/17bbe78a1dce2a9005a9f4aa7d2dc8df8df34d7b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.326ex; height:2.676ex;" alt="{\displaystyle b\,\mathrm {e} ^{2t}=0}">.</dd></dl> Da diese Gleichung für alle <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/65658b7b223af9e1acc877d848888ecdb4466560" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.84ex; height:2.009ex;" alt="{\displaystyle t}"> und damit insbesondere auch für <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>t</mi> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t=0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/43469ec032d858feae5aa87029e22eaaf0109e9c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.101ex; height:2.176ex;" alt="{\displaystyle t=0}"> gelten muss, folgt daraus durch Einsetzen von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>t</mi> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t=0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/43469ec032d858feae5aa87029e22eaaf0109e9c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.101ex; height:2.176ex;" alt="{\displaystyle t=0}">, dass <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>b</mi> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b=0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/19206e7d4dab695ccb34c502eff0741e98dbdfc2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.258ex; height:2.176ex;" alt="{\displaystyle b=0}"> sein muss. Setzt man das so berechnete <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/f11423fbb2e967f986e36804a8ae4271734917c3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.998ex; height:2.176ex;" alt="{\displaystyle b}"> wieder in die erste Gleichung ein, dann ergibt sich <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a\,\mathrm {e} ^{t}+0=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mspace width="thinmathspace" /> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">e</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msup> <mo>+</mo> <mn>0</mn> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a\,\mathrm {e} ^{t}+0=0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6518229fe15f0058c9e4db084bc11121e5ee27cb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:11.739ex; height:2.676ex;" alt="{\displaystyle a\,\mathrm {e} ^{t}+0=0}">.</dd></dl> Daraus folgt wieder, dass (für <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>t</mi> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t=0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/43469ec032d858feae5aa87029e22eaaf0109e9c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.101ex; height:2.176ex;" alt="{\displaystyle t=0}">) <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a=0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/90d476e5e765a5d77bbcff32e4584579207ec7d8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.491ex; height:2.176ex;" alt="{\displaystyle a=0}"> sein muss. Da die erste Gleichung nur für <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a=0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/90d476e5e765a5d77bbcff32e4584579207ec7d8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.491ex; height:2.176ex;" alt="{\displaystyle a=0}"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>b</mi> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b=0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/19206e7d4dab695ccb34c502eff0741e98dbdfc2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.258ex; height:2.176ex;" alt="{\displaystyle b=0}"> lösbar ist, sind die beiden Funktionen <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {e} ^{t}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">e</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {e} ^{t}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f1df06a51ce791640b5a424aaddaddfb22d4398d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.858ex; height:2.509ex;" alt="{\displaystyle \mathrm {e} ^{t}}"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {e} ^{2t}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">e</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>t</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {e} ^{2t}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6f0070b52dc5c58d36450ca8a355da9c61626565" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.68ex; height:2.676ex;" alt="{\displaystyle \mathrm {e} ^{2t}}"> linear unabhängig. <div class="sieheauch" role="navigation" style="font-style:italic;">Siehe auch: <a href="/wiki/Wronski-Determinante" title="Wronski-Determinante">Wronski-Determinante</a></div> <div class="mw-heading mw-heading3"><h3 id="Reihen">Reihen</h3>[<a href="/w/index.php?title=Lineare_Unabh%C3%A4ngigkeit&veaction=edit&section=10" title="Abschnitt bearbeiten: Reihen" class="mw-editsection-visualeditor">Bearbeiten</a> | <a href="/w/index.php?title=Lineare_Unabh%C3%A4ngigkeit&action=edit&section=10" title="Quellcode des Abschnitts bearbeiten: Reihen">Quelltext bearbeiten</a>]</div> 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}"> der Vektorraum aller reellwertigen stetigen Funktionen <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f\colon (0,1)\to \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>:</mo> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo stretchy="false">→</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f\colon (0,1)\to \mathbb {R} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ae5be92e147441163cc7bb4cd284a5d1a0ea63ca" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.773ex; height:2.843ex;" alt="{\displaystyle f\colon (0,1)\to \mathbb {R} }"> auf dem offenen Einheitsintervall. Dann gilt zwar <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {1}{1-x}}=\sum _{n=0}^{\infty }x^{n},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>1</mn> <mo>−</mo> <mi>x</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {1}{1-x}}=\sum _{n=0}^{\infty }x^{n},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/851db45af2196f0a8bf843b2ea98ae77947adf29" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:16.204ex; height:6.843ex;" alt="{\displaystyle {\frac {1}{1-x}}=\sum _{n=0}^{\infty }x^{n},}"></dd></dl> aber dennoch sind <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {1}{1-x}},1,x,x^{2},\ldots }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mrow> <mn>1</mn> <mo>−</mo> <mi>x</mi> </mrow> </mfrac> </mstyle> </mrow> <mo>,</mo> <mn>1</mn> <mo>,</mo> <mi>x</mi> <mo>,</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>,</mo> <mo>…</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {1}{1-x}},1,x,x^{2},\ldots }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/29b09e569c881d96f71c17002dfe7b23050e4139" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:15.612ex; height:3.676ex;" alt="{\displaystyle {\tfrac {1}{1-x}},1,x,x^{2},\ldots }"> linear unabhängig. Linearkombinationen aus Potenzen von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"> sind nämlich nur Polynome und keine allgemeinen Potenzreihen, insbesondere also in der Nähe von 1 beschränkt, so dass sich <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {1}{1-x}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mrow> <mn>1</mn> <mo>−</mo> <mi>x</mi> </mrow> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {1}{1-x}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d7ca34fc66513e002329160ba78445b8d54da34b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:3.877ex; height:3.676ex;" alt="{\displaystyle {\tfrac {1}{1-x}}}"> nicht als Linearkombination von Potenzen darstellen lässt. <div class="mw-heading mw-heading3"><h3 id="Zeilen_und_Spalten_einer_Matrix">Zeilen und Spalten einer Matrix</h3>[<a href="/w/index.php?title=Lineare_Unabh%C3%A4ngigkeit&veaction=edit&section=11" title="Abschnitt bearbeiten: Zeilen und Spalten einer Matrix" class="mw-editsection-visualeditor">Bearbeiten</a> | <a href="/w/index.php?title=Lineare_Unabh%C3%A4ngigkeit&action=edit&section=11" title="Quellcode des Abschnitts bearbeiten: Zeilen und Spalten einer Matrix">Quelltext bearbeiten</a>]</div> Interessant ist auch die Frage, ob die Zeilen einer <a href="/wiki/Matrix_(Mathematik)" title="Matrix (Mathematik)">Matrix</a> linear unabhängig sind oder nicht. Dabei werden die Zeilen als Vektoren betrachtet. Falls die Zeilen einer quadratischen Matrix linear unabhängig sind, so nennt man die Matrix <a href="/wiki/Regul%C3%A4re_Matrix" title="Reguläre Matrix">regulär</a>, andernfalls singulär. Die Spalten einer quadratischen Matrix sind genau dann linear unabhängig, wenn die Zeilen linear unabhängig sind. Beispiel einer Folge von regulären Matrizen: <a href="/wiki/Hilbert-Matrix" title="Hilbert-Matrix">Hilbert-Matrix</a>. <div class="mw-heading mw-heading3"><h3 id="Rationale_Unabhängigkeit">Rationale Unabhängigkeit</h3>[<a href="/w/index.php?title=Lineare_Unabh%C3%A4ngigkeit&veaction=edit&section=12" title="Abschnitt bearbeiten: Rationale Unabhängigkeit" class="mw-editsection-visualeditor">Bearbeiten</a> | <a href="/w/index.php?title=Lineare_Unabh%C3%A4ngigkeit&action=edit&section=12" title="Quellcode des Abschnitts bearbeiten: Rationale Unabhängigkeit">Quelltext bearbeiten</a>]</div> Reelle Zahlen, die über den rationalen Zahlen als Koeffizienten linear unabhängig sind, nennt man rational unabhängig oder <a href="/wiki/Inkommensurabilit%C3%A4t_(Mathematik)" title="Inkommensurabilität (Mathematik)">inkommensurabel</a>. Die Zahlen <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lbrace 1,\,{\tfrac {1}{\sqrt {2}}}\rbrace }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <mn>1</mn> <mo>,</mo> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <msqrt> <mn>2</mn> </msqrt> </mfrac> </mstyle> </mrow> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lbrace 1,\,{\tfrac {1}{\sqrt {2}}}\rbrace }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/004a6f1effa0123f40acdb72048e61d92f90e193" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:7.935ex; height:4.176ex;" alt="{\displaystyle \lbrace 1,\,{\tfrac {1}{\sqrt {2}}}\rbrace }"> sind demnach rational unabhängig oder inkommensurabel, die Zahlen <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lbrace 1,\,{\tfrac {1}{\sqrt {2}}},1+{\sqrt {2}}\rbrace }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <mn>1</mn> <mo>,</mo> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <msqrt> <mn>2</mn> </msqrt> </mfrac> </mstyle> </mrow> <mo>,</mo> <mn>1</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lbrace 1,\,{\tfrac {1}{\sqrt {2}}},1+{\sqrt {2}}\rbrace }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bf83792e6f66db3a5cd65e1754c297264e80bbac" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:16.071ex; height:4.176ex;" alt="{\displaystyle \lbrace 1,\,{\tfrac {1}{\sqrt {2}}},1+{\sqrt {2}}\rbrace }"> dagegen rational abhängig. <div class="mw-heading mw-heading2"><h2 id="Verallgemeinerungen">Verallgemeinerungen</h2>[<a href="/w/index.php?title=Lineare_Unabh%C3%A4ngigkeit&veaction=edit&section=13" title="Abschnitt bearbeiten: Verallgemeinerungen" class="mw-editsection-visualeditor">Bearbeiten</a> | <a href="/w/index.php?title=Lineare_Unabh%C3%A4ngigkeit&action=edit&section=13" title="Quellcode des Abschnitts bearbeiten: Verallgemeinerungen">Quelltext bearbeiten</a>]</div> Die Definition linear unabhängiger Vektoren lässt sich analog auf Elemente eines <a href="/wiki/Modul_(Mathematik)" title="Modul (Mathematik)">Moduls</a> anwenden. In diesem Zusammenhang werden linear unabhängige Familien auch frei genannt (siehe auch: <a href="/wiki/Freier_Modul" title="Freier Modul">freier Modul</a>). Der Begriff der linearen Unabhängigkeit lässt sich weiter zu einer Betrachtung von unabhängigen Mengen verallgemeinern, siehe dazu <a href="/wiki/Matroid" title="Matroid">Matroid</a>. <div class="mw-heading mw-heading2"><h2 id="Literatur">Literatur</h2>[<a href="/w/index.php?title=Lineare_Unabh%C3%A4ngigkeit&veaction=edit&section=14" title="Abschnitt bearbeiten: Literatur" class="mw-editsection-visualeditor">Bearbeiten</a> | <a href="/w/index.php?title=Lineare_Unabh%C3%A4ngigkeit&action=edit&section=14" title="Quellcode des Abschnitts bearbeiten: Literatur">Quelltext bearbeiten</a>]</div> <ul><li><a href="/wiki/Siegfried_Bosch" title="Siegfried Bosch">Siegfried Bosch</a>: Lineare Algebra. 5. Auflage, Springer, Berlin/Heidelberg 2014, <a href="/wiki/Spezial:ISBN-Suche/9783642552595" class="internal mw-magiclink-isbn">ISBN 978-3-642-55259-5</a>, Kapitel 1.5.</li> <li><a href="/wiki/Albrecht_Beutelspacher" title="Albrecht Beutelspacher">Albrecht Beutelsbacher</a>: Lineare Algebra: Eine Einführung in die Wissenschaft der Vektoren, Abbildungen und Matrizen. 8. Auflage, Springer, Gießen 2014, <a href="/wiki/Spezial:ISBN-Suche/9783658024123" class="internal mw-magiclink-isbn">ISBN 978-3-658-02412-3</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="Einzelnachweise">Einzelnachweise</h2>[<a href="/w/index.php?title=Lineare_Unabh%C3%A4ngigkeit&veaction=edit&section=15" title="Abschnitt bearbeiten: Einzelnachweise" class="mw-editsection-visualeditor">Bearbeiten</a> | <a href="/w/index.php?title=Lineare_Unabh%C3%A4ngigkeit&action=edit&section=15" title="Quellcode des Abschnitts bearbeiten: Einzelnachweise">Quelltext bearbeiten</a>]</div> <ol class="references"> <li id="cite_note-1"><a href="#cite_ref-1">↑</a> Ilja Nikolajewitsch Bronstein, Konstantin Adolfowitsch Semendjajew: <cite style="font-style:italic"><a href="/wiki/Taschenbuch_der_Mathematik" title="Taschenbuch der Mathematik">Taschenbuch der Mathematik</a></cite>. 5. überarbeitete und erweiterte Auflage. Verlag Harri Deutsch, Thun und Frankfurt am Main 2001, <a href="/wiki/Spezial:ISBN-Suche/3817120052" class="internal mw-magiclink-isbn">ISBN 3-8171-2005-2</a>, S. 327.<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:Lineare+Unabh%C3%A4ngigkeit&rft.au=Ilja+Nikolajewitsch+Bronstein%2C+Konstantin+Adolfowitsch+Semendjajew&rft.btitle=Taschenbuch+der+Mathematik&rft.date=2001&rft.edition=5.+%C3%BCberarbeitete+und+erweiterte&rft.genre=book&rft.isbn=3817120052&rft.pages=327&rft.place=Thun+und+Frankfurt+am+Main&rft.pub=Verlag+Harri+Deutsch" style="display:none">  </li> <li id="cite_note-:0-2">↑ <a href="#cite_ref-:0_2-0">a</a> <a href="#cite_ref-:0_2-1">b</a> <a href="/wiki/Gerd_Fischer_(Mathematiker)" title="Gerd Fischer (Mathematiker)">Gerd Fischer</a>, Boris Springborn: <cite style="font-style:italic">Lineare Algebra. Eine Einführung für Studienanfänger</cite>. 19. Auflage. Springer, Berlin 2020, <a href="/wiki/Spezial:ISBN-Suche/9783662616444" class="internal mw-magiclink-isbn">ISBN 978-3-662-61644-4</a>, S. 100.<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:Lineare+Unabh%C3%A4ngigkeit&rft.au=Gerd+Fischer%2C+Boris+Springborn&rft.btitle=Lineare+Algebra.+Eine+Einf%C3%BChrung+f%C3%BCr+Studienanf%C3%A4nger&rft.date=2020&rft.edition=19.&rft.genre=book&rft.isbn=9783662616444&rft.pages=100&rft.place=Berlin&rft.pub=Springer" style="display:none">  </li> <li id="cite_note-:1-3">↑ <a href="#cite_ref-:1_3-0">a</a> <a href="#cite_ref-:1_3-1">b</a> <a href="/wiki/Albrecht_Beutelspacher" title="Albrecht Beutelspacher">Albrecht Beutelspacher</a>: <cite style="font-style:italic">Lineare Algebra. Eine Einführung in die Wissenschaft der Vektoren, Abbildungen und Matrizen</cite>. 8. Auflage. Springer, Wiesbaden 2014, <a href="/wiki/Spezial:ISBN-Suche/9783658024123" class="internal mw-magiclink-isbn">ISBN 978-3-658-02412-3</a>, S. 67.<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:Lineare+Unabh%C3%A4ngigkeit&rft.au=Albrecht+Beutelspacher&rft.btitle=Lineare+Algebra.+Eine+Einf%C3%BChrung+in+die+Wissenschaft+der+Vektoren%2C+Abbildungen+und+Matrizen&rft.date=2014&rft.edition=8.&rft.genre=book&rft.isbn=9783658024123&rft.pages=67&rft.place=Wiesbaden&rft.pub=Springer" style="display:none">  </li> </ol></div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?type=1x1" alt="" width="1" height="1" style="border: none; position: absolute;"></noscript> <div class="printfooter" data-nosnippet="">Abgerufen von „<a dir="ltr" href="https://de.wikipedia.org/w/index.php?title=Lineare_Unabhängigkeit&oldid=245488636">https://de.wikipedia.org/w/index.php?title=Lineare_Unabhängigkeit&oldid=245488636</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=Lineare+Unabh%C3%A4ngigkeit" title="Wir ermutigen dich dazu, ein Benutzerkonto zu erstellen und dich anzumelden. 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lang="ar" hreflang="ar" data-title="استقلال خطي" data-language-autonym="العربية" data-language-local-name="Arabisch" class="interlanguage-link-target">العربية</a></li><li class="interlanguage-link interwiki-az mw-list-item"><a href="https://az.wikipedia.org/wiki/X%C9%99tti_qeyri-as%C4%B1l%C4%B1l%C4%B1q" title="Xətti qeyri-asılılıq – Aserbaidschanisch" lang="az" hreflang="az" data-title="Xətti qeyri-asılılıq" data-language-autonym="Azərbaycanca" data-language-local-name="Aserbaidschanisch" class="interlanguage-link-target">Azərbaycanca</a></li><li class="interlanguage-link interwiki-bg mw-list-item"><a href="https://bg.wikipedia.org/wiki/%D0%9B%D0%B8%D0%BD%D0%B5%D0%B9%D0%BD%D0%B0_%D0%BD%D0%B5%D0%B7%D0%B0%D0%B2%D0%B8%D1%81%D0%B8%D0%BC%D0%BE%D1%81%D1%82" title="Линейна независимост – Bulgarisch" lang="bg" hreflang="bg" data-title="Линейна независимост" data-language-autonym="Български" data-language-local-name="Bulgarisch" class="interlanguage-link-target">Български</a></li><li class="interlanguage-link interwiki-bs mw-list-item"><a href="https://bs.wikipedia.org/wiki/Linearna_nezavisnost" title="Linearna nezavisnost – Bosnisch" lang="bs" hreflang="bs" data-title="Linearna nezavisnost" data-language-autonym="Bosanski" data-language-local-name="Bosnisch" class="interlanguage-link-target">Bosanski</a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Independ%C3%A8ncia_lineal" title="Independència lineal – Katalanisch" lang="ca" hreflang="ca" data-title="Independència lineal" 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/Line%C3%A1rn%C3%AD_nez%C3%A1vislost" title="Lineární nezávislost – Tschechisch" lang="cs" hreflang="cs" data-title="Lineární nezávislost" 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/Linear_independence" title="Linear independence – Englisch" lang="en" hreflang="en" data-title="Linear independence" 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/Lineara_sendependeco" title="Lineara sendependeco – Esperanto" lang="eo" hreflang="eo" data-title="Lineara sendependeco" 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/Dependencia_e_independencia_lineal" title="Dependencia e independencia lineal – Spanisch" lang="es" hreflang="es" data-title="Dependencia e independencia lineal" data-language-autonym="Español" data-language-local-name="Spanisch" class="interlanguage-link-target">Español</a></li><li class="interlanguage-link interwiki-et mw-list-item"><a href="https://et.wikipedia.org/wiki/Lineaarne_s%C3%B5ltuvus" title="Lineaarne sõltuvus – Estnisch" lang="et" hreflang="et" data-title="Lineaarne sõltuvus" data-language-autonym="Eesti" data-language-local-name="Estnisch" class="interlanguage-link-target">Eesti</a></li><li class="interlanguage-link interwiki-eu mw-list-item"><a href="https://eu.wikipedia.org/wiki/Dependentzia_eta_independentzia_lineal" title="Dependentzia eta independentzia lineal – Baskisch" lang="eu" hreflang="eu" data-title="Dependentzia eta independentzia lineal" data-language-autonym="Euskara" data-language-local-name="Baskisch" class="interlanguage-link-target">Euskara</a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D8%A7%D8%B3%D8%AA%D9%82%D9%84%D8%A7%D9%84_%D8%AE%D8%B7%DB%8C" title="استقلال خطی – Persisch" lang="fa" hreflang="fa" data-title="استقلال خطی" data-language-autonym="فارسی" data-language-local-name="Persisch" class="interlanguage-link-target">فارسی</a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/Lineaarinen_riippumattomuus" title="Lineaarinen riippumattomuus – Finnisch" lang="fi" hreflang="fi" data-title="Lineaarinen riippumattomuus" 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/Ind%C3%A9pendance_lin%C3%A9aire" title="Indépendance linéaire – Französisch" lang="fr" hreflang="fr" data-title="Indépendance linéaire" data-language-autonym="Français" data-language-local-name="Französisch" class="interlanguage-link-target">Français</a></li><li class="interlanguage-link interwiki-gl mw-list-item"><a href="https://gl.wikipedia.org/wiki/Independencia_linear" title="Independencia linear – Galicisch" lang="gl" hreflang="gl" data-title="Independencia linear" data-language-autonym="Galego" data-language-local-name="Galicisch" class="interlanguage-link-target">Galego</a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%AA%D7%9C%D7%95%D7%AA_%D7%9C%D7%99%D7%A0%D7%99%D7%90%D7%A8%D7%99%D7%AA" title="תלות ליניארית – Hebräisch" lang="he" hreflang="he" data-title="תלות ליניארית" data-language-autonym="עברית" data-language-local-name="Hebräisch" class="interlanguage-link-target">עברית</a></li><li class="interlanguage-link interwiki-hr mw-list-item"><a href="https://hr.wikipedia.org/wiki/Linearna_nezavisnost" title="Linearna nezavisnost – Kroatisch" lang="hr" hreflang="hr" data-title="Linearna nezavisnost" data-language-autonym="Hrvatski" data-language-local-name="Kroatisch" class="interlanguage-link-target">Hrvatski</a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/Line%C3%A1ris_f%C3%BCggetlens%C3%A9g" title="Lineáris függetlenség – Ungarisch" lang="hu" hreflang="hu" data-title="Lineáris függetlenség" 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/Kebebasan_linear" title="Kebebasan linear – Indonesisch" lang="id" hreflang="id" data-title="Kebebasan linear" data-language-autonym="Bahasa Indonesia" data-language-local-name="Indonesisch" class="interlanguage-link-target">Bahasa Indonesia</a></li><li class="interlanguage-link interwiki-is mw-list-item"><a href="https://is.wikipedia.org/wiki/L%C3%ADnulegt_%C3%B3h%C3%A6%C3%B0i" title="Línulegt óhæði – Isländisch" lang="is" hreflang="is" data-title="Línulegt óhæði" data-language-autonym="Íslenska" data-language-local-name="Isländisch" class="interlanguage-link-target">Íslenska</a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Indipendenza_lineare" title="Indipendenza lineare – Italienisch" lang="it" hreflang="it" data-title="Indipendenza lineare" 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/%E7%B7%9A%E5%9E%8B%E7%8B%AC%E7%AB%8B" title="線型独立 – Japanisch" lang="ja" hreflang="ja" data-title="線型独立" 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-kk mw-list-item"><a href="https://kk.wikipedia.org/wiki/%D0%A1%D1%8B%D0%B7%D1%8B%D2%9B%D1%82%D1%8B%D2%9B_%D1%82%D3%99%D1%83%D0%B5%D0%BB%D0%B4%D1%96%D0%BB%D1%96%D0%BA" title="Сызықтық тәуелділік – Kasachisch" lang="kk" hreflang="kk" data-title="Сызықтық тәуелділік" data-language-autonym="Қазақша" data-language-local-name="Kasachisch" class="interlanguage-link-target">Қазақша</a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EC%9D%BC%EC%B0%A8_%EB%8F%85%EB%A6%BD_%EC%A7%91%ED%95%A9" title="일차 독립 집합 – Koreanisch" lang="ko" hreflang="ko" data-title="일차 독립 집합" data-language-autonym="한국어" data-language-local-name="Koreanisch" class="interlanguage-link-target">한국어</a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Lineaire_onafhankelijkheid" title="Lineaire onafhankelijkheid – Niederländisch" lang="nl" hreflang="nl" data-title="Lineaire onafhankelijkheid" data-language-autonym="Nederlands" data-language-local-name="Niederländisch" class="interlanguage-link-target">Nederlands</a></li><li class="interlanguage-link interwiki-no mw-list-item"><a href="https://no.wikipedia.org/wiki/Line%C3%A6r_uavhengighet" title="Lineær uavhengighet – Norwegisch (Bokmål)" lang="nb" hreflang="nb" data-title="Lineær uavhengighet" data-language-autonym="Norsk bokmål" data-language-local-name="Norwegisch (Bokmål)" class="interlanguage-link-target">Norsk bokmål</a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Liniowa_niezale%C5%BCno%C5%9B%C4%87" title="Liniowa niezależność – Polnisch" lang="pl" hreflang="pl" data-title="Liniowa niezależność" data-language-autonym="Polski" data-language-local-name="Polnisch" class="interlanguage-link-target">Polski</a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Independ%C3%AAncia_linear" title="Independência linear – Portugiesisch" lang="pt" hreflang="pt" data-title="Independência linear" data-language-autonym="Português" data-language-local-name="Portugiesisch" class="interlanguage-link-target">Português</a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%9B%D0%B8%D0%BD%D0%B5%D0%B9%D0%BD%D0%B0%D1%8F_%D0%BD%D0%B5%D0%B7%D0%B0%D0%B2%D0%B8%D1%81%D0%B8%D0%BC%D0%BE%D1%81%D1%82%D1%8C" title="Линейная независимость – Russisch" lang="ru" hreflang="ru" data-title="Линейная независимость" data-language-autonym="Русский" data-language-local-name="Russisch" class="interlanguage-link-target">Русский</a></li><li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://simple.wikipedia.org/wiki/Linear_independence" title="Linear independence – einfaches Englisch" lang="en-simple" hreflang="en-simple" data-title="Linear independence" data-language-autonym="Simple English" data-language-local-name="einfaches Englisch" class="interlanguage-link-target">Simple English</a></li><li class="interlanguage-link interwiki-sl mw-list-item"><a href="https://sl.wikipedia.org/wiki/Linearna_neodvisnost" title="Linearna neodvisnost – Slowenisch" lang="sl" hreflang="sl" data-title="Linearna neodvisnost" data-language-autonym="Slovenščina" data-language-local-name="Slowenisch" class="interlanguage-link-target">Slovenščina</a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Linj%C3%A4rt_oberoende" title="Linjärt oberoende – Schwedisch" lang="sv" hreflang="sv" data-title="Linjärt oberoende" data-language-autonym="Svenska" data-language-local-name="Schwedisch" class="interlanguage-link-target">Svenska</a></li><li class="interlanguage-link interwiki-ta mw-list-item"><a href="https://ta.wikipedia.org/wiki/%E0%AE%A8%E0%AF%87%E0%AE%B0%E0%AE%BF%E0%AE%AF%E0%AE%B2%E0%AF%8D_%E0%AE%9A%E0%AE%BE%E0%AE%B0%E0%AF%8D%E0%AE%AA%E0%AE%BF%E0%AE%A9%E0%AF%8D%E0%AE%AE%E0%AF%88" title="நேரியல் சார்பின்மை – Tamil" lang="ta" hreflang="ta" data-title="நேரியல் சார்பின்மை" data-language-autonym="தமிழ்" data-language-local-name="Tamil" class="interlanguage-link-target">தமிழ்</a></li><li class="interlanguage-link interwiki-tg mw-list-item"><a href="https://tg.wikipedia.org/wiki/%D0%92%D0%BE%D0%B1%D0%B0%D1%81%D1%82%D0%B0%D0%B3%D0%B8%D0%B8_%D1%85%D0%B0%D1%82%D1%82%D3%A3" title="Вобастагии хаттӣ – Tadschikisch" lang="tg" hreflang="tg" data-title="Вобастагии хаттӣ" data-language-autonym="Тоҷикӣ" data-language-local-name="Tadschikisch" class="interlanguage-link-target">Тоҷикӣ</a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/Do%C4%9Frusal_ba%C4%9F%C4%B1ms%C4%B1zl%C4%B1k" title="Doğrusal bağımsızlık – Türkisch" lang="tr" hreflang="tr" data-title="Doğrusal bağımsızlık" data-language-autonym="Türkçe" data-language-local-name="Türkisch" class="interlanguage-link-target">Türkçe</a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%9B%D1%96%D0%BD%D1%96%D0%B9%D0%BD%D0%BE_%D0%BD%D0%B5%D0%B7%D0%B0%D0%BB%D0%B5%D0%B6%D0%BD%D1%96_%D0%B2%D0%B5%D0%BA%D1%82%D0%BE%D1%80%D0%B8" title="Лінійно незалежні вектори – Ukrainisch" lang="uk" hreflang="uk" data-title="Лінійно незалежні вектори" 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/%D9%84%DA%A9%DB%8C%D8%B1%DB%8C_%D8%A2%D8%B2%D8%A7%D8%AF%DB%8C" title="لکیری آزادی – Urdu" lang="ur" hreflang="ur" data-title="لکیری آزادی" 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/%C4%90%E1%BB%99c_l%E1%BA%ADp_tuy%E1%BA%BFn_t%C3%ADnh" title="Độc lập tuyến tính – Vietnamesisch" lang="vi" hreflang="vi" data-title="Độc lập tuyến tính" 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/%E7%B7%9A%E6%80%A7%E7%84%A1%E9%97%9C" title="線性無關 – Chinesisch" lang="zh" hreflang="zh" data-title="線性無關" data-language-autonym="中文" data-language-local-name="Chinesisch" class="interlanguage-link-target">中文</a></li><li class="interlanguage-link interwiki-zh-yue mw-list-item"><a href="https://zh-yue.wikipedia.org/wiki/%E7%B7%9A%E6%80%A7%E7%8D%A8%E7%AB%8B%E6%80%A7" title="線性獨立性 – Kantonesisch" lang="yue" hreflang="yue" data-title="線性獨立性" data-language-autonym="粵語" data-language-local-name="Kantonesisch" class="interlanguage-link-target">粵語</a></li> </ul> <div class="after-portlet after-portlet-lang"><a href="https://www.wikidata.org/wiki/Special:EntityPage/Q27670#sitelinks-wikipedia" title="Links auf Artikel in anderen Sprachen bearbeiten" class="wbc-editpage">Links bearbeiten</a></div> </div> </nav> </div> </div> <footer id="footer" class="mw-footer" > <ul id="footer-info"> <li id="footer-info-lastmod"> Diese Seite wurde zuletzt am 30. 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Lineare Unabhängigkeit – Wikipedia