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<title>Rangsatz</title>

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<h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">Rangsatz</span></h1>

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<div id="mw-content-text" class="mw-body-content mw-content-ltr" lang="de" dir="ltr"><div class="mw-content-ltr mw-parser-output" lang="de" dir="ltr"><p>Der <b>Rangsatz</b> (auch <b>Dimensionssatz</b> oder <b>Kern-Bild-Satz</b><sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup>) ist ein <a href="Satz_(Mathematik)" title="Satz (Mathematik)">Satz</a> aus dem <a href="Teilgebiete_der_Mathematik" title="Teilgebiete der Mathematik">mathematischen Teilgebiet</a> der <a href="Lineare_Algebra" title="Lineare Algebra">linearen Algebra</a>. Er zeigt einen Zusammenhang zwischen den <a href="Dimension_(Mathematik)" title="Dimension (Mathematik)">Dimensionen</a> der <a href="Definitionsmenge" title="Definitionsmenge">Definitionsmenge</a>, des <a href="Kern_(Algebra)" title="Kern (Algebra)">Kerns</a> und des <a href="Bild_(Mathematik)" title="Bild (Mathematik)">Bildes</a> einer <a href="Lineare_Abbildung" title="Lineare Abbildung">linearen Abbildung</a> zwischen zwei <a href="Vektorraum" title="Vektorraum">Vektorräumen</a> auf. Damit stellt er ein wichtiges Hilfsmittel zur Untersuchung von linearen Abbildungen dar.

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<div class="mw-heading mw-heading2"><h2 id="Aussage">Aussage</h2></div>

<div class="mw-heading mw-heading3"><h3 id="Formulierung_für_lineare_Abbildungen"><span id="Formulierung_f.C3.BCr_lineare_Abbildungen"></span>Formulierung für lineare Abbildungen</h3></div>

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<annotation encoding="application/x-tex">{\displaystyle f\colon V\to W}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/fa58648b9aa9476995b2b41de27a9a591e631f56.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.149ex; height:2.509ex;" alt="{\displaystyle f\colon V\to W}" loading="lazy"></span> eine <a href="Lineare_Abbildung" title="Lineare Abbildung">lineare Abbildung</a> von einem <a href="Vektorraum" title="Vektorraum">Vektorraum</a> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V}">

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<annotation encoding="application/x-tex">{\displaystyle V}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/af0f6064540e84211d0ffe4dac72098adfa52845.svg" 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}" loading="lazy"></span> in einen Vektorraum <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle W}">

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<annotation encoding="application/x-tex">{\displaystyle W}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/54a9c4c547f4d6111f81946cad242b18298d70b7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.435ex; height:2.176ex;" alt="{\displaystyle W}" loading="lazy"></span>, dann gilt für die <a href="Dimension_(Mathematik)" title="Dimension (Mathematik)">Dimensionen</a> der <a href="Definitionsmenge" title="Definitionsmenge">Definitionsmenge</a> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V}">

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<annotation encoding="application/x-tex">{\displaystyle V}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/af0f6064540e84211d0ffe4dac72098adfa52845.svg" 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}" loading="lazy"></span>, des <a href="Kern_(Algebra)" title="Kern (Algebra)">Kerns</a> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {ker} (f)}">

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<annotation encoding="application/x-tex">{\displaystyle \mathrm {ker} (f)}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d1568a4125ff09078d4202220c8a2b99f456f670.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.259ex; height:2.843ex;" alt="{\displaystyle \mathrm {ker} (f)}" loading="lazy"></span> und des <a href="Bild_(Mathematik)" title="Bild (Mathematik)">Bildes</a> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {im} (f)}">

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<annotation encoding="application/x-tex">{\displaystyle \mathrm {im} (f)}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/5633fabe287b50cb28af88e03af812153a2d7d37.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.671ex; height:2.843ex;" alt="{\displaystyle \mathrm {im} (f)}" loading="lazy"></span> der Abbildung <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}">

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<annotation encoding="application/x-tex">{\displaystyle f}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/132e57acb643253e7810ee9702d9581f159a1c61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" loading="lazy"></span> die Gleichung

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<annotation encoding="application/x-tex">{\displaystyle \dim V=\dim \mathrm {ker} (f)+\dim \mathrm {im} (f)}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/77b5d23176a0f5f9cbbc97412dc83d9c1d378311.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:32.443ex; height:2.843ex;" alt="{\displaystyle \dim V=\dim \mathrm {ker} (f)+\dim \mathrm {im} (f)}" loading="lazy"></span>.<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup></dd></dl>

<p>Mit den Bezeichnungen <a href="Defekt_(Mathematik)" title="Defekt (Mathematik)">Defekt</a> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {def} (f)}">

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<annotation encoding="application/x-tex">{\displaystyle \mathrm {def} (f)}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/8554a2ae6e77694bedf68db20b60309fe50271d9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.278ex; height:2.843ex;" alt="{\displaystyle \mathrm {def} (f)}" loading="lazy"></span> für die Dimension des Kerns und <a href="Rang_(Lineare_Algebra)" title="Rang (Lineare Algebra)">Rang</a> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {rk} (f)}">

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<annotation encoding="application/x-tex">{\displaystyle \mathrm {rk} (f)}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e999720b285aa574397333dbe3056675513ec5c0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.227ex; height:2.843ex;" alt="{\displaystyle \mathrm {rk} (f)}" loading="lazy"></span> (von engl. rank) für die Dimension des Bildes der Abbildung <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}">

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<annotation encoding="application/x-tex">{\displaystyle f}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/132e57acb643253e7810ee9702d9581f159a1c61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" loading="lazy"></span> liest sich der Rangsatz als

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<annotation encoding="application/x-tex">{\displaystyle \dim V=\mathrm {def} (f)+\mathrm {rk} (f)}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/90c57c24896763bc7e77da3707241d306b9f5f7a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.493ex; height:2.843ex;" alt="{\displaystyle \dim V=\mathrm {def} (f)+\mathrm {rk} (f)}" loading="lazy"></span>.</dd></dl>

<p>Der Satz gilt für Vektorräume beliebiger (auch unendlicher) Dimension.

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<div class="mw-heading mw-heading3"><h3 id="Formulierung_für_Matrizen"><span id="Formulierung_f.C3.BCr_Matrizen"></span>Formulierung für Matrizen</h3></div>

<p>Jede lineare Abbildung zwischen endlichdimensionalen Vektorräumen lässt sich mithilfe einer Matrix darstellen (siehe <a href="Abbildungsmatrix" title="Abbildungsmatrix">Abbildungsmatrix</a>). Umgekehrt definiert jede Matrix <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}">

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<annotation encoding="application/x-tex">{\displaystyle A}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7daff47fa58cdfd29dc333def748ff5fa4c923e3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}" loading="lazy"></span> durch die Vorschrift <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\mapsto Ax}">

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<annotation encoding="application/x-tex">{\displaystyle x\mapsto Ax}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/cedfe896a1dd7e5e193c6d25026824149fff79e0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.017ex; height:2.176ex;" alt="{\displaystyle x\mapsto Ax}" loading="lazy"></span> eine lineare Abbildung. Aufgrund dieses engen Zusammenhangs zwischen linearen Abbildungen und Matrizen lässt sich der Rangsatz auch für Matrizen formulieren: Ist <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}">

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<annotation encoding="application/x-tex">{\displaystyle A}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7daff47fa58cdfd29dc333def748ff5fa4c923e3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}" loading="lazy"></span> eine Matrix mit <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m}">

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<annotation encoding="application/x-tex">{\displaystyle m}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0a07d98bb302f3856cbabc47b2b9016692e3f7bc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.04ex; height:1.676ex;" alt="{\displaystyle m}" loading="lazy"></span> Zeilen und <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}">

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<annotation encoding="application/x-tex">{\displaystyle n}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a601995d55609f2d9f5e233e36fbe9ea26011b3b.svg" 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}" loading="lazy"></span> Spalten, so gilt

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<annotation encoding="application/x-tex">{\displaystyle n=\dim \mathrm {ker} (A)+\dim \mathrm {im} (A)}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/388a8568db2c7081adb905d3e4fc7cdae6406b14.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:28.717ex; height:2.843ex;" alt="{\displaystyle n=\dim \mathrm {ker} (A)+\dim \mathrm {im} (A)}" loading="lazy"></span>,</dd></dl>

<p>wobei <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {ker} (A)}">

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<annotation encoding="application/x-tex">{\displaystyle \mathrm {ker} (A)}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f26b5de9e2a0709ec3561d6d2b503fd84e4aa299.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.724ex; height:2.843ex;" alt="{\displaystyle \mathrm {ker} (A)}" loading="lazy"></span> der Kern und <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {im} (A)}">

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<annotation encoding="application/x-tex">{\displaystyle \mathrm {im} (A)}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e3f02b6f4309b813a2213a68a4129acc04950ba9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.135ex; height:2.843ex;" alt="{\displaystyle \mathrm {im} (A)}" loading="lazy"></span> das Bild der Matrix ist.<sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> Die Dimension des Bildes einer Matrix ist ihr Rang. Bezeichnet man den Rang mit <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r}">

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<annotation encoding="application/x-tex">{\displaystyle r}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0d1ecb613aa2984f0576f70f86650b7c2a132538.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.049ex; height:1.676ex;" alt="{\displaystyle r}" loading="lazy"></span>, so liest sich der Rangsatz als

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<annotation encoding="application/x-tex">{\displaystyle n=\dim \mathrm {ker} (A)+r}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/234dff91c63d05ea626fb434bc54b79d4765815a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.368ex; height:2.843ex;" alt="{\displaystyle n=\dim \mathrm {ker} (A)+r}" loading="lazy"></span>.</dd></dl>

<div class="mw-heading mw-heading2"><h2 id="Beweise">Beweise</h2></div>

<div class="mw-heading mw-heading3"><h3 id="Beweis_über_den_Homomorphiesatz"><span id="Beweis_.C3.BCber_den_Homomorphiesatz"></span>Beweis über den Homomorphiesatz</h3></div>

<p>Der Satz folgt unmittelbar aus dem <a href="Homomorphiesatz" title="Homomorphiesatz">Homomorphiesatz</a>

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<annotation encoding="application/x-tex">{\displaystyle \mathrm {im} (f)\cong V/\mathrm {ker} (f)}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e8915275599d18c5055a09b45149097396f86768.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.978ex; height:2.843ex;" alt="{\displaystyle \mathrm {im} (f)\cong V/\mathrm {ker} (f)}" loading="lazy"></span>.</dd></dl>

<p>Da der <a href="Faktorraum" title="Faktorraum">Faktorraum</a> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V/\mathrm {ker} (f)}">

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<annotation encoding="application/x-tex">{\displaystyle V/\mathrm {ker} (f)}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c58967f973148ebb7c07522055eded7c706150e1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.209ex; height:2.843ex;" alt="{\displaystyle V/\mathrm {ker} (f)}" loading="lazy"></span> <a href="Isomorph" class="mw-redirect" title="Isomorph">isomorph</a> zu einem <a href="Komplement%C3%A4rraum" title="Komplementärraum">Komplementärraum</a> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U}">

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<annotation encoding="application/x-tex">{\displaystyle U}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/458a728f53b9a0274f059cd695e067c430956025.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.783ex; height:2.176ex;" alt="{\displaystyle U}" loading="lazy"></span> von <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {ker} (f)}">

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<annotation encoding="application/x-tex">{\displaystyle \mathrm {ker} (f)}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d1568a4125ff09078d4202220c8a2b99f456f670.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.259ex; height:2.843ex;" alt="{\displaystyle \mathrm {ker} (f)}" loading="lazy"></span> in <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V}">

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<annotation encoding="application/x-tex">{\displaystyle V}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/af0f6064540e84211d0ffe4dac72098adfa52845.svg" 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}" loading="lazy"></span> ist, gilt

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<annotation encoding="application/x-tex">{\displaystyle \mathrm {im} (f)\cong U}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/79006da3fe4f8b3dcb6b79c2cf84dec5763f8f18.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.552ex; height:2.843ex;" alt="{\displaystyle \mathrm {im} (f)\cong U}" loading="lazy"></span>.</dd></dl>

<p>Nachdem nun

</p>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle V=\mathrm {ker} (f)\oplus U}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9cabbd7425b6b7d36d24a591ce1e70cc139b4244.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.768ex; height:2.843ex;" alt="{\displaystyle V=\mathrm {ker} (f)\oplus U}" loading="lazy"></span></dd></dl>

<p>ist, folgt aus der Äquivalenz von Isomorphie und Gleichheit der Dimension

</p>

</mrow>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \dim V=\dim \mathrm {ker} (f)+\dim U=\dim \mathrm {ker} (f)+\dim \mathrm {im} (f)}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/05924cc9144a617feec20b79edc71d5e10e754e1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:54.949ex; height:2.843ex;" alt="{\displaystyle \dim V=\dim \mathrm {ker} (f)+\dim U=\dim \mathrm {ker} (f)+\dim \mathrm {im} (f)}" loading="lazy"></span>.</dd></dl>

<div class="mw-heading mw-heading3"><h3 id="Beweis_durch_Basisergänzung"><span id="Beweis_durch_Basiserg.C3.A4nzung"></span>Beweis durch Basisergänzung</h3></div>

<p>Ist eine Menge <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B\subset \mathrm {ker} (f)}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle B\subset \mathrm {ker} (f)}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/05cd4dce3ca6d775fa96c32b3b5f60e8c04ddef3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.122ex; height:2.843ex;" alt="{\displaystyle B\subset \mathrm {ker} (f)}" loading="lazy"></span> eine <a href="Basis_(Vektorraum)" title="Basis (Vektorraum)">Basis</a> von <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {ker} (f)}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathrm {ker} (f)}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d1568a4125ff09078d4202220c8a2b99f456f670.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.259ex; height:2.843ex;" alt="{\displaystyle \mathrm {ker} (f)}" loading="lazy"></span>, die durch eine Menge <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle A}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7daff47fa58cdfd29dc333def748ff5fa4c923e3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}" loading="lazy"></span> mit <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\cap B=\emptyset }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle A\cap B=\emptyset }</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b0814f1814eb415cc84354bf261d7f9ed62367d1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:10.351ex; height:2.509ex;" alt="{\displaystyle A\cap B=\emptyset }" loading="lazy"></span> zu einer Basis <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\cup B}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle A\cup B}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/cdb575990bcfbcdf616aa6fd76e8b30bf7fd2169.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.09ex; height:2.176ex;" alt="{\displaystyle A\cup B}" loading="lazy"></span> von <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle V}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/af0f6064540e84211d0ffe4dac72098adfa52845.svg" 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}" loading="lazy"></span> ergänzt wird (<span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle A}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7daff47fa58cdfd29dc333def748ff5fa4c923e3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}" loading="lazy"></span> ist dann eine Basis eines Komplementärraums von <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {ker} (f)}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathrm {ker} (f)}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d1568a4125ff09078d4202220c8a2b99f456f670.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.259ex; height:2.843ex;" alt="{\displaystyle \mathrm {ker} (f)}" loading="lazy"></span>), dann ist

</p>

<mrow>

<mrow>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f(A)=\left\{f(a)\mid a\in A\right\}}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/55c383d8efd7eaf7fb07faf83e3e19c2ba8a5f39.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.323ex; height:2.843ex;" alt="{\displaystyle f(A)=\left\{f(a)\mid a\in A\right\}}" loading="lazy"></span></dd></dl>

<p>eine Basis des Bildes <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {im} (f)}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathrm {im} (f)}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/5633fabe287b50cb28af88e03af812153a2d7d37.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.671ex; height:2.843ex;" alt="{\displaystyle \mathrm {im} (f)}" loading="lazy"></span>. Betrachtet man nun die <a href="Einschr%C3%A4nkung" title="Einschränkung">Einschränkung</a> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f^{\prime }}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f^{\prime }}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/af4ba7eed864135ec5838c81b7692e19556580d8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.005ex; height:2.843ex;" alt="{\displaystyle f^{\prime }}" loading="lazy"></span> von <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/132e57acb643253e7810ee9702d9581f159a1c61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" loading="lazy"></span> auf den <a href="Lineare_H%C3%BClle" title="Lineare Hülle">Spann</a> (die lineare Hülle)

</p>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle U=\mathrm {span} (A)}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/76dd485450af322da0ca2a43dc04150c065e4912.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.097ex; height:2.843ex;" alt="{\displaystyle U=\mathrm {span} (A)}" loading="lazy"></span>,</dd></dl>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f^{\prime }}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/af4ba7eed864135ec5838c81b7692e19556580d8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.005ex; height:2.843ex;" alt="{\displaystyle f^{\prime }}" loading="lazy"></span> <a href="Injektivit%C3%A4t" class="mw-redirect" title="Injektivität">injektiv</a> und

</p>

</mrow>

<msup>

</mrow>

</msup>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathrm {im} (f^{\prime })=\mathrm {im} (f)}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b1c72ecebb6a6c3d3b7f51862aae2efa320b4021.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.166ex; height:3.009ex;" alt="{\displaystyle \mathrm {im} (f^{\prime })=\mathrm {im} (f)}" loading="lazy"></span>.</dd></dl>

<p>Somit ist <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f^{\prime }}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f^{\prime }}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/af4ba7eed864135ec5838c81b7692e19556580d8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.005ex; height:2.843ex;" alt="{\displaystyle f^{\prime }}" loading="lazy"></span> ein Isomorphismus zwischen <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle U}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/458a728f53b9a0274f059cd695e067c430956025.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.783ex; height:2.176ex;" alt="{\displaystyle U}" loading="lazy"></span> und dem Bild von <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/132e57acb643253e7810ee9702d9581f159a1c61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" loading="lazy"></span>. Daher gilt

</p>

<mrow>

</mrow>

<mrow>

</mrow>

</mrow>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \dim V=\left|A\right|+\left|B\right|=\dim U+\dim \mathrm {ker} (f)=\dim \mathrm {im} (f)+\dim \mathrm {ker} (f)}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1bf6f84db5b245e14d7c92a57760ba92ec486267.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:66.982ex; height:2.843ex;" alt="{\displaystyle \dim V=\left|A\right|+\left|B\right|=\dim U+\dim \mathrm {ker} (f)=\dim \mathrm {im} (f)+\dim \mathrm {ker} (f)}" loading="lazy"></span>.</dd></dl>

<p>Der Homomorphiesatz folgt ebenfalls – durch Übergang vom Komplementärraum zum Faktorraum.

</p>

<div class="mw-heading mw-heading2"><h2 id="Folgerungen">Folgerungen</h2></div>

<p>Im endlichdimensionalen Fall lässt sich mithilfe des Rangsatzes die Dimension des Bildraums aus der Dimension des Kerns als

</p>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \dim \mathrm {im} (f)=\dim V-\dim \mathrm {ker} (f)}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b1c03d4e054bc1166638dbba23b30545eebb0004.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:32.443ex; height:2.843ex;" alt="{\displaystyle \dim \mathrm {im} (f)=\dim V-\dim \mathrm {ker} (f)}" loading="lazy"></span></dd></dl>

<p>berechnen. Entsprechend gilt umgekehrt auch

</p>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \dim \mathrm {ker} (f)=\dim V-\dim \mathrm {im} (f)}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c0c9c3e214ff3e5069656efb8f553118688b2102.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:32.443ex; height:2.843ex;" alt="{\displaystyle \dim \mathrm {ker} (f)=\dim V-\dim \mathrm {im} (f)}" loading="lazy"></span>.</dd></dl>

<p>Im unendlichdimensionalen Fall lässt sich mittels des Rangsatzes die Dimension des Bildraums nicht aus der Dimension des Kerns (oder umgekehrt) berechnen, wenn der Kern dieselbe Dimension wie der gesamte Raum besitzt. Andernfalls ist die Dimension des Bildraums <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle W}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle W}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/54a9c4c547f4d6111f81946cad242b18298d70b7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.435ex; height:2.176ex;" alt="{\displaystyle W}" loading="lazy"></span> gleich der Dimension von <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle V}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/af0f6064540e84211d0ffe4dac72098adfa52845.svg" 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}" loading="lazy"></span>.

</p>

<div class="mw-heading mw-heading2"><h2 id="Verallgemeinerung">Verallgemeinerung</h2></div>

<p>Eine weitreichende Verallgemeinerung des Rangsatzes ist die Aussage, dass die alternierende Summe der Dimensionen der einzelnen Komponenten eines <a href="Kettenkomplex" title="Kettenkomplex">Kettenkomplexes</a> gleich der alternierenden Summe der Dimensionen seiner Homologiegruppen ist. Siehe dazu die <a href="Kettenkomplex#Euler-Charakteristik" title="Kettenkomplex">Euler-Charakteristik eines Kettenkomplexes</a>.

</p>

<div class="mw-heading mw-heading2"><h2 id="Siehe_auch">Siehe auch</h2></div>

<ul><li><a href="Dimensionsformel" title="Dimensionsformel">Dimensionsformel</a></li></ul>

<div class="mw-heading mw-heading2"><h2 id="Literatur">Literatur</h2></div>

<ul><li><a href="Albrecht_Beutelspacher" title="Albrecht Beutelspacher">Albrecht Beutelspacher</a>: <i>Lineare Algebra</i>. 8. Auflage. Springer Spektrum, 2014, ISBN 978-3-658-02412-3, S. 165–166.</li>

<li>Tilo Arens et al.: <i>Mathematik</i>. 5. Auflage. Springer Spektrum. 2023, ISBN 978-3-662-64388-4, S. 629–630.</li></ul>

<ul><li><a href="Gilbert_Strang" title="Gilbert Strang">Gilbert Strang</a>: <i>Lineare Algebra</i>. Springer. Berlin / Heidelberg 2003, ISBN 978-3-540-43949-3, S. 181–187.</li>

<li><a href="Hans-Joachim_Kowalsky" title="Hans-Joachim Kowalsky">Hans-Joachim Kowalsky</a>, <a href="Gerhard_O._Michler" title="Gerhard O. Michler">Gerhard O. Michler</a>: <cite style="font-style:italic">Lineare Algebra</cite>. 12. Auflage. De Gruyter, 2003, ISBN 3-11-017963-6, <span style="white-space:nowrap">S.<span style="display:inline-block;width:.2em"> </span>58 (Satz 3.2.13)</span>, <a href="Digital_Object_Identifier" title="Digital Object Identifier">doi</a>:<span class="uri-handle" style="white-space:nowrap"><a rel="nofollow" class="external text" href="https://doi.org/10.1515/9783110200041">10.1515/9783110200041</a></span>.<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:Rangsatz&rft.au=Hans-Joachim+Kowalsky%2C+Gerhard+O.+Michler&rft.btitle=Lineare+Algebra&rft.date=2003&rft.doi=10.1515%2F9783110200041&rft.edition=12.&rft.genre=book&rft.isbn=3110179636&rft.pages=58+%28Satz+3.2.13%29&rft.pub=De+Gruyter" style="display:none"> </span></li></ul>

<div class="mw-heading mw-heading2"><h2 id="Weblinks">Weblinks</h2></div>

<ul><li>Robert Milson: <a rel="nofollow" class="external text" href="https://planetmath.org/RankNullityTheorem"><i>Rank-nullity theorem</i>.</a> In: <i><a href="PlanetMath" title="PlanetMath">PlanetMath</a>.</i> (englisch)</li>

<li>Rahmi Jackson: <a rel="nofollow" class="external text" href="https://mathworld.wolfram.com/Rank-NullityTheorem.html"><i>Rank-Nullity Theorem</i>.</a> In: <i><a href="MathWorld" title="MathWorld">MathWorld</a></i> (englisch).</li></ul>

<div class="mw-heading mw-heading2"><h2 id="Einzelnachweise">Einzelnachweise</h2></div>

<li id="cite_note-1"><span class="mw-cite-backlink"><a href="#cite_ref-1">↑</a></span> <span class="reference-text">Gerhard Merziger, Thomas Wirth: <cite style="font-style:italic">Repetitorium höhere Mathematik</cite>. 7. Auflage. Binomi Verlag, Barsinghausen 2015, ISBN 978-3-923923-32-8, <span style="white-space:nowrap">S.<span style="display:inline-block;width:.2em"> </span>189</span>.<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:Rangsatz&rft.au=Gerhard+Merziger%2C+Thomas+Wirth&rft.btitle=Repetitorium+h%C3%B6here+Mathematik&rft.date=2015&rft.edition=7.&rft.genre=book&rft.isbn=9783923923328&rft.pages=189&rft.place=Barsinghausen&rft.pub=Binomi+Verlag" style="display:none"> </span></span>

</li>

<li id="cite_note-2"><span class="mw-cite-backlink"><a href="#cite_ref-2">↑</a></span> <span class="reference-text">Siegfried Bosch: <cite style="font-style:italic">Lineare Algebra</cite>. 6. Auflage. Springer, Berlin / Heidelberg 2021, ISBN 978-3-662-62615-3, <span style="white-space:nowrap">S.<span style="display:inline-block;width:.2em"> </span>80</span>.<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:Rangsatz&rft.au=Siegfried+Bosch&rft.btitle=Lineare+Algebra&rft.date=2021&rft.edition=6.&rft.genre=book&rft.isbn=9783662626153&rft.pages=80&rft.place=Berlin+%2F+Heidelberg&rft.pub=Springer" style="display:none"> </span></span>

</li>

<li id="cite_note-3"><span class="mw-cite-backlink"><a href="#cite_ref-3">↑</a></span> <span class="reference-text">Steffen Goebbels, Stefan Ritter: <cite style="font-style:italic">Mathematik verstehen und anwenden: Differenzial- und Integralrechnung, Lineare Algebra</cite>. 4. Auflage. Springer, Berlin / Heidelberg 2023, ISBN 978-3-662-68366-8, <span style="white-space:nowrap">S.<span style="display:inline-block;width:.2em"> </span>587</span>.<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:Rangsatz&rft.au=Steffen+Goebbels%2C+Stefan+Ritter&rft.btitle=Mathematik+verstehen+und+anwenden%3A+Differenzial-+und+Integralrechnung%2C+Lineare+Algebra&rft.date=2023&rft.edition=4.&rft.genre=book&rft.isbn=9783662683668&rft.pages=587&rft.place=Berlin+%2F+Heidelberg&rft.pub=Springer" style="display:none"> </span></span>

</li>

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