- die Menge
ist Basis von ![{\displaystyle \mathbf {U} }](https://wikimedia.org/api/rest_v1/media/math/render/svg/b2141bec2344e3dc5241ff50b0fd366755e00223)
![{\displaystyle s_{1}\,{\vec {u}}_{1}+s_{2}\,{\vec {u}}_{2}+s_{3}\,{\vec {u}}_{3}\Rightarrow s_{1}\,{\begin{pmatrix}0\\2\\3\\1\end{pmatrix}}+s_{2}\,{\begin{pmatrix}0\\1\\1\\0\end{pmatrix}}+s_{3}\,{\begin{pmatrix}0\\0\\1\\0\end{pmatrix}}={\begin{pmatrix}0\\0\\0\\0\end{pmatrix}}\Rightarrow {\begin{matrix}s_{1}=0\\2s_{1}+s_{2}=0\\3s_{1}+s_{2}+s_{3}=0\end{matrix}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2a9ffb1b567751511e3452bd81c36d6e4f396186)
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Koeffizienten des linearen Gleichungssystem
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Basis
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- Schwerpunkt von Polygon
![{\displaystyle {\vec {S}}={\frac {1}{n}}\,\left({\vec {p}}_{1}+{\vec {p}}_{2}+\dots +p_{n}\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fe0ac4c628983d0d4f7ada88ea90cf068f7b693b)
- Wenn Nullvektor durch Koeffizienten ≠ 0 erzeugbar ist LGS linear abhängig.
- Beispiel 1
→ linear abhängig
- Beispiel 2
![{\displaystyle {\begin{pmatrix}0\\0\end{pmatrix}}={\begin{pmatrix}1\\2\end{pmatrix}}\,s_{1}+{\begin{pmatrix}3\\1\end{pmatrix}}\,s_{2}+{\begin{pmatrix}-3\\9\end{pmatrix}}\,s_{3}={\begin{pmatrix}s_{1}+3\,s_{2}-3\,s_{3}\\2\,s_{1}+s_{2}+9s_{3}\end{pmatrix}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/292db0edbcd7c396710a356c66bad7e30e992031)
- gleichsetzen
![{\displaystyle 0=s_{1}+3\,s_{2}-3\,s_{3}=2\,s_{1}+s_{2}+9\,s_{3}\Rightarrow s_{1}=-6\,s_{3};\ s_{2}=3\,s_{3}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f6b9a68bf1edfa63951e6ff6b4e268e31ed74538)
- für
erhält man Nullvektor → linear abhängig
- Beispiel 3
→ linear unabhängig
- Beispiel 4
→ linear unabhängig
- Beispiel 1
rt…sin(2x) gn…cos x bl…cos(2x)
![{\displaystyle \left\{0=s_{1}\,\sin \left(2\,x\right)+s_{2}\,\cos x+s_{3}\,\cos \left(2\,x\right)\ \left|\ x\in \left[0,1\right]\right.\right\}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b897e9d01108082413b3040ca19f568ebf98f62f)
- mit
![{\displaystyle x=0\Rightarrow 0=s_{2}+s_{3}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/03a6c4d4d276f1eb53a009211c3b30cace7866a4)
- mit
![{\displaystyle x={\frac {\pi }{4}}\Rightarrow 0=s_{1}+s_{2}\,{\frac {1}{\sqrt {2}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6535629b2d798bc50d6dbd4fd5fa6cdc2073be83)
- mit
![{\displaystyle x={\frac {\pi }{6}}\Rightarrow 0=s_{1}\,{\frac {\sqrt {3}}{2}}+s_{2}\,{\frac {\sqrt {3}}{2}}+s_{2}\,{\frac {1}{3}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7a602db36f938004c90463cc59735b3977c96919)
- Beispiel 2
rt…e2x gn…3·e-2x bl…4·cosh(2x)
![{\displaystyle \left\{\left.0=s_{1}\,e^{2\,x}+s_{2}\,3\,e^{-2\,x}+s_{3}\,4\,\cosh \left(2\,x\right)\ \right|\ x\in \left[0,1\right]\right\}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/02608271b751e5e2675613a7c897356bd9b03bc6)
- Beispiel 3
rt…x(x+1) gn…x(1-x) bl…x·π-1
![{\displaystyle \left\{\left.0=s_{1}\,x\,\left(x+1\right)+s_{2}\,x\,\left(1-x\right)+s_{3}\,{\frac {x}{\pi }}\ \right|\ x\in \left[0,1\right]\right\}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/72d4420a734cedac721e52849c42e9a56710f7ac)
- Matrizenmultiplikation
![{\displaystyle {\begin{pmatrix}a&b&c\end{pmatrix}}\,{\begin{pmatrix}x\\y\\z\end{pmatrix}}=a\,x+b\,y+c\,z}](https://wikimedia.org/api/rest_v1/media/math/render/svg/069d72a28157ca331e0507d8acb1f57d5d6630c9)
![{\displaystyle {\begin{pmatrix}{*}&{*}&{*}\end{pmatrix}}\,{\begin{pmatrix}{*}\\{*}\\{*}\end{pmatrix}}=*\qquad {\begin{pmatrix}{*}\\{*}\\{*}\end{pmatrix}}\,{\begin{pmatrix}{*}&{*}&{*}\end{pmatrix}}={\begin{pmatrix}{*}&{*}&{*}\\{*}&{*}&{*}\\{*}&{*}&{*}\end{pmatrix}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e977b417246965478b3425420ccd04bf7aa02fd2)
- Inverse Matrix
![{\displaystyle \mathbf {M} \,\mathbf {M} ^{-1}=\mathbf {I} }](https://wikimedia.org/api/rest_v1/media/math/render/svg/f2a42ead7fd12335a29a22724514856d352d02fb)
mit Element von ![{\displaystyle \mathbf {A} ^{-1}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1e4a37527a4700b3ad51e31a1fba1dbfc2a5af22)
- folglich:
(bei 2×2)
(bei 3×3)
- Eigenwert
![{\displaystyle \det \left(\mathbf {M} -\lambda \,\mathbf {I} \right)=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b67a35231a3398405f7d3bee68a9277c984b75eb)
- Eigenwerte sind Nullstellen des resultierenden Polynoms
- Eigenvektor
![{\displaystyle \left(\mathbf {M} -\lambda \,\mathbf {I} \right)\,{\vec {x}}={\vec {0}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bf96e511e6b74c02948b60a36d7455f278d1fcba)
- Berechne x für alle λ
- Körper der Matrix
![{\displaystyle \mathbf {M} \,{\vec {x}}=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f6dc5a2cf95d33386649242079104c0266304185)
- Körper ist Menge aller
.
- Berechnung
- Laplace'scher Entwicklungssatz
- Regel von Sarrus (bis 3×3)
- Rechenregeln
![{\displaystyle \left|\mathbf {A} \right|\,\left|\mathbf {B} \right|=\left|\mathbf {A} \,\mathbf {B} \right|}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8e43a1cad102c734239630a61ba9ce81afc3f0e3)
wenn A n×n
![{\displaystyle \left|\mathbf {A} ^{-1}\right|=\left|\mathbf {A} \right|^{-1}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f200af54e4782ba5e58b645c35b19307b00cb9e7)
![{\displaystyle \left|\mathbf {A} \right|=\left|\mathbf {A} ^{\mathrm {T} }\right|}](https://wikimedia.org/api/rest_v1/media/math/render/svg/792fd2f21a692d66d9f178e7ae06784b734c3e63)
(A und B sind ähnlich)
- Cramer'sche Regel
![{\displaystyle x_{i}={\frac {\left|\mathbf {C} _{i}\right|}{\left|\mathbf {A} \right|}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c0c8926b9c71cf2b6c7ffb23a6970144d0807fc1)
- Spatprodukt
![{\displaystyle {\begin{vmatrix}a_{11}&a_{12}&a_{13}\\a_{21}&a_{22}&a_{23}\\a_{31}&a_{32}&a_{33}\end{vmatrix}}\Leftrightarrow {\vec {a}}_{1}\cdot \left({\vec {a}}_{2}\times {\vec {a}}_{3}\right)\Leftrightarrow \left({\vec {a}}_{1}\times {\vec {a}}_{2}\right)\cdot {\vec {a}}_{3}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/55c221a8305be850d3eb996cc708ad093fb2eb4d)
![{\displaystyle {\vec {a}}\cdot {\vec {b}}=\left|{\vec {a}}\right|\cdot \left|{\vec {b}}\right|\cdot \cos {\alpha }}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2f6d2e28e8b46d3756e21844549b1dd873a1b8e9)
- siehe auch: Kosinussatz
- Orthogonalprojektion von Vektoren
![{\displaystyle \cos {\alpha }={\frac {{\vec {a}}\cdot {\vec {b}}}{\left|{\vec {a}}\right|\cdot \left|{\vec {b}}\right|}}\cdot \left|{\vec {a}}\right|={\frac {{\vec {a}}\cdot {\vec {b}}}{\left|{\vec {b}}\right|}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/451cb1bf5b28abc4e36bf93cf0ed81337bdfb037)
![{\displaystyle \operatorname {Projektion} \left({\vec {a}}\to {\vec {b}}\right)=\operatorname {P} _{\vec {a}}\left({\vec {b}}\right)={\frac {{\vec {a}}\cdot {\vec {b}}}{\left|{\vec {b}}\right|}}\cdot {\frac {\vec {b}}{\left|{\vec {b}}\right|}}={\frac {{\vec {a}}\cdot {\vec {b}}}{\left|{\vec {b}}\right|^{2}}}\cdot {\vec {b}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2e2320d6923be412c94f3ceb695d54806d2f74e4)
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