Benutzer:AlexTheGer/Vektoren

${\overrightarrow {AB}}$ ${\overrightarrow {BC}}$ ${\overrightarrow {AB}}+{\overrightarrow {BC}}:={\overrightarrow {AC}}$

$\lambda \cdot {\begin{pmatrix}x\\y\end{pmatrix}}={\begin{pmatrix}{\lambda \cdot x}\\{\lambda \cdot y}\end{pmatrix}}$

${\begin{Vmatrix}{\begin{pmatrix}x\\y\\z\end{pmatrix}}\end{Vmatrix}}={\sqrt {x^{2}+y^{2}+z^{2}}}$

$(x_{1},x_{2},...,x_{n})+(y_{1},y_{2},...,y_{n})=(x_{1}+y_{1},...,x_{n}+y_{n})\$

$Lin(M)=\{{\vec {V}}|{\vec {v}}=\lambda _{1}{\vec {v}}_{1}+...+\lambda _{k}{\vec {v}}_{k},\quad \lambda _{1},...,\lambda _{k}\in K,\quad {\vec {v}}_{1},...,{\vec {v}}_{k}\in M,\quad k\in \mathbb {N} \}$

$\{{\vec {v}}_{1},...,{\vec {v}}_{k}\}\$ $0=\lambda _{1}\cdot {\vec {v}}_{1}+...+\lambda _{k}\cdot {\vec {v}}_{k}$ $\lambda _{i}\neq 0$

${\vec {v}}_{1}=(3,2,1),{\vec {v}}_{2}=(0,3,1),{\vec {v}}_{3}=(6,1,1)$ $2\cdot {\vec {v}}_{1}+(-1)\cdot {\vec {v}}_{2}+(-1)\cdot v_{3}=(0,0,0)$

${\vec {v}}_{1},...,{\vec {v}}_{k}$ ${\vec {w}}_{1},...,{\vec {w}}_{e}$

${\vec {v}}_{i}$ ${\vec {w}}_{j}$ $\{{\vec {v}}_{1},...,{\vec {v}}_{i-1},{\vec {w}}_{j},{\vec {v}}_{i+1},...,{\vec {v}}_{n}\}$

$A={\begin{pmatrix}a_{11}&a_{12}&\cdots &a_{1n}\\a_{21}&a_{22}&\cdots &a_{2n}\\\vdots &&&\vdots \\a_{m1}&a_{m2}&\cdots &a_{mn}\end{pmatrix}}=(a_{ij})$

${\begin{pmatrix}a_{11}&\cdots &a_{1n}\\\vdots &&\vdots \\a_{m1}&\cdots &a_{mn}\end{pmatrix}}+{\begin{pmatrix}b_{11}&\cdots &b_{1n}\\\vdots &&\vdots \\b_{m1}&\cdots &b_{mn}\end{pmatrix}}={\begin{pmatrix}a_{11}+b_{11}&\cdots &a_{1n}+b_{1n}\\\vdots &&\vdots \\a_{m1}+b_{m1}&\cdots &a_{mn}+b_{mn}\end{pmatrix}}$

${\begin{pmatrix}a_{11}&\cdots &a_{1n}\\\vdots &&\vdots \\a_{m1}&\cdots &a_{mn}\end{pmatrix}}\cdot {\begin{pmatrix}b_{1}\\\vdots \\b_{n}\end{pmatrix}}={\begin{pmatrix}a_{11}\cdot b_{1}+a_{12}\cdot b_{2}+\cdots +a_{1n}+b_{n}\\\vdots \\\cdots (Zeile\cdot Spalte)\end{pmatrix}}$

${\begin{pmatrix}a_{11}&\cdots &a_{1n}\\\vdots &&\vdots \\a_{i1}&\cdots &a_{in}\\\vdots &&\vdots \\a_{m1}&\cdots &a_{mn}\end{pmatrix}}\cdot {\begin{pmatrix}b_{11}&\cdots &b_{1j}&\cdots &b_{1r}\\&&\vdots &\\b_{n1}&\cdots &b_{nj}&\cdots &b_{nr}\end{pmatrix}}={\begin{pmatrix}c_{11}&\cdots &c_{1r}\\\vdots &c_{ij}&\vdots \\c_{m1}&\cdots &c_{mr}\end{pmatrix}}\in M(mxr,k)$

$c_{ij}=a_{i1}\cdot b_{1j}+a_{i2}\cdot b_{2j}+\dots +a_{in}\cdot b_{nj}=\sum _{k=1}^{n}a_{ik}\cdot b_{kj}$

${\begin{pmatrix}a_{11}&&&\\0&\ddots &&*\\&&a_{rr}\\0&0&0&0\end{pmatrix}}\quad {\begin{matrix}a_{11}\neq 0\\a_{22}\neq 0\\\vdots \\a_{rr}\neq 0\end{matrix}}$