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Definition

A vector space $(V,+,\cdot)$ over Körper $({\mathop{\mathbb K}},+,\cdot),\ {\boldsymbol v} \in {\mathop{\mathbb K}}^n$

Linear Unabhängig: Vektoren heißen linear unabhängig, wenn aus:

$\alpha_1 {\boldsymbol v}_1 + \alpha_2 {\boldsymbol v}_2 + \ldots + \alpha_n {\boldsymbol v}_n = {\boldsymbol 0}$ folgt, dass alle $\alpha_i = 0$

Basis ${\boldsymbol{B}}=\{{\boldsymbol b}_1, {\boldsymbol b}_2, ...\}$: $n$ Vektoren, linear unabhängig, erzeugen $V$

Betrag (Norm): ${\left\lVert{{\boldsymbol a}}\right\rVert} = \sqrt{{\boldsymbol a} \cdot {\boldsymbol a}} = \sqrt{a_1^2+a_2^2+\ldots +a_n^2}$

Skalarprodukt: ${\boldsymbol v} \cdot {\boldsymbol w} = {\boldsymbol v}^\top\!\! \cdot {\boldsymbol w} = \sum v_i w_i = {\left\lVert{{\boldsymbol a}}\right\rVert}{\left\lVert{{\boldsymbol b}}\right\rVert} \cos(\measuredangle {\boldsymbol a},{\boldsymbol b})$

$\left\langle{{\boldsymbol v}}{{\boldsymbol w}}\right\rangle_{{\boldsymbol{A}}} = {\boldsymbol v}^\top {\boldsymbol{A}} {\boldsymbol w}$ (quadr., symm., pos. definite Matrix ${\boldsymbol{A}}$)

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