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Definition

A vector space (V,+,\cdot) over Körper (\K,+,\cdot),\ \boldsymbol v \in \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})