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Definition

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

Linear Unabhängig: Vektoren heißen linear unabhängig, wenn aus:

α1𝐯1+α2𝐯2++αn𝐯n=0\alpha_1 \boldsymbol v_1 + \alpha_2 \boldsymbol v_2 + \ldots + \alpha_n \boldsymbol v_n = \boldsymbol 0 folgt, dass alle αi=0\alpha_i = 0

Basis 𝐁={𝐛1,𝐛2,...}\boldsymbol{B}=\{\boldsymbol b_1, \boldsymbol b_2, ...\}: nn Vektoren, linear unabhängig, erzeugen VV

Betrag (Norm): 𝐚=𝐚𝐚=a12+a22++an2\left\lVert{\boldsymbol a}\right\rVert = \sqrt{\boldsymbol a \cdot \boldsymbol a} = \sqrt{a_1^2+a_2^2+\ldots +a_n^2}

Skalarprodukt: 𝐯𝐰=𝐯𝐰=viwi=𝐚𝐛cos(𝐚,𝐛)\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)

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