halvvejs færdig med emne 5
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beviser.lyx
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beviser.lyx
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@ -1680,7 +1680,7 @@ For en samling af elementer
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\begin_inset Formula $V$
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\end_inset
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gælder det at
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gælder det at
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\end_layout
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\begin_layout Enumerate
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@ -1921,7 +1921,11 @@ Hvis
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kan
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\emph on
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udvides
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ud
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\emph default
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koordinattransformationsmatricer
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\emph on
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vides
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\emph default
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til en basis for
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\begin_inset Formula $V$
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@ -2249,5 +2253,230 @@ Noter
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Overvej at droppe Lemma 7.2 fra dispositionen.
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\end_layout
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\begin_layout Standard
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\begin_inset Newpage newpage
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\end_inset
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\end_layout
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\begin_layout Section
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Matrixrepræsentationer
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\end_layout
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\begin_layout Subsection
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Definition 8.3 (
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\emph on
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Koordinatvektor
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\emph default
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)
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\end_layout
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\begin_layout Standard
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\begin_inset Formula $\mathcal{V}=(\boldsymbol{v}_{1},\boldsymbol{v}_{2},\dots,\boldsymbol{v}_{n})$
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\end_inset
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er en basis for et
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\begin_inset Formula $\mathbb{F}$
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\end_inset
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-vektorrum
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\begin_inset Formula $V$
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\end_inset
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.
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\emph on
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Koordinatvektoren
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\emph default
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for et element
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\begin_inset Formula $\boldsymbol{v}\in V$
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\end_inset
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mht.
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basen
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\begin_inset Formula $\mathcal{V}$
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\end_inset
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menes elementet
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\begin_inset Formula $L_{\mathcal{V}}^{-1}(\boldsymbol{v})\in\mathbb{F}^{n}$
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\end_inset
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.
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Koordinatvektoren kan også betegnes med
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\begin_inset Formula $\left[\boldsymbol{v}\right]_{\mathcal{V}}$
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\end_inset
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\end_layout
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\begin_layout Standard
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Koordinatvektoren er den vektor
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\begin_inset Formula
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\[
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\begin{pmatrix}\alpha_{1}\\
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\alpha_{2}\\
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\vdots\\
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\alpha_{n}
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\end{pmatrix}\in\mathbb{F}^{n}
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\]
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\end_inset
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som opfylder relationen
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\begin_inset Formula
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\[
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\boldsymbol{v}=\alpha_{1}\cdot\boldsymbol{v}_{1}+\alpha_{2}\cdot\boldsymbol{v}_{2}+\cdots+\alpha_{n}\cdot\boldsymbol{v}_{n}.
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\]
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\end_inset
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\end_layout
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\begin_layout Subsection
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Definition 8.6 (
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\emph on
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Koordinattransformationsmatricen
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\emph default
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)
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\end_layout
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\begin_layout Standard
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Lad
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\begin_inset Formula $\mathcal{V}=(\boldsymbol{v}_{1},\boldsymbol{v}_{2},\dots,\boldsymbol{v}_{n})$
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\end_inset
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og
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\begin_inset Formula $\mathcal{W}=(\boldsymbol{w}_{1},\boldsymbol{w}_{2},\dots,\boldsymbol{w}_{n})$
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\end_inset
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være baser for det samme
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\begin_inset Formula $\mathbb{F}$
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\end_inset
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-vektorrum
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\begin_inset Formula $V$
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\end_inset
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.
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\emph on
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Koordinattransformationsmatricen for overgangen fra
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\begin_inset Formula $\mathcal{W}$
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\end_inset
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-basen til
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\begin_inset Formula $\mathcal{V}$
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\end_inset
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-basen defineres som matricen
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\begin_inset Formula
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\[
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_{\underset{til}{\underbrace{\mathcal{V}}}}\left[\boxempty\right]_{\underset{fra}{\underbrace{\mathcal{W}}}}=\begin{pmatrix}\vline & \vline & & \vline\\
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\left[\boldsymbol{w}_{1}\right]_{\mathcal{V}} & \left[\boldsymbol{w}_{2}\right]_{\mathcal{V}} & \cdots & \left[\boldsymbol{w}_{n}\right]_{\mathcal{V}}\\
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\vline & \vline & & \vline
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\end{pmatrix}\in{\rm Mat_{n}(\mathbb{F})}
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\]
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\end_inset
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\end_layout
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\begin_layout Subsection
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Definition 8.9 (
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\emph on
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Matrixrepræsentation
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\emph default
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)
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\end_layout
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\begin_layout Standard
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Lad
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\begin_inset Formula $L:\:W\rightarrow V$
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\end_inset
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betegne en lineær afbildning mellem
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\begin_inset Formula $\mathbb{F}$
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\end_inset
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-vektorrum
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\begin_inset Formula $W$
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\end_inset
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og
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\begin_inset Formula $V$
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\end_inset
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med baser hhv.
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\begin_inset Formula $\mathcal{W}=(\boldsymbol{w}_{1},\boldsymbol{w}_{2},\dots,\boldsymbol{w}_{n})$
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\end_inset
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og
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\begin_inset Formula $\mathcal{V}=(\boldsymbol{v}_{1},\boldsymbol{v}_{2},\dots,\boldsymbol{v}_{n})$
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\end_inset
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.
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\emph on
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Matrixrepræsentationen
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\emph default
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for
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\begin_inset Formula $L$
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\end_inset
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mht.
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til baserne
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\begin_inset Formula $\mathcal{W}$
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\end_inset
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og
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\begin_inset Formula $\mathcal{V}$
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\end_inset
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defineres da som matricen
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\end_layout
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\begin_layout Standard
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\emph on
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\begin_inset Formula
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\[
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_{\underset{til}{\underbrace{\mathcal{V}}}}\left[L\right]_{\underset{fra}{\underbrace{\mathcal{W}}}}=\begin{pmatrix}\vline & \vline & & \vline\\
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\left[L(\boldsymbol{w}_{1})\right]_{\mathcal{V}} & \left[L(\boldsymbol{w}_{2})\right]_{\mathcal{V}} & \cdots & \left[(\boldsymbol{w}_{n})\right]_{\mathcal{V}}\\
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\vline & \vline & & \vline
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\end{pmatrix}\in{\rm Mat_{n}(\mathbb{F})}
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\]
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\end_inset
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\end_layout
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\begin_layout Subsection
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Proposition 8.10(1) (Matrixrepræsentationer og koordinatvektorer)
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\end_layout
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\begin_layout Standard
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\begin_inset Formula
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\[
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\left[L(\boldsymbol{v})\right]_{\mathcal{W}}={}_{\mathcal{V}}\left[L\right]_{\mathcal{W}}\cdot\left[\boldsymbol{v}\right]_{\mathcal{V}}.
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\]
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\end_inset
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\end_layout
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\begin_layout Subsection
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Lemma 8.19
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\end_layout
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\begin_layout Standard
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\end_layout
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\end_body
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\end_document
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