halvvejs færdig med emne 5

beviser.lyx

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