afl.tex

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% http://anthony.liekens.net/index.php/LaTeX/VectorNormNotation
\newcommand{\vectornorm}[1]{\left\|#1\right\|}


\title{Projekt B}

\begin{document}
\maketitle
\pagebreak


\section*{Opgave 2}
I opgaven er nedenstående givet

$$
V_1 = 
\begin{pmatrix}
1\\
2\\
4\\
2
\end{pmatrix}
$$

Og

$$
V_2 = 
\begin{pmatrix}
-3\\
4\\
3\\
4
\end{pmatrix}
$$

\subsection*{(a)}
Ved at benytte Gram–Schmidt proceduren søges en ortonormal basis ${q_1, q_2}$ for $V$.

$$
q_1 = \frac{1}{\vectornorm{v_1}}v_1 = \frac{1}{5} \begin{pmatrix}1\\ 2\\ 4\\ 2\\\end{pmatrix} = \begin{pmatrix}\frac{1}{5}\\ \frac{2}{5}\\ \frac{4}{5}\\ \frac{2}{5}\\\end{pmatrix}
$$

$$
q_2 = v_2 - Proj_{v_1} v_2 = v_2 - \left(v_2\cdot q_1\right) q_1
$$

$$
 = \begin{pmatrix}-3\\ 4\\ 3\\ 4\end{pmatrix} - \left(\begin{pmatrix}-3\\ 4\\ 3\\ 4\end{pmatrix}\cdot \begin{pmatrix}\frac{1}{5}\\ \frac{2}{5}\\ \frac{4}{5}\\ \frac{2}{5}\\\end{pmatrix}\right) \begin{pmatrix}\frac{1}{5}\\ \frac{2}{5}\\ \frac{4}{5}\\ \frac{2}{5}\\\end{pmatrix}
$$

$$
 = \begin{pmatrix}-3\\ 4\\ 3\\ 4\end{pmatrix} - 5\begin{pmatrix}\frac{1}{5}\\ \frac{2}{5}\\ \frac{4}{5}\\ \frac{2}{5}\\\end{pmatrix}
 = \begin{pmatrix}-3\\ 4\\ 3\\ 4\end{pmatrix} - \begin{pmatrix}1\\ 2\\ 4\\ 2\end{pmatrix}
 = \begin{pmatrix}-4\\ 2\\ -1\\ 2\end{pmatrix}
$$

$q_2$ skal dog lige normaliseres før denne, sammen med $q_1$, udgør ortonormal basis for $v$.

$$
q_2 = \frac{q_2}{\vectornorm{q_2}} = \frac{q_2}{5} = \begin{pmatrix}-4/5\\ 2/5\\ -1/5\\ 2/5\end{pmatrix}
$$








\subsection*{(b)}

\subsection*{(c)}



\end{document}