\documentclass{article}
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\newcommand{\R}{\mathbb{R}}


\begin{document}
\title{Correction exercices applications linéaires}

\section{}
\subsection{}

$f := \begin{pmatrix} x\\y \end{pmatrix} \mapsto \begin{pmatrix} ax+by \\ cx + dy \end{pmatrix} = \begin{pmatrix} \lambda x \\ \lambda y \end{pmatrix}  $ où $\lambda\in \R$

\emph{On identifie} $\begin{cases}
	a &= \lambda \\
	b&= 0 \\
	c&= 0 \\
	d&= \lambda \\
\end{cases}$ 

Notons $A = \begin{pmatrix} a & b \\ c & d \end{pmatrix} = \begin{pmatrix} \lambda & 0 \\ 0 & \lambda \end{pmatrix} = \lambda I_2$

\subsection{}

$f$ est la symmétrie d'axe $\Delta$ 

\subsection{}

Notons $z = x + iy$
On sait que $z' = x' + iy' = e^{i\theta}z$

\begin{align*}
	f\begin{pmatrix} x\\y \end{pmatrix} &= \begin{pmatrix} \Re(e^{i\theta}(x+iy)) \\ \Im(e^{i\theta}(x+iy))  \end{pmatrix} \\
					    &= \begin{pmatrix} ax + by \\ cx + dy \end{pmatrix}  \\
	e^{i\theta}(x+iy) &= (\cos \theta + i \sin\theta)(x+iy) \\
	&= x \cos\theta + i x \sin\theta - iy\cos\theta - y \sin\theta \\
	ax+by &= x\cos\theta -y\sin\theta \\
	cx+dy &= x\sin\theta + y\cos\theta \\
\end{align*}

$\begin{cases}
	a &= \cos\theta \\
	b&= -\sin\theta \\
	c &= \sin\theta \\
	d &= \cos\theta \\
\end{cases}$ et $A = \begin{pmatrix} \cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \end{pmatrix} $

\paragraph{Meth 2}

\emph{cours} : $f: \begin{pmatrix} x\\y \end{pmatrix} \mapsto A \begin{pmatrix} x\\y \end{pmatrix} = \begin{pmatrix} a&b\\c&d \end{pmatrix} \begin{pmatrix} x\\y \end{pmatrix} $


où

\begin{itemize}
	\item $\begin{pmatrix} a\\c \end{pmatrix} = f(e_1) = f\begin{pmatrix} 0\\1 \end{pmatrix} $ 
	\item $\begin{pmatrix} b\\d \end{pmatrix} = f(e_2) = f\begin{pmatrix} 0\\1 \end{pmatrix} $
\end{itemize}

\begin{enumerate}
	\item $f$ homotéthie de rapport $\lambda$
	\begin{itemize}
		\item $f(e_1) = f\begin{pmatrix} 1\\0 \end{pmatrix} =\begin{pmatrix} \lambda \\ 0 \end{pmatrix} = \begin{pmatrix} a\\c \end{pmatrix} $
		\item $f(e_2) = f\begin{pmatrix} 0\\1 \end{pmatrix} =\begin{pmatrix} 0 \\ \lambda \end{pmatrix} = \begin{pmatrix} b\\d \end{pmatrix} $
	\end{itemize}
	\item $f$ symmétrie d'axe $\Delta$
	\begin{itemize}
		\item $f(e_1) = f\begin{pmatrix} 1\\0 \end{pmatrix} = e_2 = \begin{pmatrix} 0\\1 \end{pmatrix} = \begin{pmatrix} a\\ c \end{pmatrix} $
		\item $f(e_2) = f\begin{pmatrix} 0\\1 \end{pmatrix} = e_1 = \begin{pmatrix} 1\\0 \end{pmatrix} = \begin{pmatrix} b\\d \end{pmatrix} $
	\end{itemize}
	\item $f$ rotation d'angle $\theta$ 
	\begin{itemize}
		\item $f(e_1) = \begin{pmatrix} \cos\theta\\ \sin\theta \end{pmatrix} =\begin{pmatrix} a\\c \end{pmatrix} $ 
		\item $f(e_2) = \begin{pmatrix} -\sin\theta \\ \cos\theta \end{pmatrix} =\begin{pmatrix} b\\d \end{pmatrix} $
	\end{itemize}
\end{enumerate}



\end{document}