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\def \RM{\mathbb{R}}%        corps des reels
\def \NM{\mathbb{N}}%        entiers naturels
\def \ZM{\mathbb{Z}}%        entiers relatifs
\def \CM{\mathbb{C}}%        nombres complexes
\def \QM{\mathbb{Q}}%        nombres rationnels
\def \PM{\mathbb{P}}%        la droite projective
\def \UM{\mathbb{U}}%        groupe unitaire

\def\rg{\mathop{\rm rg}\nolimits}
\def\Id{\mathop{\rm Id}\nolimits}
 
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\begin{document}

%\maketitle

{\bf Pr{\'e}pa CAPES UPMC 200{\`e}-2008} \hfill Emmanuel FERRAND, Laurent KOELBLEN
\bigskip

\centerline{\bf TD d'Alg{\`e}bre du mercredi 28 novembre 2007}

\bigskip


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\section{} Soient $E,F,G$ trois espaces vectoriels de dimension finie,
et $f:E \rightarrow F$, $g: F \rightarrow\nobreak G$ deux applications lin{\'e}aires.
Montrer que $\rg(f)+\rg(g)-\dim(F) \leq \rg(g \circ f) \leq \min\{\rg(f), \rg(g)\}$

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\section{} Montrer que l'ensemble des matrices inversibles est un
ouvert dense connexe de $\mathcal M_n(\CM)$.

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\section{} Soit $E$ un $\CM$-espace vectoriel de dimension finie, $f$ un
endomorphisme de $E$ et $P$ et $Q$ deux polynômes de $\CM[X]$ de pgcd
$D$ et de ppcm $M$.

\subsection{} Montrer que $\ker P(f)  \cap \ker Q(f)  = \ker D(f) $

\subsection{} Montrer que $\ker P(f)  + \ker Q(f)  = \ker M(f) $


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\section{} Soit $A$ une matrice carr{\'e}e {\`a} coefficient complexes. Montrer
que $A$ est conjugu{\'e}e {\`a} sa transpos{\'e}e.

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\section{} Soit $A$ une matrice triangulaire sup{\'e}rieure, n'ayant que
des $1$ sur la diagonale. A quelle condition est-elle diagonalisable? 

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\section{} Soit $A$ la matrice carr{\'e}e $n \times n$ dont tous les
coefficients sont nuls, sauf ceux de la derni{\`e}re ligne et de la
derni{\`e}re colonne, qui sont {\'e}gaux {\`a} $1$. Est-elle diagonalisable sur
$\CM$, sur $\RM$?
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\section{} \subsection{} 
Soit $P \in \CM[X]$ un polynome de degr{\'e} 2, ayant deux racines $a$ et $b$ distinctes.
soit $n \in  \NM$. Faire la division eucldienne de $X^n$ par $P$.

\subsection{} 
Soit $f$ un endomorphisme de $\CM^2$ ayant $P$ pour
annulateur. Pour tout $n \in \NM$, exprimer $f^n$ en fonction de $n,a,b,f$.



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\section{}
\subsection{} Soit $J$ une matrice carr{\'e}e dont tous les coefficients sont
{\'e}gaux {\`a} $1$. Calculer $J^k$, pour tout $k \in \NM$.

\subsection{} Soit $a \in \RM$. D{\'e}velopper $P(X)=\det(X \!\cdot \Id+a \cdot
J)$. 

\subsection{} Soit $D$ une matrice diagonale, calculer $\det(J+D)$.


\end{document}