% espaces vectoriels
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%%%     créée le Mon, 01 Jun 2009 21:45:13 +0200                 %%%
%%%     sur http://allken-bernard.org/pierre/phpmyexercices      %%%
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% Apparence des exercices
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\rhead{\textbf{\thepage/\pageref{fin}}}
\lfoot{\small \textit{Lycée Joachim du Bellay}}
\cfoot{\small \textit{Mathématiques, prépa ECE1}}
\rfoot{\small \textit{http://allken-bernard.org/pierre/ece}}
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%%%     Exercice 5
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\begin{exercice}
\nopagebreak[5]On pose~:
\[v=\begin{pmatrix} 1\\ -1  \end{pmatrix}\]
\'Ecrire le vecteur $v$ comme une combinaison linéaire des vecteurs~:
\[u_1=\begin{pmatrix} 2\\ 2  \end{pmatrix}\quad
u_2=\begin{pmatrix} 1\\ 2\end{pmatrix}\]
\end{exercice}


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%%%     Exercice 6
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\begin{exercice}
\nopagebreak[5]Dire, pour chaque famille $(u_1,u_2,u_3)$ de vecteurs, si elle est libre ou liée~:
\[u_1=\begin{pmatrix} 2\\ 1\\ 1  \end{pmatrix}\quad
u_2=\begin{pmatrix} 1\\ 2\\ 1\end{pmatrix}\quad
u_3=\begin{pmatrix} 1\\ 1\\ 2  \end{pmatrix}\]
\[u_1=\begin{pmatrix} 1\\ -2\\ 1  \end{pmatrix}\quad
u_2=\begin{pmatrix} 0\\ 3\\ -1\end{pmatrix}\quad
u_3=\begin{pmatrix} 2\\ -1\\ 1  \end{pmatrix}\]
\[u_1=\begin{pmatrix} 1\\ 2\\ 3  \end{pmatrix}\quad
u_2=\begin{pmatrix} 4\\ 5\\ 6\end{pmatrix}\quad
u_3=\begin{pmatrix} 7\\ 8\\ 9  \end{pmatrix}\]
\end{exercice}


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%%%     Exercice 7
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\begin{exercice}
\nopagebreak[5]Soient $a$, $b$, $c$ trois réels, tous différents. Démontrer que les vecteurs~:
\[u=\begin{pmatrix} 1\\ a\\ a^2  \end{pmatrix}\quad
v=\begin{pmatrix} 1\\ b\\ b^2\end{pmatrix}\quad
w=\begin{pmatrix} 1\\ c\\ c^2  \end{pmatrix}\]
forment une famille libre.
\end{exercice}


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%%%     Exercice 8
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\begin{exercice}
\nopagebreak[5]Démontrer que les vecteurs~:
\[u=\begin{pmatrix} 0\\ 1\\ 1  \end{pmatrix}\quad
v=\begin{pmatrix} 1\\ 0\\ 1\end{pmatrix}\quad
w=\begin{pmatrix} 1\\ 1\\ 0  \end{pmatrix}\]
engendrent $\mathbf R^3$.
\end{exercice}


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%%%     Exercice 9
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\begin{exercice}
\nopagebreak[5]Pour quelle(s) valeur(s) de $a$ la famille suivante est-elle libre~?
\[u=\begin{pmatrix} a\\ 1\\ 1  \end{pmatrix}\quad
v=\begin{pmatrix} 1\\ a\\ 1\end{pmatrix}\quad
w=\begin{pmatrix} 1\\ 1\\ a  \end{pmatrix}\]
\end{exercice}


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%%%     Exercice 10
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\begin{exercice}
\nopagebreak[5]Soit $F$ l'ensemble des vecteurs $\begin{pmatrix} x\\ y\\ z  \end{pmatrix}$ tels que~:
\[x+y+z=0\]
Soient~:
\[u=\begin{pmatrix} 1\\ -1\\ 0  \end{pmatrix}\quad
v=\begin{pmatrix} 0\\ 1\\ -1\end{pmatrix}\]
Démontrer que $F=\mathrm{Vect}(u,v)$.
\end{exercice}


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%%%     Exercice 11
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\begin{exercice}
\nopagebreak[5]Soit $m$ un réel. On considère les vecteurs~:
\[\begin{pmatrix} m\\ 1\\ 0\\ 0  \end{pmatrix}
\;\begin{pmatrix} 1\\ m\\ 1\\ 0  \end{pmatrix}
\;\begin{pmatrix} 0\\ 1\\ m\\ 1  \end{pmatrix}
\;\begin{pmatrix} 0\\ 0 \\ 1 \\ m  \end{pmatrix}\]
La famille formée par ces vecteurs est-elle libre~?
\end{exercice}


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%%%     Exercice 12
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\begin{exercice}
\nopagebreak[5]Dire (en justifiant), pour chacun des ensembles suivants, si c'est un sous-espace vectoriel de $\mathbf R^2$. Dans tout les cas, en faire un dessin.
\[A=\left\{\begin{pmatrix} x\\ y\end{pmatrix}\in\mathbf R^2,\; y=x^2\right\}\]
\[B=\left\{\begin{pmatrix} x\\ y\end{pmatrix}\in\mathbf R^2,\; y=2x\right\}\]
\[C=\left\{\begin{pmatrix} x\\ x\end{pmatrix}\in\mathbf R^2,\; x\in\mathbf R\right\}\]
\[D=\left\{\begin{pmatrix} x\\ y\end{pmatrix}\in\mathbf R^2,\; x^2+y^2=0\right\}\]
\[E=\left\{\begin{pmatrix} x\\ y\end{pmatrix}\in\mathbf R^2,\; y-|x|=0\right\}\]
\[F=\left\{\begin{pmatrix} 1\\ t \end{pmatrix}\in\mathbf R^2,\; t\in\mathbf R\right\}\]

\end{exercice}


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%%%     Exercice 13
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\begin{exercice}
\nopagebreak[5]Démontrer que les ensembles suivants sont des sous-espace vectoriel de $\mathbf R^3$. \`A chaque fois, en donner une base.
\[A=\left\{\begin{pmatrix} x\\ y\\ z \end{pmatrix}\in\mathbf R^3,\; x-2y+3z=0\right\}\]
\[B=\left\{\begin{pmatrix} x+y \\ 2x-y\\ -3x+2y\end{pmatrix}\in\mathbf R^3,\; x\in\mathbf R,\; y\in\mathbf R\right\}\]
\[C=\left\{\begin{pmatrix} x\\ y\\ z\end{pmatrix}\in\mathbf R^3,\; x+y+z=0\textrm{ et }2x-y+z=0\right\}\]
\end{exercice}


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%%%     Exercice 14
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\begin{exercice}
\nopagebreak[5]On pose~:
\[E=\left\{\begin{pmatrix} x\\ y\\ z\\ t\end{pmatrix}\in\mathbf R^4,\; x+y+z+t=0\right\}\]
\[F=\left\{\begin{pmatrix} x\\ y\\ z\\ t\end{pmatrix}\in\mathbf R^4,\; x-y+z-t=0\right\}\]
Démontrer que $E$ et $F$ sont des sous-espaces vectoriels de $\mathbf R^4$.
Donner une base pour chacun des sous-espaces vectoriels $E$, $F$, $E\cap F$.
\end{exercice}


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%%%     Exercice 15
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\begin{exercice}
\nopagebreak[5]Soit $f$ l'application linéaire $\mathbf R^3\to\mathbf R^3$ dont la matrice dans les bases canoniques est~:
\[A=\begin{pmatrix}
0 & 1 & 1 \\
1 & -1 & 0\\
1 & 0 & 1\\
\end{pmatrix}\]
\begin{enumerate}
\item Calculer $f\!\!\begin{pmatrix} x\\ y\\ z\end{pmatrix}$.
\item Déterminer une base de $\mathrm{Ker}(f)$ 
\item Déterminer une base de $\mathrm{Im}(f)$
\item La fonction $f$ est-elle injective~?
\item Est-elle surjective~?
\end{enumerate}
\end{exercice}


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%%%     Exercice 16
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\begin{exercice}
\nopagebreak[5]Soit $g:\mathbf R^m\to \mathbf R^n$ l'application linéaire dont la matrice dans les bases canoniques est~:
\[B=\begin{pmatrix}
2 & 1 \\
1 & 1\\
1 & 0\\
\end{pmatrix}\]
\begin{enumerate}
\item Que valent $m$ et $n$~?
\item Déterminer une base de $\mathrm{Ker}(g)$.
\item Déterminer une base de $\mathrm{Im}(g)$.
\item \'Etudier l'injectivité et la surjectivité de $g$.
\end{enumerate}
\end{exercice}


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%%%     Exercice 17
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\begin{exercice}
\nopagebreak[5]On définit une fonction $f:\mathbf R^2\to \mathbf R^2$ en posant~:
\[f\!\!\begin{pmatrix} x\\ y\end{pmatrix}=\begin{pmatrix} x+y \\ x-y\end{pmatrix}\]
\begin{enumerate}
\item Démontrer que $f$ est linéaire.
\item Déterminer la matrice de $f$ dans les bases canoniques.
\item \'Etudier l'injectivité de $f$.
\item \'Etudier la surjectivité de $f$.
\end{enumerate}
\end{exercice}


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%%%     Exercice 18
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\begin{exercice}
\nopagebreak[5]On définit une fonction $f:\mathbf R^2\to \mathbf R^2$ en posant~:
\[f\!\!\begin{pmatrix} x\\ y\end{pmatrix}=\begin{pmatrix} x^2+y^2 \\ x^2-y^2\end{pmatrix}\]
Démontrer que $f$ n'est pas linéaire.
\end{exercice}


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%%%     Exercice 19
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\begin{exercice}
\nopagebreak[5]Soient $A\in\mathcal M_n(\mathbf R)$ une matrice et $f:\mathbf R^n\to\mathbf R^n$ l'application linéaire définie par $f(X)=AX$. On suppose $A$ inversible. Démontrer que $f$ est bijective et que $f^{-1}$ est l'application linéaire $\mathbf R^n\to\mathbf R^n$ définie par $f^{-1}(Y)=A^{-1} Y$.
\end{exercice}


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%%%     Exercice 20
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\begin{exercice}
\nopagebreak[5]Soit~:
\[A=\begin{pmatrix}
0 & 2\\
1 & 1
\end{pmatrix}\]
Soit $f:\mathbf R^2\to \mathbf R^2$ l'application linéaire dont la matrice dans la base canonique est $A$.
Soit $a\in\mathbf R$ tel qu'il existe un vecteur non nul $X\in\mathbf R^2$ tel que $f(X)=aX$.
Démontrer que $a=-1$ ou $a=2$.
\end{exercice}


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%%%     Exercice 21
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\begin{exercice}
\nopagebreak[5]Soit $f:\mathbf R^3\to\mathbf R^3$ l'application linéaire dont la matrice dans les bases canonique est~:
\[\begin{pmatrix}
-1-t & 0 & 1\\
-3 & 4-t & 0\\
0 & 0 & 2-t 
\end{pmatrix}\]
Pour quelles valeurs de $t$ l'application $f$ est-elle injective~?
\end{exercice}


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%%%     Exercice 29
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\begin{exercice}
\nopagebreak[5]Soit $T:\mathbf R_2[X]\to\mathbf R_2[X]$ l'application définie par $T(P(X))=P(X+1)-P(X)$.
\begin{enumerate}
\item Démontrer que $T$ est un endomorphisme de $\mathbf R_2[X]$.
\item Déterminer la matrice de $T$ dans la base canonique de $\mathbf R_2[X]$.
\item Déterminer le noyau de $T$.
\item Déterminer l'image de $T$.
\item $T$ est-il injectif ? Surjectif ?
\end{enumerate}
\end{exercice}


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%%%     Exercice 30
%%%     Le numéro correspond à la base de données :
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\begin{exercice}
\nopagebreak[5]Soit $T:\mathbf R_2[X]\to\mathbf R_2[X]$ l'application définie par $T(P(X))=P'(X)$.
\begin{enumerate}
\item Démontrer que $T$ est un endomorphisme de $\mathbf R_2[X]$.
\item Déterminer la matrice de $T$ dans la base canonique de $\mathbf R_2[X]$.
\item Déterminer le noyau de $T$.
\item Déterminer l'image de $T$.
\item $T$ est-il injectif ? Surjectif ?
\end{enumerate}
\end{exercice}

\label{fin}
\end{document}