grandes-ecoles 2014 QI.A.1

grandes-ecoles · France · centrale-maths2__pc Invariant lines and eigenvalues and vectors Compute eigenvectors or eigenspaces

Let $F$ and $G$ be two supplementary subspaces of $E$ and $s$ the symmetry with respect to $F$ parallel to $G$. a) Show that $F = F_s$ and $G = G_s$. b) Show that $s \circ s = \operatorname{Id}_E$. Deduce that $s$ is an automorphism of $E$. c) Determine the eigenvalues and eigenspaces of $s$. We will discuss according to the subspaces $F$ and $G$.

Let $F$ and $G$ be two supplementary subspaces of $E$ and $s$ the symmetry with respect to $F$ parallel to $G$.\\
a) Show that $F = F_s$ and $G = G_s$.\\
b) Show that $s \circ s = \operatorname{Id}_E$. Deduce that $s$ is an automorphism of $E$.\\
c) Determine the eigenvalues and eigenspaces of $s$. We will discuss according to the subspaces $F$ and $G$.

Paper Questions

QI.A.1 QI.A.2 QI.B.1 QI.C.1 QI.C.2 QI.C.3 QI.D.1 QI.D.2 QI.E.1 QI.E.2 QI.E.3 QII.A.1 QII.A.2 QII.A.3 QII.B.1 QII.B.2 QIII.A.1 QIII.A.2 QIII.B.1 QIII.B.2 QIII.B.3 QV.A QV.B.1 QV.B.2 QV.B.3 QV.B.4