grandes-ecoles 2014 QI.A.2

grandes-ecoles · France · centrale-maths2__pc Invariant lines and eigenvalues and vectors Diagonalizability determination or proof

Let $s$ be an endomorphism of $E$ such that $s \circ s = \operatorname{Id}_E$. We set $F = \operatorname{Ker}(s - \operatorname{Id}_E)$ and $G = \operatorname{Ker}(s + \operatorname{Id}_E)$. a) Show that $F$ and $G$ are two supplementary subspaces of $E$. b) Deduce that $s$ is a symmetry and specify its elements.

Let $s$ be an endomorphism of $E$ such that $s \circ s = \operatorname{Id}_E$. We set $F = \operatorname{Ker}(s - \operatorname{Id}_E)$ and $G = \operatorname{Ker}(s + \operatorname{Id}_E)$.\\
a) Show that $F$ and $G$ are two supplementary subspaces of $E$.\\
b) Deduce that $s$ is a symmetry and specify its elements.

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