grandes-ecoles 2010 QI.A.2

grandes-ecoles · France · centrale-maths2__psi Groups Group Homomorphisms and Isomorphisms

Let $h \in \mathscr{L}(E)$ be an endomorphism of $E$. Show that the following properties are equivalent: i) $h$ is an element of $\operatorname{Sim}(E)$; ii) $h^{*}h$ is collinear to $\operatorname{Id}_{E}$; iii) the matrix of $h$ in an orthonormal basis of $E$ is collinear to an orthogonal matrix.

Let $h \in \mathscr{L}(E)$ be an endomorphism of $E$. Show that the following properties are equivalent:\\
i) $h$ is an element of $\operatorname{Sim}(E)$;\\
ii) $h^{*}h$ is collinear to $\operatorname{Id}_{E}$;\\
iii) the matrix of $h$ in an orthonormal basis of $E$ is collinear to an orthogonal matrix.

Paper Questions

QI.A.1 QI.A.2 QI.B.1 QI.B.2 QI.B.3 QI.B.4 QI.C.1 QI.C.2 QI.C.3 QI.C.4 QI.C.5 QI.D.1 QI.D.2 QI.D.3 QI.D.4 QII.A.1 QII.A.2 QII.B.1 QII.B.2 QII.C.1 QII.C.2 QII.C.3 QII.C.4 QII.C.5 QII.C.6 QII.D QII.E