grandes-ecoles 2010 QII.C.2

grandes-ecoles · France · centrale-maths2__psi Groups Symplectic and Orthogonal Group Properties
In this section, the dimension of $E$ is 12. We assume that there exists in $\mathscr{L}(E)$, a family $(f_{1}, f_{2}, f_{3}, f_{4})$ of antisymmetric orthogonal automorphisms satisfying: $\forall i \neq j, f_{i}f_{j} + f_{j}f_{i} = 0$. Show that $f_{1}f_{2}f_{3}$ is an orthogonal automorphism, symmetric and not collinear to $\operatorname{Id}_{E}$. 
In this section, the dimension of $E$ is 12. We assume that there exists in $\mathscr{L}(E)$, a family $(f_{1}, f_{2}, f_{3}, f_{4})$ of antisymmetric orthogonal automorphisms satisfying: $\forall i \neq j, f_{i}f_{j} + f_{j}f_{i} = 0$. Show that $f_{1}f_{2}f_{3}$ is an orthogonal automorphism, symmetric and not collinear to $\operatorname{Id}_{E}$.
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