grandes-ecoles 2013 Q22

grandes-ecoles · France · x-ens-maths1__mp Matrices Linear Transformation and Endomorphism Properties

Let $\mathcal{U}_q$ be the set of endomorphisms $\phi \in \mathcal{L}(V)$ that are compatible with $P_a$, and $\Psi_a : \mathcal{U}_q \rightarrow \mathcal{L}(W_{\ell})$ the unique algebra morphism such that $\Psi_a(\phi) \circ P_a = P_a \circ \phi$ for all $\phi \in \mathcal{U}_q$. Let $W$ be a non-zero subspace of $W_{\ell}$ stable under $\Psi_a(H)$.
22a. Show that $W$ contains at least one of the vectors $v_i$.
22b. What can be said if $W$ is moreover stable under $\Psi_a(E)$?

Let $\mathcal{U}_q$ be the set of endomorphisms $\phi \in \mathcal{L}(V)$ that are compatible with $P_a$, and $\Psi_a : \mathcal{U}_q \rightarrow \mathcal{L}(W_{\ell})$ the unique algebra morphism such that $\Psi_a(\phi) \circ P_a = P_a \circ \phi$ for all $\phi \in \mathcal{U}_q$. Let $W$ be a non-zero subspace of $W_{\ell}$ stable under $\Psi_a(H)$.

22a. Show that $W$ contains at least one of the vectors $v_i$.

22b. What can be said if $W$ is moreover stable under $\Psi_a(E)$?

Paper Questions

Q1a Q1b Q2 Q3 Q4 Q5 Q6 Q7 Q8 Q9 Q10 Q11 Q12 Q13 Q14 Q15 Q16 Q17 Q18 Q19 Q20 Q21 Q22 Q23