grandes-ecoles 2013 Q17

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

Let $\ell, W_{\ell}, a, P_a$ be as in Part II. We say that an element $\phi$ of $\mathcal{L}(V)$ is compatible with $P_a$ if $P_a \circ \phi \circ P_a = P_a \circ \phi$.
17a. Show that if $\phi \in \mathcal{L}(V)$ commutes with $P_a$, then $\phi$ is compatible with $P_a$.
17b. Show that $H$ and $H^{-1}$ are compatible with $P_a$.

Let $\ell, W_{\ell}, a, P_a$ be as in Part II. We say that an element $\phi$ of $\mathcal{L}(V)$ is compatible with $P_a$ if $P_a \circ \phi \circ P_a = P_a \circ \phi$.

17a. Show that if $\phi \in \mathcal{L}(V)$ commutes with $P_a$, then $\phi$ is compatible with $P_a$.

17b. Show that $H$ and $H^{-1}$ are compatible with $P_a$.

Paper Questions

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