Let $r$ and $m$ be strictly positive integers with $r \leq m$. We consider a subspace $V$ of the $\mathbf { C }$-vector space $M _ { m } ( \mathbf { C } )$ such that every element of $V$ is a matrix of rank at most $r$, and we assume $V$ contains the block matrix $A = \left( \begin{array} { c c } I _ { r } & 0 \\ 0 & 0 \end{array} \right)$. By question V.2a), every element $B$ of $V$ has the block form $B = \left( \begin{array} { c c } B_{11} & B_{12} \\ B_{21} & 0 \end{array} \right)$. We denote by $W$ the intersection of $V$ with the subspace of $M _ { m } ( \mathbf { C } )$ consisting of block matrices of the form $$\left( \begin{array} { c c } 0 & 0 \\ B _ { 21 } & 0 \end{array} \right)$$ We define a linear application $\varphi$ from $M _ { m } ( \mathbf { C } )$ to $M _ { r , m } ( \mathbf { C } )$ by $$\varphi : \left( \begin{array} { l l } B _ { 11 } & B _ { 12 } \\ B _ { 21 } & B _ { 22 } \end{array} \right) \mapsto \left( \begin{array} { l l } B _ { 11 } & B _ { 12 } \end{array} \right)$$ (with the notations of V.2a)). a) We write any matrix $C$ of $M _ { r , m } ( \mathbf { C } )$ in the form of a block matrix $C = \left( \begin{array} { l l } C _ { 11 } & C _ { 12 } \end{array} \right)$ with $C _ { 11 } \in M _ { r } ( \mathbf { C } )$ and $C _ { 12 } \in M _ { r , m - r } ( \mathbf { C } )$. Let $\psi$ be the linear map from $W$ to $M _ { r , m } ( \mathbf { C } ) ^ { \vee }$ which sends $B = \left( \begin{array} { c c } 0 & 0 \\ B _ { 21 } & 0 \end{array} \right)$ to the linear form $C \mapsto \operatorname { Tr } \left( B _ { 21 } C _ { 12 } \right)$. Let $s = \operatorname { dim } W$. Using the map $\psi$, show that $\operatorname { dim } ( \varphi ( V ) ) \leq m r - s$. b) Deduce that $\operatorname { dim } V \leq m r$.