Exercise 21, § 8

Let $K$ be an infinite commutative field, $m, n, r$ three integers $\geq 0$ such that $r \leq n \leq m$, $\mathbf{M}_{m,n}(K)$ the vector $K$-space (of dimension $mn$) consisting of the matrices of type $(m, n)$ over $K$ and $V$ a vector subspace of $\mathbf{M}_{m,n}(K)$ such that $r$ is the greatest value of the rank of matrices $X \in V$. We propose to prove the inequality

(*) $\dim(V) \leq mr.$

(a) Show that we can assume that $m = n$ and that the matrix

belongs to V. Deduce that every matrix $X \in V$ is then of the form

(where $X_{11}$ is of order $r$) with the condition $X_{21}X_{12} = 0$ (use the fact that for all $\xi \in K$, the matrix $\xi X_0 + X$ is of rank $\leq r$).

(b) Deduce from (a) that, for two matrices $X, Y$ in V,

(c) For every matrix $X \in V$, we write $F(X) = (X_{11}X_{12}) \in M_{r,n}(K); f$ is a linear mapping and $\dim(V) = \dim(\operatorname{Im}(f)) + \dim(\operatorname{Ker}(f))$. On the other hand, let $u_x$ denote the linear form $(Y_{11}Y_{12}) \mapsto \operatorname{Tr}(X_{21}Y_{12})$ on $M_{r,n}(K)$. Show that the linear mapping $X \mapsto u_X$ of $\operatorname{Ker}(f)$ into the dual $(M_{r,n}(K))^*$ is injective; complete the proof of (*) by noting that the image under $X \mapsto u_X$ of $\operatorname{Ker}(f)$ is contained in the orthogonal of $\operatorname{Im}(f)$.