Space of all matrices with given column space
Let $V$ be the vector space of all $m \times n$ (assume $m \leq n$) matrices $M$ over a field $\mathbb{F}$ whose column space satisfies $\operatorname{colsp}(M) \subseteq W$, where $W \subseteq \mathbb{F}^m$ is a given subspace. Prove that $\operatorname{dim}_\mathbb{F}(V) = n \operatorname{dim}_\mathbb{F}(W)$.
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