Space of matrices with bounded row space

Let $V$ be a vector space of $m \times n$ matrices over a field $\mathbb{F}$ with the property that, for all $M \in V, \ \operatorname{rowsp}(M) \subseteq W$, where $W \subseteq \mathbb{F}^n$ is a subspace which is a row space of some $M \in V$. Prove that $\operatorname{dim}_\mathbb{F}(V) = \operatorname{max}\{n,m\} \operatorname{dim}_\mathbb{F}(W)$.

Linear Algebra
Ribs Ribs
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  • Martin Martin
    0

    If you want equality for the dimension you need to also assume that W is itself the row space of some M in V. Otherwise V=0 is a counterexample.

    • Ribs Ribs
      +1

      You're right, edited, thanks

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Martin Martin
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