Let $Y$ and $Z$ be subspaces of the finite dim vector spaces $V$ and $W$, respectively. Let $R=\{α\in L(V,W):α(Y)\subset Z\}$. What is $\dim R$?

Question

How would I go about proving this? I tried by considering maps from $Y$ to $Z$ and the space of all such maps would have $\dim Y\times\dim Z$ but I have no idea how to extend it to the entire vector space.

My guess is that it would be $\dim Y\times\dim Z + (\dim(V/Y)\times\dim W$ but I’m not sure about this.

Answer 1

Hint As is often the case, it can be helpful to pick a basis adapted to the constraints of the problem. So, pick a basis $\mathcal E_0 = \{e_1, \ldots, e_{\dim Y}\}$ of $Y$ and extend it to a basis $\mathcal E = \{e_1, \ldots, e_{\dim V}\}$ of $V$, and likewise extend a basis $\mathcal F_0$ of $Z$ to a basis $\mathcal F$ of $W$. In terms of our bases, $\alpha(Y) \subseteq Z$ if and only if $\alpha(\mathcal E_0) \subseteq \operatorname{span} \mathcal F_0$, that is, if its matrix representation has the form $$[\alpha]_{\mathcal E, \mathcal F} = \pmatrix{\ast & \ast \\ 0 & \ast},$$ where the upper-left block has size $\dim Z \times \dim Y$.