There is this answer which very intuitively answers the question:
But I would appreciate help with Axler's proof (3.95) on page 101 of "Linear Algebra Done Right", 3rd ed. There he gives a one liner referring to a prior result: Dimension of $\mathcal L(V,W) =(\dim V)(\dim W)$. I am puzzeled how to apply this as we know $\dim V$, but we are trying to determine, in this case, $\dim W$.
So how do we know $\dim \mathcal L(V,W)$ a priori?