Let $F$ be a field, and suppose $V$ and $W$ are vector spaces over $F$. What is the dimension (meaning cardinality of any basis) of the space of linear maps from $V$ to $W$? I hope there is an answer in terms of the dimension of $V$, the dimension of $W$, and the cardinality of $F$.
If $V$ is finite dimensional, the answer is the product of the dimension of $V$ and the dimension of $W$. This recent question shows the dimension is at least continuum if $V$ has countably infinite dimension and $W=F$.
