Consider the set of all linear transformations from $V$ to $W$ to be a vector space over $F$. What is the dimension of vector space? Demonstrate an explicit basis. You may use usual matrix arithmetic without proof.
I've been working on this question for a while and have made no progress. How should I proceed? Thanks for all of your help.
Best Answer
I am guessing you mean finite dimensional vector spaces. Let $\dim(V)=n$ and $\dim(W)=m$.
Hint: after you fix a basis for $V$ and $W$, each linear transformation is expressible as an $n\times m$ matrix acting on the right of row vectors of length $n$.