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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.

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Linear transformations of what? – Neal Oct 29 '12 at 0:59
@Neal Sorry about that. I edited the question. – Juan Gomez Oct 29 '12 at 1:01
What do you know about the representation of linear maps as matrices? – wj32 Oct 29 '12 at 1:08
@wj32 Not much at all to be honest. – Juan Gomez Oct 29 '12 at 1:09
What do you get if multiply a matrix $A$ by $(1,0,0,..)^T$ from the right? – Berci Oct 29 '12 at 1:12

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 $n$.

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