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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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    @JuanGomez: $T$ means transpose. Also, take a look at this: http://www.math.dartmouth.edu/archive/m24w07/public_html/Lecture12.pdf2012-10-29

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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 $n$.