Linear Maps from a finite space to an infinite space Suppose V is finite dimensional with dim V > 0. Prove that if W is infinite dimensional then $L(V, W)$ is infinite dimensional.
Help? I really have no idea how to go about this one?
I'm assuming I need to use the fact that if a space is infinite dimensional then there is no list that spans it.
Maybe something like if dim V = n, then there are infinitely many maps $Tv_j = w_j$ that map to different linearly independent lists of length n in W, assuming that $v_1,...v_n$ is a basis for V. 

Maybe it's easier to prove the contrapositive: if $L(V,W)$ is finite-dimensional, then $W$ is finite-dimensional.
Let $f_1,...,f_n : V \rightarrow W$ be linear functions that span $L(V,W).$ Fix any nonzero $v \in V$. Then $f_1(v),...,f_n(v)$ span $W$, so $W$ is finite-dimensional.
The reason for this is that, for any $w \in W,$ you can find a linear function $f : V \rightarrow W$ such that $f(v) = w,$ and you can write $f$ as a combination of the $f_1,...,f_n.$
