Revisited: How is $\phi:{\cal{L}}(V,W)\rightarrow M_{m\times n}(F)$ an isomophism of vector spaces? I'm told in lecture that if $V,W$ are vector spaces over $F$ and ${\cal{L}}(V,W)$ is the vector space of all linear maps $V\rightarrow W$ and ${\scr{B}}$ and ${\scr{C}}$ are bases for $V$ and $W$ respectively, then $\phi:{\cal{L}}(V,W)\rightarrow M_{m\times n}(F)$ defined by $\phi(T)=[T]^{\scr{C}}_{\scr{B}}$ is an isomorphism of vector spaces, where $T \in {\cal{L}}(V,W)$ and $[T]^{\scr{C}}_{\scr{B}}\in M_{m\times n}(F)$.
Now, I'm trying to understand how this theorem completes the proof for the following theorem:
Let $\{v_1,\dots,v_n\}$ be linearly independent set in a finite dimensional vector space $V$ and let $w_1,\dots,w_n$ be arbitrary vectors in a vector space $W$, then there is a linear map $T:V\rightarrow W$ such that $T(v_1)=w_1, T(v_2)=w_2, \dots$, or simply $T(v_i)=w_i$ for all $i=1,2,\dots,n$.
My professor concludes at the end of the proof for the second theorem that "[b]y theorem, such [a] $T$ exists and is unique." Any help? Furthermore, how does the below outline conclude such a theorem?

Let ${\scr{B}}=\{v_1,\dots,v_n\}$ be a basis for $V$ and ${\scr{C}}=\{u_1,\dots,u_m\}$ a basis for $W$. Then
\begin{eqnarray}
w_1 & = & a_{11}u_1+a_{21}u_2+\cdots+a_{m1}u_m & = & T(v_1)\\
w_1 & = & a_{12}u_1+a_{22}u_2+\cdots+a_{m2}u_m & = & T(v_2)\\
\vdots &&&&\vdots\\
w_n & = & a_{1n}u_1+a_{2n}u_2+\cdots+a_{mn}u_m & = & T(v_n),
\end{eqnarray}
which can be rewritten as
\begin{eqnarray}
\begin{pmatrix}
u_1&\cdots&u_m
\end{pmatrix}
\begin{pmatrix}
a_{11}&\cdots&a_{1n}\\
\vdots&\ddots&\vdots\\
a_{m1}&\cdots&a_{mn}
\end{pmatrix}=
\begin{pmatrix}
T(v_1)\\ \vdots \\ T(v_n)
\end{pmatrix},
\end{eqnarray}
thus such a $T$ exists.

Does this really suffice to prove?

Since every $m\times{n}$ matrix arises as the matrix of a linear transformation from V to W with respect to bases $\mathscr{B}$ and $\mathscr{C}$ by your given $\phi(T)$ we know $\phi$ is surjective.  And since a linear transformation is uniquely determined by its images on a basis, $\phi$ is injective.  Thus, $\phi$ is a bijection and therefore $\mathcal{L}(V,W)$ is isomorphic to $M_{m\times{n}}(F)$.
That was the isomorphism proof.  The second proof does not even need that, because it follows naturally from the definition of linear transformation.  The only thing that T could be would be (given scalars $c_j$, $j=1,2,...,n$ )
$$T(c_1v_1+c_2v_2+...c_nv_n) = c_1w_1+c_2w_2+...+c_nw_n$$
This is well defined and thus it is a linear transformation (I'm not putting the well defined proof here....)
