Need help intuitively understand a theorem This theorem is un-named in my class notes. However, I cannot wrap my head around what is going on, maybe someone can provide a concrete example?
Let $B=\{v_{1},...,v_{n}\},\; C =\{w_{1},...,w_{m}\},\; D=\{x_{1},...,x_{p}\}$ be bases for vector spaces $V$, $W$, and $X$ respectively over the same field $F$.
Then the notes claim:
$(1)\;$ The function $f: L(V,W) \rightarrow M_{m,n}(F)$ which sends a linear transformation $T$ to the matrix $A_{T}$ (with respect to $B$ and $C$) is an isomorphism.
$(2)\;$ Let $S: V \rightarrow W, T: W \rightarrow X$ be linear and let $A_{S}$ (respectively, $A_{T}$) be the matrix of $S$ (respectively, $T$) with respect to  $B$ and $C$ (respectively, $C$ and $D$) Then the matrix $A_{T \circ S}$ of $T\circ S$ with respect to $B$ and $D$ is the product $A_{T}A_{S}$
Now I understand what each of the terms mean, but when put together in long sentences I find it difficult to understand. I'm not asking for a proof here just maybe some concrete examples.
 A: The theorem is saying the in finite dimensional vector spaces, linear maps can really be thought of as matrices.  That is, if $V, W, X$ are finite dimensional vector spaces of dimension $n,m,p$ respectively, then your theorem says:
(1) Giving a linear map $T: V\to W$ is same thing as giving a $m\times n$ matrix (i.e. there is a one-to-one correspondence between the two).  More precisely, there is a bijection between the two sets $\mathcal{L}(V,W)$ and $M_{n,m}(\mathbb{F})$.  In fact, the correspondence between the two is more than a bijection: it is an isomorphism: If $T, S: V\to W$ linear maps, and matrices $A, B$ correspond to them respectively, then the linear map $T+S:V\to W$ exactly corresponds to $A+B$.
(2) Even better: if $T:V\to W$ and $S:W\to X$ are linear maps and $A, B$ are matrices that correspond to them, then the linear map $S\circ T$ in fact corresponds to $BA$.
In sum, what the theorem is saying is that once you have chosen the basis for $V$, $W$, and $X$, any linear maps between them has a matrix representation, and to find matrix representation of sums and composition of linear maps (which is also a linear map), we can just compute using matrices corresponding to those linear maps.
