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Matrix representation of linear transformations
Once bases are chosen, a linear transformation can be represented by a matrix. Let
$$T:V\to W$$
be linear, with ordered basis $B=(\mathbf b_1,\ldots,\mathbf b_n)$ for $V$ and basis $C$ for $W$.
There is a matrix $A$ such that
$$[T(\mathbf v)]_C=A[\mathbf v]_B$$
for every $\mathbf v\in V$.
Building the matrix from a basis
Because a linear transformation is determined by its action on a basis, the columns of $A$ are the coordinates of the transformed basis vectors:
$$A= \begin{pmatrix} [T(\mathbf b_1)]_C & \cdots & [T(\mathbf b_n)]_C \end{pmatrix}.$$
For example, if $T:\mathbb R^2\to\mathbb R^2$ satisfies
$$T(1,0)=(2,1),\qquad T(0,1)=(-1,3),$$
then in the standard basis
$$A=\begin{pmatrix}2&-1\1&3\end{pmatrix}.$$
The matrix depends on the bases
The transformation $T$ is the underlying map. Its matrix is a coordinate description of that map and changes when the input or output basis changes.
This distinction is essential: different matrices can represent the same linear transformation in different coordinate systems.
Composition becomes multiplication
If two linear transformations are composed and compatible bases are used, the matrix of the composition is the product of their matrices. Matrix multiplication therefore mirrors composition of linear actions.