In Subsection LT.NLTFO we built new linear transformations from other linear transformations. Sums, scalar multiples and compositions. These new linear transformations will have matrix representations as well. How do the new matrix representations relate to the old matrix representations? Here are the three theorems.
Suppose that $\ltdefn{T}{U}{V}$ and $\ltdefn{S}{U}{V}$ are linear transformations, $B$ is a basis of $U$ and $C$ is a basis of $V$. Then \begin{equation*} \matrixrep{T+S}{B}{C}=\matrixrep{T}{B}{C}+\matrixrep{S}{B}{C} \end{equation*}
Suppose that $\ltdefn{T}{U}{V}$ is a linear transformation, $\alpha\in\complex{\null}$, $B$ is a basis of $U$ and $C$ is a basis of $V$. Then \begin{equation*} \matrixrep{\alpha T}{B}{C}=\alpha\matrixrep{T}{B}{C} \end{equation*}
The vector space of all linear transformations from $U$ to $V$ is now isomorphic to the vector space of all $m\times n$ matrices.
Suppose that $\ltdefn{T}{U}{V}$ and $\ltdefn{S}{V}{W}$ are linear transformations, $B$ is a basis of $U$, $C$ is a basis of $V$, and $D$ is a basis of $W$. Then \begin{equation*} \matrixrep{\compose{S}{T}}{B}{D}=\matrixrep{S}{C}{D}\matrixrep{T}{B}{C} \end{equation*}
This is the second great surprise of introductory linear algebra. Matrices are linear transformations (functions, really), and matrix multiplication is function composition! We can form the composition of two linear transformations, then form the matrix representation of the result. Or we can form the matrix representation of each linear transformation separately, then multiply the two representations together via Definition MM. In either case, we arrive at the same result.
A diagram, similar to ones we have seen earlier, might make the importance of this theorem clearer,
One of our goals in the first part of this book is to make the definition of matrix multiplication (Definition MVP, Definition MM) seem as natural as possible. However, many of us are brought up with an entry-by-entry description of matrix multiplication (Theorem EMP) as the definition of matrix multiplication, and then theorems about columns of matrices and linear combinations follow from that definition. With this unmotivated definition, the realization that matrix multiplication is function composition is quite remarkable. It is an interesting exercise to begin with the question, “What is the matrix representation of the composition of two linear transformations?” and then, without using any theorems about matrix multiplication, finally arrive at the entry-by-entry description of matrix multiplication. Try it yourself (Exercise MR.T80).