If you give me a matrix, then I can quickly build you a linear transformation. Always. First a motivating example and then the theorem.
So the multiplication of a vector by a matrix “transforms” the input vector into an output vector, possibly of a different size, by performing a linear combination. And this transformation happens in a “linear” fashion. This “functional” view of the matrix-vector product is the most important shift you can make right now in how you think about linear algebra. Here is the theorem, whose proof is very nearly an exact copy of the verification in the last example.
Suppose that $A$ is an $m\times n$ matrix. Define a function $\ltdefn{T}{\complex{n}}{\complex{m}}$ by $\lt{T}{\vect{x}}=A\vect{x}$. Then $T$ is a linear transformation.
So Theorem MBLT gives us a rapid way to construct linear transformations. Grab an $m\times n$ matrix $A$, define $\lt{T}{\vect{x}}=A\vect{x}$ and Theorem MBLT tells us that $T$ is a linear transformation from $\complex{n}$ to $\complex{m}$, without any further checking.
We can turn Theorem MBLT around. You give me a linear transformation and I will give you a matrix.
Example MFLT was not an accident. Consider any one of the archetypes where both the domain and codomain are sets of column vectors (Archetype M through Archetype R) and you should be able to mimic the previous example. Here is the theorem, which is notable since it is our first occasion to use the full power of the defining properties of a linear transformation when our hypothesis includes a linear transformation.
Suppose that $\ltdefn{T}{\complex{n}}{\complex{m}}$ is a linear transformation. Then there is an $m\times n$ matrix $A$ such that $\lt{T}{\vect{x}}=A\vect{x}$.
So if we were to restrict our study of linear transformations to those where the domain and codomain are both vector spaces of column vectors (Definition VSCV), every matrix leads to a linear transformation of this type (Theorem MBLT), while every such linear transformation leads to a matrix (Theorem MLTCV). So matrices and linear transformations are fundamentally the same. We call the matrix $A$ of Theorem MLTCV the matrix representation of $T$.
We have defined linear transformations for more general vector spaces than just $\complex{m}$. Can we extend this correspondence between linear transformations and matrices to more general linear transformations (more general domains and codomains)? Yes, and this is the main theme of Chapter R. Stay tuned. For now, let us illustrate Theorem MLTCV with an example.