Matrix Multiplication vs. Transformation Composition

We already know that matrices are vital tools for understanding and manipulating special transformations known as linear transformations. What role does matrix multiplication play?

It turns out that it has a very natural meaning; matrix multiplication simply results in the composition of linear transformations!

Suppose that we have a vector $V_1$ and two linear transformations $T_1$ and $T_2$.

Let $V_2=T_1(V_1)$ and $V_3=T_2(V_2)$. Then we have: $$V_3=T_2(T_1(V_1))=T_2\circ T_1(V_1)$$ Think about what this means in terms of matrix multiplication.

In order to obtain $V_3$, we first multiply the matrix for $T_1$ by the vector $V_1$, and then we multiply the matrix for $T_2$ by our new vector.

But wait! Matrix multiplication is associative! Although I will not prove it here, we have the following property for any three matrices $A$, $B$, and $C$, given that the products are defined: $$(AB)C=A(BC)$$ Thus, we may first multiply the matrices for $T_2$ and $T_1$, and then multiply by the vector!

By multiplying the matrices for the transformations, we have created a single matrix that, when multiplied by the vector, does the work of both transformations at once. This is exactly the definition of composition!

Thus, when you multiply matrices, you create a composition of linear transformations.