Composition of matrix transformations is the matrix transformation of the product

Statement

Let $T \colon \mathbb{R}^n \to \mathbb{R}^m$ be a matrix transformation given by an $m \times n$ matrix $A$. Let $S \colon \mathbb{R}^m \to \mathbb{R}^p$ be a matrix transformation given by a $p \times m$ matrix $B$. Then the composition $S \circ T \colon \mathbb{R}^n \to \mathbb{R}^p$ is again a matrix transformation. Its matrix is the matrix product $BA$.