## Associativity of Linear Transformations and Matrix Multiplication

Suppose we have well defined linear transformations
$R: \mathbf{V}_1 \rightarrow \mathbf{V}_2 , \:S: \mathbf{V}_ \rightarrow \mathbf{V}_3 , \: T: \mathbf{V}_3 \rightarrow \mathbf{V}_4$
with associated matrices
$R, \: S \: T$
respectively. Then composition of linear transformations, and matrix multiplication, is associative. That is
$T \circ (S \circ R) \mathbf{v}_1=(T \circ S) \circ R \mathbf{v}_1$

$T(SR) \mathbf{v}_1=(TS)R \mathbf{v}_1$

Considering the left hand side of the first one above
$S \circ R$
sends
$\mathbf{V}_1$
to
$\mathbf{v}_3$
then
$T$
sends
$\mathbf{v}_3$
to
$\mathbf{v}_4$
.
On the right hand side
$R$
sends
$\mathbf{v}_1$
to
$\mathbf{v}_2$
then
$T \circ S$
sends
$\mathbf{v}_2$
to
$\mathbf{v}_4$
.
Hence the first equation holds. Since the matrices represent the corresponding transformations, the second also holds, and in fact,
$T(SR) \mathbf{v}_1=(TS)R \mathbf{v}_1 =(TSR) \mathbf{v}_1$