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,
\[T(SR) \mathbf{v}_1=(TS)R \mathbf{v}_1 =(TSR) \mathbf{v}_1\]

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