Chapter 3.6, Problem 35EQ

To determine

$\begin{array}{l}\text{To verify the Theorem 3}\text{.2 which states that}\\ {\text{if T:R}}^m\to {\text{R}}^n{\text{ and T:R}}^n\to {\text{R}}^p\text{ be linear transformations}\text{. Then S}\circ {\text{T:R}}^m\to {\text{R}}^p\\ \text{is a linear transformtaion}\text{. And , their standard matrices are related by}\\ \left[\text{S}\circ \text{T}\right]\text{=[S][T]}\\ \text{Given T}\left[\begin{array}{c}{\text{x}}_1\\ \begin{array}{c}{\text{x}}_2\\ {\text{x}}_3\end{array}\end{array}\right]=\left[\begin{array}{c}{\text{x}}_1+{\text{x}}_2\\ \begin{array}{c}{\text{x}}_2+{\text{x}}_3\\ {\text{x}}_1+{\text{x}}_3\end{array}\end{array}\right],\text{ S}\left[\begin{array}{c}{\text{y}}_1\\ \begin{array}{c}{\text{y}}_2\\ {\text{y}}_3\end{array}\end{array}\right]=\left[\begin{array}{c}{\text{y}}_1-{\text{y}}_2\\ \begin{array}{c}{\text{y}}_2-{\text{y}}_3\\ -{\text{y}}_1+{\text{y}}_3\end{array}\end{array}\right]\\ \text{To verify the theorem, by finding the matrix of S}\circ \text{T (a) by direct substitution and (b) by }\\ \text{matrix multiplication of [S][T]}\text{.}\end{array}$