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To solve these exercises and verify Theorem 3.32, we need to perform the following steps as indicated in the image:

1. **Step (a): Direct Substitution**  
We will apply \( S \circ T \) by directly substituting the result of \( T(x_1, x_2) \) into \( S \).

2. **Step (b): Matrix Multiplication**  
We will compute the matrix multiplication of \( [S] \) and \( [T] \) to obtain the matrix for \( S \circ T \).

Let's break down both parts for each problem. Starting with problem 30:

### Problem 30
- \( T \left[ \begin{array}{c} x_1 \\ x_2 \end{array} \right] = \left[ \begin{array}{c} x_1 - x_2 \\ x_1 + x_2 \end{array} \right] \)
- \( S \left[ \begin{array}{c} y_1 \\ y_2 \end{array} \right] = \left[ \begin{array}{c} 2y_1 \\ -y_2 \end{array} \right] \)

#### (a) Direct Substitution
Apply \( S(T(x_1, x_2)) \):

1. First, calculate \( T(x_1, x_2) = \left[ \begin{array}{c} x_1 - x_2 \\ x_1 + x_2 \end{array} \right] \)
2. Then substitute into \( S \): 
   \[
   S \left( \left[ \begin{array}{c} x_1 - x_2 \\ x_1 + x_2 \end{array} \right] \right) = \left[ \begin{array}{c} 2(x_1 - x_2) \\ -(x_1 + x_2) \end{array} \right]
   \]
Thus, \( S \circ T = \left[ \begin{array}{c} 2(x_1 - x_2) \\ -(x_1 + x_2) \end{array} \right] \).

#### (b) Matrix Multiplication
Write the matrices for \( S \) and \( T \):
\[
[T] = \left[ \begin{array}{cc} 1 & -1 \\ 1 & 1 \end{array} \right], \quad [S] = \left[ \begin{array}{cc} 2 & 0 \\ 0 & -1 \end{array} \right]
\]

Now, perform the multiplication \( [S][T] \):
\[
[S][T] = \left[ \begin{array}{cc} 2 & 0 \\ 0 & -1 \end{array} \right] \left[ \begin{array}{cc} 1 & -1 \\ 1 & 1 \end{array} \right] = \left[ \begin{array}{cc} 2 & -2 \\ -1 & -1 \end{array} \right]
\]
Thus, the matrix for \( S \circ T \) is:
\[
\left[ \begin{array}{cc} 2 & -2 \\ -1 & -1 \end{array} \right]
\]

### Problem 31
- \( T \left[ \begin{array}{c} x_1 \\ x_2 \end{array} \right] = \left[ \begin{array}{c} x_1 + 2x_2 \\ -3x_1 + x_2 \end{array} \right] \)
- \( S \left[ \begin{array}{c} y_1 \\ y_2 \end{array} \right] = \left[ \begin{array}{c} y_1 + 3y_2 \\ y_1 - y_2 \end{array} \right] \)

#### (a) Direct Substitution
Apply \( S(T(x_1, x_2)) \):

1. First, calculate \( T(x_1, x_2) = \left[ \begin{array}{c} x_1 + 2x_2 \\ -3x_1 + x_2 \end{array} \right] \)
2. Then substitute into \( S \): 
   \[
   S \left( \left[ \begin{array}{c} x_1 + 2x_2 \\ -3x_1 + x_2 \end{array} \right) \right] = \left[ \begin{array}{c} (x_1 + 2x_2) + 3(-3x_1 + x_2) \\ (x_1 + 2x_2) - (-3x_1 + x_2) \end{array} \right]
   \]
Simplifying the results:
\[
S(T(x_1, x_2)) = \left[ \begin{array}{c} x_1 + 2x_2 - 9x_1 + 3x_2 \\ x_1 + 2x_2 + 3x_1 - x_2 \end{array} \right] = \left[ \begin{array}{c} -8x_1 + 5x_2 \\ 4x_1 + x_2 \end{array} \right]
\]
Thus, \( S \circ T = \left[ \begin{array}{c} -8x_1 + 5x_2 \\ 4x_1 + x_2 \end{array} \right] \).

#### (b) Matrix Multiplication
Write the matrices for \( S \) and \( T \):
\[
[T] = \left[ \begin{array}{cc} 1 & 2 \\ -3 & 1 \end{array} \right], \quad [S] = \left[ \begin{array}{cc} 1 & 3 \\ 1 & -1 \end{array} \right]
\]

Now, perform the multiplication \( [S][T] \):
\[
[S][T] = \left[ \begin{array}{cc} 1 & 3 \\ 1 & -1 \end{array} \right] \left[ \begin{array}{cc} 1 & 2 \\ -3 & 1 \end{array} \right] = \left[ \begin{array}{cc} -8 & 5 \\ 4 & 1 \end{array} \right]
\]
Thus, the matrix for \( S \circ T \) is:
\[
\left[ \begin{array}{cc} -8 & 5 \\ 4 & 1 \end{array} \right]
\]

Would you like further details on any of these calculations, or do you have any questions?

### Related Questions:
1. What is the general form of verifying a theorem like Theorem 3.32 in matrix algebra?
2. How do we interpret the composition of linear transformations geometrically?
3. Can matrix multiplication always be used to represent the composition of two transformations?
4. What are the key properties of linear maps that allow direct substitution to work?
5. How would these transformations change if we modified the \( S \) or \( T \) matrices?

**Tip**: Always verify the dimensions of matrices before performing multiplication to ensure compatibility.

Verificar el Teorema 3.32 al encontrar la matriz de S ∘ T mediante (a) sustitución directa y (b) multiplicación matricial para las siguientes funciones lineales: T y S.

In this exercise, we verify Theorem 3.32 by performing both direct substitution and matrix multiplication for the linear transformations T and S. Learn how to compose matrices and compute transformations step by step.

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