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To solve this problem, we need to compose the linear transformations \( T_1 \) and \( T_2 \).

Given:
\[ T_1: \mathbb{R}^2 \to \mathbb{R}^2 \text{ and } T_2: \mathbb{R}^2 \to \mathbb{R}^2 \]

Defined as follows:
\[ T_1 \left( \begin{bmatrix} x_1 \\ x_2 \end{bmatrix} \right) = \begin{bmatrix} ax_2 \\ bx_1 \end{bmatrix} \]
\[ T_2 \left( \begin{bmatrix} x_1 \\ x_2 \end{bmatrix} \right) = \begin{bmatrix} cx_1 \\ dx_2 \end{bmatrix} \]

We need to find:
\[ (T_2 \circ T_1) \left( \begin{bmatrix} x \\ y \end{bmatrix} \right) \]

### Step-by-Step Solution

1. **Apply \( T_1 \) to \( \begin{bmatrix} x \\ y \end{bmatrix} \):**

\[ T_1 \left( \begin{bmatrix} x \\ y \end{bmatrix} \right) = \begin{bmatrix} ay \\ bx \end{bmatrix} \]

2. **Apply \( T_2 \) to the result of \( T_1 \):**

\[ T_2 \left( \begin{bmatrix} ay \\ bx \end{bmatrix} \right) = \begin{bmatrix} c(ay) \\ d(bx) \end{bmatrix} = \begin{bmatrix} cay \\ dbx \end{bmatrix} \]

Thus, the result of the composition \( (T_2 \circ T_1) \left( \begin{bmatrix} x \\ y \end{bmatrix} \right) \) is:

\[ \begin{bmatrix} cay \\ dbx \end{bmatrix} \]

Would you like more details or have any questions about this solution?

Here are some additional related questions:
1. How do you determine the matrix representation of a linear transformation?
2. Can you find the matrix representation of \( T_1 \) and \( T_2 \) in this problem?
3. What properties do linear transformations preserve?
4. How do you compose multiple linear transformations?
5. What is the kernel of a linear transformation?
6. How do you determine if a linear transformation is invertible?
7. What is the significance of eigenvalues and eigenvectors in linear transformations?
8. How can you use linear transformations to solve systems of linear equations?

**Tip:** When working with compositions of linear transformations, always apply the transformations step-by-step to ensure accuracy.

Explore how to compose linear transformations T1 and T2 step-by-step. This problem involves understanding matrix representations and the composition of functions in linear algebra. Suitable for advanced high school students and beyond.

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