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.