To solve the problem, we are tasked with finding a vector $\mathbf{x} = \begin{bmatrix} x_1 \\ x_2 \end{bmatrix} \in \mathbb{R}^2$ such that the linear transformation $T(\mathbf{x}) = A \mathbf{x} = \mathbf{b}$, where:

$A = \begin{bmatrix} 2 & -4 \\ -3 & 5 \\ 1 & -3 \end{bmatrix}, \quad \mathbf{b} = \begin{bmatrix} 10 \\ -10 \\ 10 \end{bmatrix}.$

This involves solving the matrix equation:

$A \mathbf{x} = \mathbf{b}.$

Step 1: Write the equation in component form

$\begin{bmatrix} 2 & -4 \\ -3 & 5 \\ 1 & -3 \end{bmatrix} \begin{bmatrix} x_1 \\ x_2 \end{bmatrix} = \begin{bmatrix} 10 \\ -10 \\ 10 \end{bmatrix}.$

In expanded form, this results in the following system of equations:

1. $2x_1 - 4x_2 = 10$,
2. $-3x_1 + 5x_2 = -10$,
3. $x_1 - 3x_2 = 10$.

Step 2: Solve the system

The system is overdetermined, so we solve the first two equations and verify consistency with the third.

Solve the first two equations:

1. From $2x_1 - 4x_2 = 10$, divide through by 2: $x_1 - 2x_2 = 5.$

2. Rewrite as: $x_1 = 5 + 2x_2.$

3. Substitute $x_1 = 5 + 2x_2$ into the second equation $-3x_1 + 5x_2 = -10$: $-3(5 + 2x_2) + 5x_2 = -10.$

4. Simplify: $-15 - 6x_2 + 5x_2 = -10.$

   $-15 - x_2 = -10.$

   $x_2 = -5.$

5. Substitute $x_2 = -5$ into $x_1 = 5 + 2x_2$: $x_1 = 5 + 2(-5) = 5 - 10 = -5.$

Thus, $\mathbf{x} = \begin{bmatrix} -5 \\ -5 \end{bmatrix}$.

Step 3: Verify the solution

Substitute $\mathbf{x} = \begin{bmatrix} -5 \\ -5 \end{bmatrix}$ into the third equation $x_1 - 3x_2 = 10$: $-5 - 3(-5) = -5 + 15 = 10.$

The solution satisfies all three equations.

Final Answer:

$\mathbf{x} = \begin{bmatrix} -5 \\ -5 \end{bmatrix}.$

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Would you like a detailed step-by-step explanation of any part?
Here are 5 related questions to expand on this topic:

1. How do you determine if an overdetermined system has a solution?
2. What is the geometric interpretation of a linear transformation $T: \mathbb{R}^2 \to \mathbb{R}^3$?
3. How do you find the rank of a matrix?
4. How can we confirm if a given matrix is invertible?
5. What are the implications of consistency in a linear system?

Tip: Always check the rank of the coefficient matrix to ensure the system is consistent before solving.