The image appears to involve a linear transformation problem. Let me break down the steps to solve it.

1. Problem Statement: The task is to define a transformation $T: \mathbb{R}^4 \to \mathbb{R}^3$ by $T(\mathbf{x}) = A\mathbf{x}$, where $A$ is a $3 \times 4$ matrix, and find a vector $\mathbf{x}$ such that $T(\mathbf{x})$ equals a given vector.

2. Matrix $A$: The matrix $A$ given in the image is:

   2 & -1 & -3 & 2 \\ 3 & 4 & -2 & -3 \\ 5 & -2 & 4 & -1 \end{pmatrix}$$

3. Vector $T(\mathbf{x})$: The equation $A\mathbf{x} = \begin{pmatrix} 20 \\ 15 \\ 100 \end{pmatrix}$ is given. We need to solve for $\mathbf{x}$.

4. Method: Solve the system of linear equations $A\mathbf{x} = \mathbf{b}$, where $\mathbf{b} = \begin{pmatrix} 20 \\ 15 \\ 100 \end{pmatrix}$.

I will now calculate the solution for $\mathbf{x}$.The solution to the system $A\mathbf{x} = \mathbf{b}$ is the vector:

$\mathbf{x} = \begin{pmatrix} 14.41 \\ -4.95 \\ 4.47 \\ -0.18 \end{pmatrix}$

This is the vector $\mathbf{x}$ such that $T(\mathbf{x}) = A\mathbf{x} = \begin{pmatrix} 20 \\ 15 \\ 100 \end{pmatrix}$.

Do you need further details, or have any questions?

Here are some related questions to explore:

1. How do we know that the matrix equation $A\mathbf{x} = \mathbf{b}$ has a solution?
2. What methods can be used to solve systems of linear equations like this?
3. Can we check if $A\mathbf{x} = \mathbf{b}$ by performing matrix multiplication?
4. What does the least squares solution mean in this context?
5. How does the rank of matrix $A$ affect the solution?

Tip: Always check the rank of a matrix to determine if a unique solution exists or if the system has infinitely many solutions.