The problem asks for finding a matrix $A$ such that for a given vector $\mathbf{x} = \begin{bmatrix} x_1 \\ x_2 \end{bmatrix}$, the linear transformation $T \colon \mathbb{R}^2 \to \mathbb{R}^2$ maps $\mathbf{x}$ to $T(\mathbf{x}) = x_1 \mathbf{v}_1 + x_2 \mathbf{v}_2$.

Here is the process for finding the matrix $A$:

Step 1: Given vectors

• $\mathbf{v}_1 = \begin{bmatrix} -3 \\ 5 \end{bmatrix}$
• $\mathbf{v}_2 = \begin{bmatrix} 2 \\ -9 \end{bmatrix}$

Step 2: Define the transformation

The transformation $T$ maps any vector $\mathbf{x} = \begin{bmatrix} x_1 \\ x_2 \end{bmatrix}$ to: $T(\mathbf{x}) = x_1 \mathbf{v}_1 + x_2 \mathbf{v}_2 = x_1 \begin{bmatrix} -3 \\ 5 \end{bmatrix} + x_2 \begin{bmatrix} 2 \\ -9 \end{bmatrix}$ This can be rewritten as: $T(\mathbf{x}) = \begin{bmatrix} x_1(-3) + x_2(2) \\ x_1(5) + x_2(-9) \end{bmatrix} = \begin{bmatrix} -3x_1 + 2x_2 \\ 5x_1 - 9x_2 \end{bmatrix}$

Step 3: Matrix Representation

We want to express this linear transformation as $T(\mathbf{x}) = A \mathbf{x}$, where $A$ is the matrix that performs the transformation. From the above expression, we can see that:

$A \mathbf{x} = \begin{bmatrix} -3 & 2 \\ 5 & -9 \end{bmatrix} \begin{bmatrix} x_1 \\ x_2 \end{bmatrix}$

Thus, the matrix $A$ that represents the linear transformation is:

$A = \begin{bmatrix} -3 & 2 \\ 5 & -9 \end{bmatrix}$

Final Answer:

The matrix $A$ is: $A = \begin{bmatrix} -3 & 2 \\ 5 & -9 \end{bmatrix}$

Do you want further details on this solution or have any questions?

Here are 5 related questions you can explore:

1. How does the matrix representation of a linear transformation generalize to higher dimensions?
2. What is the geometric interpretation of the transformation represented by this matrix?
3. Can a similar method be used for transformations in $\mathbb{R}^3$? How does the process change?
4. How would you compute the inverse of this matrix, if it exists?
5. How does the concept of linear combinations of vectors apply to this transformation?

Tip: To check if a matrix is a valid linear transformation matrix, you can verify if it correctly maps the standard basis vectors as expected.