We are tasked with determining the matrix $T$ that represents a linear transformation based on the given vectors and their images under the transformation.

From the image, we are given:

$T\left( \begin{bmatrix} 2 \\ -1 \end{bmatrix} \right) = \begin{bmatrix} -1 \\ 6 \end{bmatrix}, \quad T\left( \begin{bmatrix} -6 \\ -5 \end{bmatrix} \right) = \begin{bmatrix} -5 \\ -2 \end{bmatrix}.$

Step 1: Express the Transformation as a Matrix

A linear transformation $T(\mathbf{v})$ can be written as a matrix multiplication:

$T(\mathbf{v}) = A \mathbf{v}$

where $A$ is the transformation matrix we need to find. The columns of $A$ will be the images of the standard basis vectors under the transformation.

Let’s call the two given vectors as $\mathbf{v}_1 = \begin{bmatrix} 2 \\ -1 \end{bmatrix}$ and $\mathbf{v}_2 = \begin{bmatrix} -6 \\ -5 \end{bmatrix}$, and their corresponding images as $\mathbf{w}_1 = \begin{bmatrix} -1 \\ 6 \end{bmatrix}$ and $\mathbf{w}_2 = \begin{bmatrix} -5 \\ -2 \end{bmatrix}$.

Step 2: Set up the System of Equations

We want to solve for the matrix $A$ such that:

$A \begin{bmatrix} 2 \\ -1 \end{bmatrix} = \begin{bmatrix} -1 \\ 6 \end{bmatrix} \quad \text{and} \quad A \begin{bmatrix} -6 \\ -5 \end{bmatrix} = \begin{bmatrix} -5 \\ -2 \end{bmatrix}.$

The matrix $A$ is of the form:

$A = \begin{bmatrix} a & b \\ c & d \end{bmatrix}$

Now we can substitute the given transformations into this equation.

Step 3: Solve for the Elements of the Matrix

Using the first vector transformation:

$A \begin{bmatrix} 2 \\ -1 \end{bmatrix} = \begin{bmatrix} -1 \\ 6 \end{bmatrix}$

This gives us two equations: