https://file.aiquickdraw.com/mathbot/file/172835946419516rdnb97.jpg

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:

\[
\begin{aligned}
2a - b &= -1 \\
2c - d &= 6
\end{aligned}
\]

Using the second vector transformation:

\[
A \begin{bmatrix} -6 \\ -5 \end{bmatrix} = \begin{bmatrix} -5 \\ -2 \end{bmatrix}
\]

This gives us two more equations:

\[
\begin{aligned}
-6a - 5b &= -5 \\
-6c - 5d &= -2
\end{aligned}
\]

### Step 4: Solve the System of Equations

We now have the following system of four equations:

1. \( 2a - b = -1 \)
2. \( 2c - d = 6 \)
3. \( -6a - 5b = -5 \)
4. \( -6c - 5d = -2 \)

Let me solve these equations to find the values of \( a \), \( b \), \( c \), and \( d \).The solution to the system of equations gives the values:

\[
a = 0, \quad b = 1, \quad c = 2, \quad d = -2
\]

Thus, the matrix \( A \) representing the linear transformation \( T \) is:

\[
A = \begin{bmatrix} 0 & 1 \\ 2 & -2 \end{bmatrix}
\]

### Final Expression:
For any vector \( \mathbf{v} \in \mathbb{R}^2 \), the linear transformation \( T(\mathbf{v}) \) is given by:

\[
T(\mathbf{v}) = \begin{bmatrix} 0 & 1 \\ 2 & -2 \end{bmatrix} \mathbf{v}
\]

Would you like to explore further details or ask questions about this solution? Here are five related questions to expand your understanding:

1. How does this transformation affect the orientation of vectors in \( \mathbb{R}^2 \)?
2. Can you find the inverse of the matrix \( A \), and under what conditions does it exist?
3. How would this transformation act on the standard basis vectors \( \mathbf{e}_1 \) and \( \mathbf{e}_2 \)?
4. What is the geometric interpretation of the linear transformation \( T \) in terms of scaling, rotation, or reflection?
5. How can we generalize this approach to higher-dimensional spaces?

**Tip:** Always verify that a linear transformation preserves vector addition and scalar multiplication to confirm its properties.

Suppose that T is a linear transformation such that T([2, -1]) = [-1, 6] and T([-6, -5]) = [-5, -2]. Write T as a matrix transformation.

Find the matrix representation of a linear transformation T given by specific vector mappings in \( \mathbb{R}^2 \). This problem covers key linear algebra concepts, including linear transformations, matrix representations, and systems of equations. Suitable for undergraduate students.

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