https://file.aiquickdraw.com/mathbot/file/17255624429330d52gwvn.jpg

The image describes a linear transformation \( T: \mathbb{R}^2 \to \mathbb{R}^2 \) defined by its action on two basis vectors, such that:

\[
T\left( \begin{bmatrix} 1 \\ 0 \end{bmatrix} \right) = \begin{bmatrix} 3 \\ -2 \end{bmatrix}, \quad T\left( \begin{bmatrix} 0 \\ 1 \end{bmatrix} \right) = \begin{bmatrix} -4 \\ 3 \end{bmatrix}
\]

We want to find the matrix representation of the transformation \( T \) and then use it to determine the image of another vector under \( T \).

### Step 1: Find the Matrix Representation of \( T \)

The matrix representation of the linear transformation \( T \) can be determined by noting how \( T \) acts on the standard basis vectors:

\[
T\left( \begin{bmatrix} 1 \\ 0 \end{bmatrix} \right) = \begin{bmatrix} 3 \\ -2 \end{bmatrix}, \quad T\left( \begin{bmatrix} 0 \\ 1 \end{bmatrix} \right) = \begin{bmatrix} -4 \\ 3 \end{bmatrix}.
\]

This means that the columns of the matrix representation of \( T \) are precisely these vectors:

\[
[T] = \begin{bmatrix} 3 & -4 \\ -2 & 3 \end{bmatrix}.
\]

### Step 2: Apply the Matrix to the Vector

We are asked to find:

\[
T\left( \begin{bmatrix} 9 \\ -1 \end{bmatrix} \right).
\]

To do this, we multiply the matrix representation of \( T \) by the given vector:

\[
\begin{bmatrix} 3 & -4 \\ -2 & 3 \end{bmatrix} \begin{bmatrix} 9 \\ -1 \end{bmatrix}.
\]

Let's compute this product:

\[
\begin{bmatrix} 3 \cdot 9 + (-4) \cdot (-1) \\ -2 \cdot 9 + 3 \cdot (-1) \end{bmatrix} = \begin{bmatrix} 27 + 4 \\ -18 - 3 \end{bmatrix} = \begin{bmatrix} 31 \\ -21 \end{bmatrix}.
\]

### Conclusion

The result of the transformation is:

\[
T\left( \begin{bmatrix} 9 \\ -1 \end{bmatrix} \right) = \begin{bmatrix} 31 \\ -21 \end{bmatrix}.
\]

Would you like more details on this process or have any other questions?

### Related Questions:

1. What are some properties of linear transformations in \( \mathbb{R}^n \)?
2. How can you determine if a linear transformation is invertible?
3. How do you find the inverse of a given linear transformation matrix?
4. What is the geometric interpretation of a linear transformation matrix?
5. How can eigenvalues and eigenvectors of a matrix help understand its linear transformation?

### Tip:

When finding the matrix of a linear transformation, remember that the columns of the matrix correspond to the images of the standard basis vectors.

Find the linear transformation T: R^2 → R^2 where T([1, 0]) = [3, -2], T([0, 1]) = [-4, 3], and apply it to the vector [9, -1].

Learn how to determine the matrix representation of a linear transformation and apply it to a vector. This step-by-step solution demonstrates matrix multiplication using the transformation T([1, 0]) = [3, -2] and T([0, 1]) = [-4, 3] to compute T([9, -1]).

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