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.