The image contains a problem about linear transformations, specifically asking to find the matrix representation of a transformation in a given basis.

Here is a summary of what the image discusses:

1. Problem: Given a linear transformation $T: \mathbb{R}^2 \rightarrow \mathbb{R}^2$, with $T(1,0) = (1,2)$ and $T(0,1) = (-3, 4)$, the task is to find the matrix representation of $T$ in the standard basis.

2. Steps provided:

   • The transformation vectors $T(1,0) = (1,2)$ and $T(0,1) = (-3,4)$ are written in matrix form. This implies: $T(1,0) = \begin{bmatrix} 1 \\ 2 \end{bmatrix}, \quad T(0,1) = \begin{bmatrix} -3 \\ 4 \end{bmatrix}$
   • These vectors are placed as columns in the matrix representation of the linear transformation $T$.

3. Resulting matrix: The matrix representation of the transformation $T$ is: $[T] = \begin{bmatrix} 1 & -3 \\ 2 & 4 \end{bmatrix}$

Would you like me to explain more details or clarify any part?

Here are 5 follow-up questions related to this problem:

1. How does a matrix represent a linear transformation?
2. Why do the columns of the transformation matrix correspond to the images of the basis vectors?
3. How would you compute the image of any arbitrary vector using the transformation matrix?
4. What does it mean if a transformation matrix is invertible?
5. Can this matrix be used to solve systems of linear equations involving the transformation?

Tip: In linear transformations, the columns of the matrix are crucial because they define how the transformation maps each basis vector in your space.