To solve this problem, we need to determine the image of the matrix $\begin{bmatrix} 1 & -1 \\ 4 & 3 \end{bmatrix}$ under the linear transformation $T$. We know $T$ is a linear transformation from $M_{2,2}$ into $M_{2,2}$, and it is defined by its action on the basis matrices of $M_{2,2}$:

1. $T\left(\begin{bmatrix} 1 & 0 \\ 0 & 0 \end{bmatrix}\right) = \begin{bmatrix} -1 & 2 \\ 0 & 1 \end{bmatrix}$
2. $T\left(\begin{bmatrix} 0 & 1 \\ 0 & 0 \end{bmatrix}\right) = \begin{bmatrix} 0 & 2 \\ 1 & 1 \end{bmatrix}$
3. $T\left(\begin{bmatrix} 0 & 0 \\ 1 & 0 \end{bmatrix}\right) = \begin{bmatrix} 1 & 2 \\ 0 & 1 \end{bmatrix}$
4. $T\left(\begin{bmatrix} 0 & 0 \\ 0 & 1 \end{bmatrix}\right) = \begin{bmatrix} 3 & -1 \\ 1 & 0 \end{bmatrix}$

Step-by-Step Solution

1. Express the matrix $\begin{bmatrix} 1 & -1 \\ 4 & 3 \end{bmatrix}$ as a linear combination of the basis matrices: $\begin{bmatrix} 1 & -1 \\ 4 & 3 \end{bmatrix} = a \begin{bmatrix} 1 & 0 \\ 0 & 0 \end{bmatrix} + b \begin{bmatrix} 0 & 1 \\ 0 & 0 \end{bmatrix} + c \begin{bmatrix} 0 & 0 \\ 1 & 0 \end{bmatrix} + d \begin{bmatrix} 0 & 0 \\ 0 & 1 \end{bmatrix}$

   Solving for $a$, $b$, $c$, and $d$, we get:

   • $a = 1$
   • $b = -1$
   • $c = 4$
   • $d = 3$

2. Apply $T$ to the linear combination: $T\left(\begin{bmatrix} 1 & -1 \\ 4 & 3 \end{bmatrix}\right) = a T\left(\begin{bmatrix} 1 & 0 \\ 0 & 0 \end{bmatrix}\right) + b T\left(\begin{bmatrix} 0 & 1 \\ 0 & 0 \end{bmatrix}\right) + c T\left(\begin{bmatrix} 0 & 0 \\ 1 & 0 \end{bmatrix}\right) + d T\left(\begin{bmatrix} 0 & 0 \\ 0 & 1 \end{bmatrix}\right)$

3. Substitute the values of $a$, $b$, $c$, and $d$ and the images of the basis matrices: $T\left(\begin{bmatrix} 1 & -1 \\ 4 & 3 \end{bmatrix}\right) = 1 \begin{bmatrix} -1 & 2 \\ 0 & 1 \end{bmatrix} + (-1) \begin{bmatrix} 0 & 2 \\ 1 & 1 \end{bmatrix} + 4 \begin{bmatrix} 1 & 2 \\ 0 & 1 \end{bmatrix} + 3 \begin{bmatrix} 3 & -1 \\ 1 & 0 \end{bmatrix}$

4. Perform the matrix multiplications and additions: $= \begin{bmatrix} -1 & 2 \\ 0 & 1 \end{bmatrix} + \begin{bmatrix} 0 & -2 \\ -1 & -1 \end{bmatrix} + \begin{bmatrix} 4 & 8 \\ 0 & 4 \end{bmatrix} + \begin{bmatrix} 9 & -3 \\ 3 & 0 \end{bmatrix}$

5. Adding these matrices together: $= \begin{bmatrix} -1 + 0 + 4 + 9 & 2 - 2 + 8 - 3 \\ 0 - 1 + 0 + 3 & 1 - 1 + 4 + 0 \end{bmatrix}$

   Simplifying, we get: $= \begin{bmatrix} 12 & 5 \\ 2 & 4 \end{bmatrix}$

Final Answer:

$T\left(\begin{bmatrix} 1 & -1 \\ 4 & 3 \end{bmatrix}\right) = \begin{bmatrix} 12 & 5 \\ 2 & 4 \end{bmatrix}$

Would you like a detailed breakdown of any specific step?

1. What is a linear transformation?
2. How do you determine a linear transformation using basis matrices?
3. What is the purpose of expressing a matrix as a linear combination of basis matrices?
4. How do you calculate the image of a matrix under a given transformation?
5. Why is matrix addition important in transformations?

Tip: When working with transformations, always check if you’re using the correct basis.