To find $T \begin{pmatrix} 1 \\ -4 \end{pmatrix}$, we first need to determine the linear transformation $T$ based on the given information.

The linear map $T$ is defined by: $T \begin{pmatrix} 2 \\ 1 \end{pmatrix} = \begin{pmatrix} 2 \\ 3 \\ -3 \end{pmatrix}$ $T \begin{pmatrix} 1 \\ 2 \end{pmatrix} = \begin{pmatrix} -2 \\ 4 \\ 0 \end{pmatrix}$

Any vector $\begin{pmatrix} x \\ y \end{pmatrix}$ in $\mathbb{R}^2$ can be written as a linear combination of $\begin{pmatrix} 2 \\ 1 \end{pmatrix}$ and $\begin{pmatrix} 1 \\ 2 \end{pmatrix}$. Thus: $\begin{pmatrix} x \\ y \end{pmatrix} = a \begin{pmatrix} 2 \\ 1 \end{pmatrix} + b \begin{pmatrix} 1 \\ 2 \end{pmatrix}$

We need to solve for $a$ and $b$ in the equation: $\begin{pmatrix} 1 \\ -4 \end{pmatrix} = a \begin{pmatrix} 2 \\ 1 \end{pmatrix} + b \begin{pmatrix} 1 \\ 2 \end{pmatrix}$

This gives us the system of linear equations: $2a + b = 1 \quad \text{(1)}$ $a + 2b = -4 \quad \text{(2)}$

We can solve this system using substitution or elimination. Let's use elimination:

Multiply equation (1) by 2: $4a + 2b = 2 \quad \text{(3)}$

Subtract equation (2) from equation (3): $(4a + 2b) - (a + 2b) = 2 - (-4)$ $3a = 6$ $a = 2$

Substitute $a = 2$ into equation (1): $2(2) + b = 1$ $4 + b = 1$ $b = -3$

Thus, we have $a = 2$ and $b = -3$. Therefore: $\begin{pmatrix} 1 \\ -4 \end{pmatrix} = 2 \begin{pmatrix} 2 \\ 1 \end{pmatrix} - 3 \begin{pmatrix} 1 \\ 2 \end{pmatrix}$

We now use the linearity of $T$: $T \begin{pmatrix} 1 \\ -4 \end{pmatrix} = 2T \begin{pmatrix} 2 \\ 1 \end{pmatrix} - 3T \begin{pmatrix} 1 \\ 2 \end{pmatrix}$ $= 2 \begin{pmatrix} 2 \\ 3 \\ -3 \end{pmatrix} - 3 \begin{pmatrix} -2 \\ 4 \\ 0 \end{pmatrix}$ $= \begin{pmatrix} 4 \\ 6 \\ -6 \end{pmatrix} - \begin{pmatrix} -6 \\ 12 \\ 0 \end{pmatrix}$ $= \begin{pmatrix} 4 + 6 \\ 6 - 12 \\ -6 - 0 \end{pmatrix}$ $= \begin{pmatrix} 10 \\ -6 \\ -6 \end{pmatrix}$

Therefore, $T \begin{pmatrix} 1 \\ -4 \end{pmatrix}$ is equal to $\begin{pmatrix} 10 \\ -6 \\ -6 \end{pmatrix}$.

None of the provided answer choices match $\begin{pmatrix} 10 \\ -6 \\ -6 \end{pmatrix}$, so it seems there may have been an error in the provided options.

Do you want more details or have any questions?

Here are five more questions you could ask:

1. How do you find the kernel of a linear transformation?
2. Can you explain how to compute the determinant of a 3x3 matrix?
3. How do you solve a system of linear equations using matrix inversion?
4. What is the significance of eigenvalues and eigenvectors in linear algebra?
5. How do you determine if a set of vectors forms a basis for a vector space?

Tip: When solving systems of linear equations, it is often helpful to use matrix methods such as Gaussian elimination for efficiency and clarity.