We have already determined that: $T \begin{pmatrix} 1 \\ -4 \end{pmatrix} = 2T \begin{pmatrix} 2 \\ 1 \end{pmatrix} - 3T \begin{pmatrix} 1 \\ 2 \end{pmatrix}$

Using the given transformations: $T \begin{pmatrix} 2 \\ 1 \end{pmatrix} = \begin{pmatrix} 2 \\ 3 \\ -3 \end{pmatrix}$ $T \begin{pmatrix} 1 \\ 2 \end{pmatrix} = \begin{pmatrix} -2 \\ 4 \\ 0 \end{pmatrix}$

Substituting these into our equation: $T \begin{pmatrix} 1 \\ -4 \end{pmatrix} = 2 \begin{pmatrix} 2 \\ 3 \\ -3 \end{pmatrix} - 3 \begin{pmatrix} -2 \\ 4 \\ 0 \end{pmatrix}$ $= \begin{pmatrix} 4 \\ 6 \\ -6 \end{pmatrix} - \begin{pmatrix} -6 \\ 12 \\ 0 \end{pmatrix}$ $= \begin{pmatrix} 4 + 6 \\ 6 - 12 \\ -6 - 0 \end{pmatrix}$ $= \begin{pmatrix} 10 \\ -6 \\ -6 \end{pmatrix}$

Therefore, $T \begin{pmatrix} 1 \\ -4 \end{pmatrix} = \begin{pmatrix} 10 \\ -6 \\ -6 \end{pmatrix}$, which corresponds to option c.

Thus, the correct answer is:

c. $\begin{pmatrix} 10 \\ -6 \\ -6 \end{pmatrix}$

Do you want more details or have any questions?

Here are five more questions you could ask:

1. How do you find the image of a linear transformation?
2. What is the difference between linear independence and linear dependence?
3. How do you calculate the rank of a matrix?
4. What is the method to find the inverse of a 2x2 matrix?
5. How do you solve a system of equations using Cramer's rule?

Tip: When dealing with linear transformations, always express the vector you need to transform as a linear combination of the given basis vectors. This makes it easier to apply the transformation.