<s>[INST] Find the matrix of a linear transformation If T : $\mathbb R^{3}$$\mapsto$$\mathbb R^{3}$ is a linear transformation such that 
T $\begin{pmatrix} 
        1 \\
        0 \\ 
        0 \\
        \end{pmatrix}$ = $\begin{pmatrix} 
        3 \\
        1 \\ 
        4 \\
        \end{pmatrix}$, $T$ $\begin{pmatrix} 
        0 \\
        1 \\ 
        0 \\
        \end{pmatrix}$ = $\begin{pmatrix} 
        -1 \\
        -1\\ 
        3\\
        \end{pmatrix}$ , $T$ $\begin{pmatrix} 
        0 \\
        0 \\ 
        1 \\
        \end{pmatrix}$ = $\begin{pmatrix} 
        4 \\
        -3 \\ 
        -1\\
        \end{pmatrix}$ then $T$ $\begin{pmatrix} 
        -5\\
        4 \\ 
        4 \\
        \end{pmatrix}$ = $\begin{pmatrix} 
        ? \\
        ? \\ 
        ? \\
        \end{pmatrix}$

I do not know where to start with this question, I tried doing RREF of the transformation numbers but I feel that is wrong. 
 [/INST] Define $\vec e_i = (0, \dots, 1, 0, \dots)$ where the 1 is in the $i^\text{th}$ position. Then note that your question is
$$ T(-5\vec e_1 + 4 \vec e_2 + 4 \vec e_3) = -5\pmatrix{3 \\ 1 \\ 4} + 4\pmatrix{-1 \\ -1 \\ 3} + 4\pmatrix{4 \\ -3 \\ -1} = \pmatrix{-3 \\ -21 \\ -12}.$$
</s>