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. 

To expand on my comment above since @Gregory already gave the answer using one method:
Given information on how $T$ acts on the standard basis, knowing $T\begin{bmatrix}1\\0\\0\end{bmatrix}=\begin{bmatrix}3\\1\\4\end{bmatrix}$ and so on, this tells you exactly how to represent $T$ as a matrix.  The columns of $T$ are quite simply the corresponding results of $T$ being applied to the standard basis vectors.
We get then $T=\begin{bmatrix}3&-1&4\\1&-1&-3\\4&3&-1\end{bmatrix}$ and so all that remains to your problem is completing the matrix arithmetic necessary.
