linear transformation $\Bbb R^3 \to \Bbb R^2$ For linear transform $T ( 1,1,1 ) = (1 , 2),\,\, T ( 1, 1, 0) = (2 , 3)$ and $\,\,T ( 1,0,0 ) = ( 3 , 4)$,from $\mathbb{R}^3$ to $\mathbb{R}^2$, the function used was : Choose one : 
a. $T ( x, y, z) = (x - y - 3z , 4x - y - 2z )$ 
b. $T ( x, y, z) = (3x + y + z, 4x + y + z)$ 
c. $T ( x, y, z) = (3x - y - z, 4x - y -z )$
d. $T ( x, y, z) = (x - 3y - z, 4x - y -z )$
e. $T ( x, y, z) = (3x + y - z, 4x - y + z)$

The answer is (c). One can rewrite the given equations in matrix form as follows:
$$ T \cdot\begin{bmatrix} 1 & 1 & 1 \\ 1 & 1 & 0 \\ 1 & 0 & 0 \end{bmatrix} = \begin{bmatrix} 1 & 2 & 3 \\ 2 & 3 & 4 \end{bmatrix}.$$
Hence, we have 
$$ T = \begin{bmatrix} 1 & 2 & 3 \\ 2 & 3 & 4 \end{bmatrix} \cdot\begin{bmatrix} 1 & 1 & 1 \\ 1 & 1 & 0 \\ 1 & 0 & 0 \end{bmatrix}^{-1} = \begin{bmatrix} 1 & 2 & 3 \\ 2 & 3 & 4 \end{bmatrix} \cdot \begin{bmatrix} 0 & 0 & 1 \\ 0 & 1 & -1 \\ 1 & -1 & 0 \end{bmatrix} = \begin{bmatrix} 3 & -1 & -1 \\ 4 & -1 & -1 \end{bmatrix}.$$
