Is there a linear transformation $T$ from $ R^3$ into $R^2$ such that $T(1,-1,1) = (1,0)$ and $T(1,1,1)= (0,1)$ . Is there a linear transformation $T$ from $ R^3$ into $R^2$ such that
$T(1,-1,1) = (1,0)$ and $T(1,1,1)= (0,1)$ .

Sure: $$T=\frac{1}{2}\begin{pmatrix}1 & -1 & 0 \\1 & 1 & 0\end{pmatrix}.$$
More generally: given $n$ independent vectors you can always find a linear transformation mapping these to $n$ linear independent vectors.
