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 $T(1,1,1)= (0,1)$ .

Hint
$$(a,b,c)=\frac{a-b}{2}(1,-1,1) +\frac{a-b}{2}(1,1,1)+(c-a)(0,0,1)$$
Then $$T(a,b,c)=\frac{a-b}{2}T(1,-1,1) +\frac{a-b}{2}T(1,1,1)+(c-a)T(0,0,1)$$
Picking any value you want for $T(0,0,1)$ gives you the desired transformation.
