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

A linear transformation is uniquely specified by its action on a basis. We can extend the set of linearly independent vectors $\{(1, -1, 1), (1,1,1) \}$ to a basis for $\mathbb{R}^3$ by adjoining some vector $v\in \mathbb{R}^3$ to the set. The required linear transformation can then be specified by setting $T(v)$ to be any vector in $\mathbb{R}^2$.
