Is there a linear transformation \(T\) from \(\RR^3\) into \(\RR^2\) such that \(T(1, -1, 1) = (1, 0)\) and \(T(1, 1, 1) = (0, 1)\)?
Solution
Yes. The vectors \[ v_1=(1,-1,1), \qquad v_2=(1,1,1) \] are linearly independent. Extend them to a basis \(v_1,v_2,v_3\) of \(\RR^3\). Define \(T\) on this basis by \[ T(v_1)=(1,0), \qquad T(v_2)=(0,1), \qquad T(v_3)=(0,0), \] and extend linearly. This produces a linear transformation \(T:\RR^3\to\RR^2\) with the required values.