The difficulty is that thereissuch a linear transformation! The two vectors (1, -1, 1) and (1, 1, 1) are independent- you can map them into anything. Further, (0, 0, 1) is independent of both so {(1, -1, 1), (1, 1, 1), (0, 0, 1)} is a basis for . Define T(0, 0, 1) to be whatever you like, say, T(0, 0, 1)= (0, 0), and, together with the first two equations, you have defined a linear transformation from to such that T(1, -1, 1)= (1, 0) and T(1, 1, 1)= (0,1)