U is the image of a linear map from R2 to R3 given by . It's pretty easy to see that f is zero only at (0,0), (i.e. f has trivial kernel), so that the image (U) has the same dimension as the domain (R2). Whenever that happens, you can take a basis of the domain, and the image of the basis set forms a basis of the image. So {f(1,0), f(0,1)} is a basis for U.