Let δ = {1, x, x 2} be the standard basis for P2 and consider the linear transformation T : P2 → R 3 defined by T(f) = [f]δ, where [f]δ is the coordinate vector of f with respect to δ. Now, β is a basis for P2 if and only if T(β) = 1 1 0 , 1 0 1 , 0 1 1 is a basis for R 3 . (This follows from Theorem 8 on page 244. For further explanation, see Exercises 25 and 26 on page 249.) To check T(β) is a basis for R 3 , it suffices to put the three columns in 3 × 3 matrix and show that the rref of this matrix is the identity matrix. (This computation is trivial, so I won’t reproduce it here!)