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!)
