Find a basis for the subspace R of P2,

The derivative of p(x) = a + bx + cx² is p'(x) = b + 2cx. Thus, if p ∈ R, then p'(0) = b+2c(0) = 0, so we must have b = 0.

We see that we can rewrite R as R = { p(x) = a + cx² | a, c ∈ C }.

A linearly independent set that spans R is B = { 1, x² }, and B is a basis of R.