Finish verifying that P2 is a vector space (see Example 6.3).
Example 6.3
Let P2 denote the set of all polynomials of degree 2 or less with real coefficients. Define addition and scalar multiplication in the usual way. (See Appendix D.) If
p(x) = a0 + a1x + a2x2 and q(x) = b0 + b1x + b2x2
are in p2, then
p(x) + q(x) = (a0 + b0) + (a1 + b1)x + (a2 + b2)x2
has degree at most 2 and so is in P2. If c is a scalar, then
cp(x) = ca0 + ca1x + ca2x2
is also in P2. This verifies axioms 1 and 6.