Subspace of P4 problem

1. Feb 25, 2008

amolv06

1. The problem statement, all variables and given/known data

Determine whether the following is a subspace of $$P_{4}_$$

(a) The set of polynomials in $$P_{4}$$ of even degree.

2. Relevant equations

$$P_{4} = ax^{3}+bx^{2}+cx+d$$

3. The attempt at a solution

$$(p+q(x)) = p(x) + q(x)$$
$$(\alpha p)(x) = p(\alpha x)$$

If p and q are both of degree 2 then both scalar multiplication and vector addition should return a polynomial of degree 2 as far as I can tell, however my book states that this is not a subspace of $$P_{4}$$. I can't tell why it wouldn't be.

2. Feb 25, 2008

HallsofIvy

Staff Emeritus
The set of "polynomials of degree 2" are specifically those of the form ax2+ bx+ c with a non-zero. In particular, that does not include the 0 vector. Further, if p1(x)= x2+ 2x+ 1 and p2(x)= -ax2+ 3x+ 1, thenthe sum is NOT a "polynomial of degree 2". The polynomials in P4 that are of even degree are either of degree 2 or degree 0. In any case, the example I just gave answers your question.