Originally Posted by

**cope** So I'm kind of running into a wall with this one. It should have pretty simple solutions but I'm just not seeing it.

Let X be the the set of all polynomials with degree less than or equal to n. X is a vector space. A polynomial p e X has the form:

$\displaystyle p(x) = \sum_{k=1}^n a_kx^k$

Are the following sets subspaces of X for all x e R?

$\displaystyle W_1 = \{p \epsilon X | p(x) = p(-x)\}$

$\displaystyle W_2 = \{p \epsilon X | p(x) = |p(x)|\}$

$\displaystyle W_3 = \{p \epsilon X | p(x) = -p(-x)\}$

For W1 I've shown that:

$\displaystyle q_1 + q_2 = \sum_{k=1}^n (a_k + b_k)(-x)^k$

$\displaystyle \lambda q_1 = \sum_{k=1}^n \lambda a_kx^k$

So it must be a subspace. But I'm not sure how to proceed with W2 and W3. Intuition tells me W2 is a subspace too, but I'm not sure how to show that mathematically. Any tips?