Subspaces of polynomial vector space

Nov 2009
4
0
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?
 
Oct 2009
4,261
1,836
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?


\(\displaystyle p(x)=1\in W_2\,,\,\,but\,\,\,-p(x)\in W_2\) ?

\(\displaystyle W_3\) is a subspace .

Tonio
 
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Nov 2009
4
0
Wow, that really was simple. And W3 is a subspace because we've already shown p(-x) to be a subspace, and -p(-x) just switches the signs of the coefficients. All coefficients from R are in W3, so -p(-x) must be in W3 too.

Thanks