# Subspaces of polynomial vector space

#### 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?

#### tonio

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

• cope

#### cope

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