# Thread: Vector Spaces, Subspaces and Polynomials.

1. ## Vector Spaces, Subspaces and Polynomials.

I'm hoping this problem hasn't been posted before. I couldn't find a similar problem in the search so I'm posting a new thread.

Right, the following question was giving to us in one of our worksheets:

Let V be the set of polynomials p(x) satisfying

$\displaystyle \int_{-1}^1\frac{p(x)}{\sqrt{1-x^2}}\, dx=0$

Prove that V is a vector space by showing that it is a subspace of a larger vector space.
Now I know that, in order to prove that a certain vector space (V) is a subspace of another vector space (H), you have to go through the following three axioms:

1) The zero vector of V is in H.
2) H is closed under vector addition.
3) H is closed under scalar multiplication.

The bit that's confusing me is the whole "integration" thing. I'm assuming the the range between -1 and 1 is reducing the size of the space of V... so am I supposed to integrate first or... I dunno, I just need general help with this problem.

2. Originally Posted by Gojinn
I'm hoping this problem hasn't been posted before. I couldn't find a similar problem in the search so I'm posting a new thread.

Right, the following question was giving to us in one of our worksheets:

Now I know that, in order to prove that a certain vector space (V) is a subspace of another vector space (H), you have to go through the following three axioms:

1) The zero vector of V is in H.

Is it true that $\displaystyle \int\limits_{-1}^1\frac{0}{\sqrt{1-x^2}}dx=0$ ?

2) H is closed under vector addition.

Is it true that if $\displaystyle \int\limits_{-1}^1\frac{p(x)}{\sqrt{1-x^2}}dx=0\,\,\,and\,\,\,\int\limits_{-1}^1\frac{g(x)}{\sqrt{1-x^2}}dx=0$ then also $\displaystyle \int\limits_{-1}^1\frac{p(x)+g(x)}{\sqrt{1-x^2}}dx=0$ ?

3) H is closed under scalar multiplication.

Is is true that if $\displaystyle \int\limits_{-1}^1\frac{p(x)}{\sqrt{1-x^2}}dx=0$ then also $\displaystyle \int\limits_{-1}^1\frac{kp(x)}{\sqrt{1-x^2}}dx=0\,\,\,\forall\,k\in\mathbb{F}=$ the definition field?

Tonio

The bit that's confusing me is the whole "integration" thing. I'm assuming the the range between -1 and 1 is reducing the size of the space of V... so am I supposed to integrate first or... I dunno, I just need general help with this problem.
.

3. ... of course! I'm thinking too deeply. Thanks for the help.

4. I think I've done it right, but I was just hoping to get some confirmation on whether or not I've done the proof right.

For 1) I've substituted the zero vector into the condition and proved it to be true. I'm happy with that one.

For 2) do I just need to show, using the rules of integrals and assuming each individual function f(x) and g(x) are true, that both the required functions can be formed? Similar method for condition 3).

5. hi
let $\displaystyle p(x)\in V,s(x)\in V$
which means $\displaystyle \int_{-1}^{1}\frac{s(x)}{\sqrt{1-x^2}}dx=0$ and $\displaystyle \int_{-1}^{1}\frac{p(x)}{\sqrt{1-x^2}}dx=0$
prove that $\displaystyle (p+s)(x) \in V$.

6. Ahhh, so I was along the right lines! Just didn't completely finish it. Thanks mate.