# Inner product/subspace help!

#### Pixel

Hi, I have this problem and don't know how to go about it:

(P2 being the set of all polynomials of degree less than or equal to 2)

Let p(x), q(x) Є P2. You may assume that:

<p(x), q(x)> = $$\displaystyle \int p(x)q(x)dx$$ (with limits from -1 to 1) defines an inner product on P2.

(A) Find a basis for the subspace of P2:
V= {a+b+ax+bx^2|a,b Є R}

(B) Using the inner product defined above and the basis vectors found in (A), use the Gram-Schmidt procedure to find an orthonormal basis for V.

Thank you.

#### roninpro

What is it that you are having trouble with?

#### Pixel

Finding a basis for the subspace of P2, to get started, given that I have a rule: V= {a+b+ax+bx^2|a,b Є R}. No idea?

#### roninpro

Let's rearrange some of the terms in that set.

$$\displaystyle \{a(x+1)+b(x^2+1)\ |\ a,b\in \mathbb{R}\}$$

Can you see how to pick the basis now?

Pixel

#### Pixel

I'm sorry,

What do we do the procedure on? What inner product?
What's the basis got to do with it?

#### roninpro

I'm just looking at part A of your problem. It just asks you to find a basis for your set. So far, it has nothing to do with inner product space or Gram-Schmidt.

Pixel

#### Pixel

For (A), is the basis {(0,1,1),(1,0,1)}

As in: x+1 = (0,1,1)
x^2 + 1 = (1,0,1)

#### roninpro

Looks good. Now are you able to do Gram-Schmidt?