Inner product/subspace help!

May 2010
5
0
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.
 
Nov 2009
485
184
What is it that you are having trouble with?
 
May 2010
5
0
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?
 
Nov 2009
485
184
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?
 
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May 2010
5
0
I'm sorry,

What do we do the procedure on? What inner product?
What's the basis got to do with it?
 
Nov 2009
485
184
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.
 
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May 2010
5
0
For (A), is the basis {(0,1,1),(1,0,1)}

As in: x+1 = (0,1,1)
x^2 + 1 = (1,0,1)
 
Nov 2009
485
184
Looks good. Now are you able to do Gram-Schmidt?