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Linear Algebra, Adjoint of a Linear Operator and Inner Product Spaces

  • #1

Homework Statement


For each of the following inner product spaces V (over F) and linear transformation g:=V [tex]\rightarrow[/tex] F, find a vector y such that g(x) = <x,y> for all x element of V.

The particular case I'm having trouble with is:

V=P2(R), with <f,h>=[tex]\int_0^{1}[/tex] f(t)h(t)dt , g(f)=f(0)+f'(1).

From the textbook Linear Algebra (fourth edition) by Friedburg and Friends.
p. 365 problem #2. c)

Homework Equations



From the theorem given in the book, I know that:
y=[tex]\sum_{i=1}^{n}[/tex][tex]\overline{g(v_i)}[/tex]vi

is a vector such that g(x)=<x,y> for all x element of V
where B={v1,v2...vn} is an orthonormal basis for V.

I also know that the answer (printed in the back of the book) is y=210x2 - 204x + 33 but I have no idea how to get this answer!

The Attempt at a Solution


I assigned B={1,x,x2}={e1,e2,e3} as an orthornormal basis for P2(R) (the space of polynomials of degree less than or equal to 2 over R).
Then I did:
y=g(1)e1 + g(x)e2 + g(x2)e3 because the field is R, I know that [tex]\overline{g(anything)}[/tex] = g(anything) so I don't worry about the conjugate of g(anything)
=(1+0)e1 + (0+1)e2 + (0+2)e3
=1e1 + 1e2 + 2e3
= 1 + x + 2x2

But this isn't the answer from the book... I also have no idea why the inner product was defined for me, I assumed to test my y vector?
I did I similar problem (part a) over R3 using the technique I applied above, and it worked fine.
Any help would be greatly appreciated.
Thanks :)
 

Answers and Replies

  • #2
matt grime
Science Advisor
Homework Helper
9,395
3
If, as you state, {e_1,e_2,e_3}= {1,x, x^2} are an orthonormal basis then it would follow that

[tex]\int_0^1 e_i e_j dx[/tex]

would be 0 or 1.

The inner product was given to you so that you would know what the inner product was, since there are infinitely many inner products. For example, if in your R^3 example I defined {x,y} as <x,y>/2, then you would find a different answer for the question 'find y such that g(x) = {x,y}' than 'find y such that g(x)=<x,y>'.
 
  • #3
Thank you :)
So I should Gram-Schmidt the basis I have (and then normalize them) to find an orthonormal basis, and then do what I was doing, right?
 

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