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## 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=P

_{2}(

*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]v

_{i}

is a vector such that g(x)=<x,y> for all x element of V

where B={v

_{1},v

_{2}...v

_{n}} is an orthonormal basis for V.

I also know that the answer (printed in the back of the book) is y=210x

^{2}- 204x + 33 but I have no idea how to get this answer!

## The Attempt at a Solution

I assigned B={1,x,x

^{2}}={e

_{1},e

_{2},e

_{3}} as an orthornormal basis for P

_{2}(R) (the space of polynomials of degree less than or equal to 2 over R).

Then I did:

y=g(1)e

_{1}+ g(x)e

_{2}+ g(x

^{2})e

_{3}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)e

_{1}+ (0+1)e

_{2}+ (0+2)e

_{3}

=1e

_{1}+ 1e

_{2}+ 2e

_{3}

= 1 + x + 2x

^{2}

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 R

^{3}using the technique I applied above, and it worked fine.

Any help would be greatly appreciated.

Thanks :)