Solving for the Unique Vector h(x) in the Polynomial Space of Degree 2

[torquerotates]

Homework Statement

For the given inner product space V=polynomial space of degree 2. Find a the unique vector h(x) such that <f,h>=int(f*h) from 0 to 1. and g(f)=f(0)+f'(1).

Homework Equations

Theorem: let V be a finite dimensional inner product space over F and let g:V->F be a linear transformation. Then there exists an unique vector, y in V such that g(x)=<x,y>

The Attempt at a Solution

well, <f,h>=g(f)

=> <f,h>=f(0)+f'(1)
=>int(f*h) from 0 to 1=f(0)+f'(1)

I have no clue how to solve this final step

 

[Dick]

Be explicit. Let h(x)=h0+h1*x+h2*x^2, similarly for f(x). Now explicitly work out <f,g>=f(0)+f'(0). Hint: if the equation must be true for ALL f, then it must be true for say, f0=1, f1=0 and f2=0. Try some other choices for the coefficients of f, until you get enough equations to solve for the h's.