# Thread: Orthogonal basis of a polynomial and scalar product

1. ## Orthogonal basis of a polynomial and scalar product

a) says transform (1,x,x^2) into an orthogonal basis. I normally know how to do this with simple problems but this problem makes me so confused I dont even know where to start. can someone do it and show me the steps?

b) I need to give the projection of the polynomial x^2 under the sub basis of W=vect(1,x)

2. ## Re: Orthogonal basis of a polynomial and scalar product

The space spanned by {1, x, x^2} is the space of all quadratic polynomials, ax^2+ bx+ c. In order to talk about "orthogonal" you have to have an inner product defined. What inner product are you using?

3. ## Re: Orthogonal basis of a polynomial and scalar product

b) find the projection of vector $x^2$ on the subspace $W$ spanned by $\{1,x\}$

the projection is a vector of the form $a x+b$ such that

$\displaystyle x^2-a x -b$ is orthogonal to both $1$ and $x$

$\displaystyle \int_0^1 \left(x^2-a x -b\right) \, dx=0$

and

$\displaystyle \int _0^1\left(x^2-a x -b\right)xdx =0$