Finding an orthogonal complement without an explicitly defined inner product

[Idioteqnician]

Homework Statement

P5 is an inner product space with an inner product. We applied the Gram Schmidt process to the basis {1,x,x^2,x^3,x^4} and obtained the following result. {f1,f2,f3,f4,x^4+2}

What is the orthogonal complement of P3 in P5 with respect to this inner product?

Homework Equations

http://tutorial.math.lamar.edu/Classes/LinAlg/OrthonormalBasis.aspx

has everything on the gram schmidt process

Definition of orthogonal complement:
Suppose that W is a subspace of an inner product space V. We say that a vector u from V is orthogonal to W if it is orthogonal to every vector in W. The set of all vectors that are orthogonal to W is called the orthogonal complement of W.

The Attempt at a Solution

I'm not really sure. I feel like I need a defined inner product to actually find which vectors are orthogonal. I feel like I need to do something with the x^4+2, but honestly I am entirely lost.

 

[Dick]

Of course you need to know what the inner product is to say what is orthogonal. They must have told you at some point.

 

[Idioteqnician]

Nope, I copied this question straight from my homework page. We are given the result of the Gram Schmidt process, which is a set of orthonormalized vectors. Considering the orthogonal complement is the set of all vectors orthogonal to some subspace W, does that mean my answer is just a linear combination of the given result of the gram schmidt process? They are orthogonal vectors, and the only way I could think to get every orthogonal vector from that information would be to linearly combine them.

 

[Dick]

Ok, maybe I'm beginning to see what the question is. So f1 is a constant term, f2 has the form a+bx, f3 is a quadratic, d+e*x+f*x^2 etc. Does that give you a hint?

 

[Idioteqnician]

I think so. Thank you