Proving Inner Product Space: x not in W, y in W(perp)

[jbear12]

Let V be an inner product space, and let W be a finite-dimensional subspace of V. If x\notin W, prove that there exists y\in V such that y \in W(perp), but <x,y>\neq 0.

I don't have a clue...
Thanks

 

[lanedance]

you could start by using something along the lines of gram-schmidt to decompose to x into a sum of vector from W & one from W perp...

 

[jbear12]

Umm..I don't really get it. Can you explain more specifically? Thank you.

 

[lanedance]

what don't you get?

first you need to assume x is non-zero

x is not contained in W, and as its non-zero, this means it must have a component in W perp , (as V = W + W perp by defintion of W perp, sloppy notation here, but hopefully you get the idea)

now consider the dot product of x with the component of x in W perp