# Orthogonal vector

• Oct 29th 2013, 10:19 AM
zokomoko
Orthogonal vector
Hey, I wrecked my head around this one and I hope you could help me.
The questions is as follows:
"Let V be an inner product space, and U,W be subspaces of V so that dim(U)<dim(W)
Prove that exists a vector w in W (that is not the zero vector) so that it is orthogonal to all the vectors that are in U"

• Oct 29th 2013, 08:49 PM
chiro
Re: Orthogonal vector
Hey zokomoko.

Hint: Use the identity that if V < W (i.e. V is a subspace of W) then V + V_perp = W where V_perp is perpendicular to V.
• Oct 29th 2013, 11:24 PM
zokomoko
Re: Orthogonal vector
Got it!
Can't believe it was as simple as that :)
• Oct 30th 2013, 07:04 AM
HallsofIvy
Re: Orthogonal vector
Another way to do this: choose a basis for U. Then extend it to a basis for W. Let v be any vector such that its linear combination of those basis vectors has coefficient 0 for vectors that are basis vectors for U, non-zero coefficient for at least one of the basis vectors in the extension.