If S is a subspace of V, is S⊥ ⊇ V⊥?

[bodensee9]

Homework Statement

Can someone help me with this proof? I'm supposed to show that if a subspace S is contained in subspace V, then S perp contains V perp.

Homework Equations

None, or the dimensions must add up, so S + S perp = some dimension N, and V + V perp equals the same dimension N, if S and V are both subspaces of N.

The Attempt at a Solution

Am I supposed to show that the dimensions don't add up? Can anyone provide suggestions? Thanks.

 

[AKG]

S perp contains V perp iff x in V perp implies x in S perp.
x in V perp iff for all y in V, <x,y> = 0.
x in S perp iff for all y in S, <x,y> = 0.

So you want to show (for all y in V, <x,y> = 0) implies (for all y in S, <x,y> = 0). Well S is a subset of V, so this is obvious.