For any two subspaces of an inner product space..
Prove that for any two subspaces W1 and W2 of an inner product space V, we have
a) (W1 + W2)^⊥ = W1^⊥ ∩ W2^⊥ and
b) (W1 ∩ W2)^⊥ = W1^⊥ + W2^⊥
Hint: The second follows from the first- you may use the fact that (W^⊥)^⊥ = W.
Any help would be greatly appreciated!
Re: For any two subspaces of an inner product space..
Hint: The perpendicular space of W1 + W2 will not include anything that spans along that space (figure out the plane that spans the two spaces and get the perpendicular space of that).