For any two subspaces of an inner product space..

Oct 2012
48
0
Ohio
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!
 

chiro

MHF Helper
Sep 2012
6,608
1,263
Australia
Hey TimsBobby2.

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).