Let V be an inner product space, and let U and W be subspaces of V. Show that
and
here, U* denotes the orthogonal complement of U.
having trouble even finding where to start this one.
Let V be an inner product space, and let U and W be subspaces of V. Show that
and
here, U* denotes the orthogonal complement of U.
having trouble even finding where to start this one.
Sorry, I hadn't noticed that LiveJournal page was "friends only". Here's a copy of the relevant comment. The "perp" symbol ⊥ (denoting an orthogonal complement) has come out on the line instead of as a superscript, which makes it a bit hard to read.
You need to know that U⊥⊥ = U. Also, if G and H are subspaces with G⊆H, then H⊥⊆G⊥.
If x = y+z with y∈U⊥ and z∈W⊥ then it is easy to see that y and z both lie in (U∩W)⊥, hence so does x. Therefore U⊥+W⊥⊆(U∩W)⊥.
For the converse inclusion, if x∈(U⊥+W⊥)⊥ then x∈U⊥⊥ = U, and similarly x∈W⊥⊥ = W. Thus (U⊥+W⊥)⊥⊆U∩W. Take the perp of each side to see that (U∩W)⊥⊆U⊥+W⊥.
Thus (U∩W)⊥ = U⊥+W⊥. You get the other identity by taking the perp of both sides and replacing U with U⊥ and W with W⊥.