Proof Annihilator: W1 & W2 Intersection | W1 & W2 Annihilator

[guroten]

Let W1 and W2 be subspaces of a finite dimension.
Prove that A(W1 intersection with W2) = A(W1) + A(W2)
Where A(W1) is the annihilator of W1.
I can prove one way, but not the other. How do I prove the rhs is contained in the lhs?

 

[VeeEight]

Try to find the dimension of A(W1) + A(W2) and A(W1 intersect. W2) to show that they are the same space.
It will help to show that [A(W1) intersect. A(W2)] = A(W1 + W2)

 

[Dick]

If you have an operator 'a' that vanishes on W1 intersect W2, then you have have to construct two operators a1 in A(W1) and a2 in A(W2) such that a=a1+a2. I'd suggest you pick an appropriate basis for W1+W2.