Questions related to vector spaces and rank of a matrix

Hello, I'm new here (Hi)

I collected some tough questions regarded vector spaces and rank of a matrix that I couldn't answer, and ask your help, please.

1. let U, W be different sub-spaces of a vector space V; dim U= dimW = n-1, dim V = n; find dim(U intersect W).

2. Prove: if B = {v1,...,vn} is a basis of V and if U is a subspace of V with dimension k, k<=n, then there are k vectors in B that spans U.

3. **Prove or disprove**: if A,B are 3x3 matrices such that rank(A)=rank(B)=2 then AB != 0. I think it's not true but I can't come up with an example that disproves this.