The answer to your first question can be found on my blog post here. Any ideas for the second? What if (where is the full space) so that the LHS is but ? Can that happen?
I'm stuck with two questions on vector subspaces and projections.
1. Let and be subspaces of and let Under what conditions
Well, for this one I see that it will work if one of and contains the other and also if , but this doesn't seem like a good answer.
2. Let be subspaces. Does in general? If not, does it hold if and are linearly independent?
Thanks in advance,