Math Help - Vector subspaces and projection questions

1. Vector subspaces and projection questions

I'm stuck with two questions on vector subspaces and projections.

1. Let $V_1$ and $V_2$ be subspaces of $\Omega$ and let $V_0 = V_1 \cap V_2$ Under what conditions $P_{V_0} =P_{V_1} P_{V_2}$

Well, for this one I see that it will work if one of $V_1$ and $V_2$ contains the other and also if $V_1 \perp V_2$ , but this doesn't seem like a good answer.

2. Let $V_1,V_2,V_3$ be subspaces. Does $V_1 \cap (V_2 + V_3) = V_1\cap V_2 + V_1\cap V_3$ in general? If not, does it hold if $V_2$ and $V_3$ are linearly independent?

Thanks in advance,

2. Re: Vector subspaces and projection questions

Originally Posted by akolman
I'm stuck with two questions on vector subspaces and projections.

1. Let $V_1$ and $V_2$ be subspaces of $\Omega$ and let $V_0 = V_1 \cap V_2$ Under what conditions $P_{V_0} =P_{V_1} P_{V_2}$

Well, for this one I see that it will work if one of $V_1$ and $V_2$ contains the other and also if $V_1 \perp V_2$ , but this doesn't seem like a good answer.

2. Let $V_1,V_2,V_3$ be subspaces. Does $V_1 \cap (V_2 + V_3) = V_1\cap V_2 + V_1\cap V_3$ in general? If not, does it hold if $V_2$ and $V_3$ are linearly independent?

Thanks in advance,
The answer to your first question can be found on my blog post here. Any ideas for the second? What if $V_2+V_3=V$ (where $V$ is the full space) so that the LHS is $V$ but $V_1\cap V_2=V_1\cap V_3=\varnothing$? Can that happen?