# Thread: Vector subspaces and projection questions

1. ## Vector subspaces and projection questions

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

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

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

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

2. ## Re: Vector subspaces and projection questions

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

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

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

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

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