Calculate null space of two vector spaces (specific method)

Question

I'm working on a Mathematica exercise, but I encountered a problem that has more to do with my mathematical knowledge. This is the exercise along with the solution: So I basically have 4 vectors. Vector $u_1$ and $u_2$ span vectorspace $V$, and vectors $w_1$ and $w_2$ span $W$. We are looking for the intersection of $V$ and $W$. Now, I don't understand why they compute the null space of both vector spaces.

I understand the solution provided here by user Daryl; How to find an intersection of a 2 vector subspace?.

But I don't understand why one would compute the null space first? Could someone explain this first step to me? Thanks in advance.

Answer 1

The nullspace is the set of vectors perpendicular to the vectors. (Because all those dot products are zero.) So you join the two nullspaces and then ask for the nullspace of that, which gives you all vectors perpendicular to both nullspaces, which means that they lie in both original spaces.

Imagine it with $V$ and $W$ being two planes in 3-space. Then the two nullspaces are normal lines to both planes. The join of those lines is the plane they span. The nullspace of that plane is a line perpendicular to both of the normal lines, and that happens to be the intersection of the planes.