Question

In the lecture 6, Prof talks about Union and intersection of subspaces.
I didn't understand how intersection of 2 Line Spaces (i.e. NullSpace) classifies as a subspace.
Consider, 2 line spaces consisting of vectors(0,0,1) and (1,0,0). Linear combination of this 2 vectors won't always lie on the null space.
I understand that Null Space by itself is a subspace, but when taken in this context i.e. as an intersection of 2 line spaces, it's not a subspace.

Answer 1

subspace is one point, (0,0,0)

Answer 2

All the linear combinations of your 2 vectors is a column space, which is a subspace of R3.
Their intersection is the null vector, which is, as you state, a subspace.
Does that help?