Question

so if you have a 0 in the pivot position such as the following matrix:

0 1 2 3 4
0 0 0 1 2
0 0 0 0 0

how would you go about finding the basis for all four subspaces?

Answer 1

If all elements below pivot that is diagonal element is zero than matrix becomes zero

Answer 2

Let A be the original 3 x 5 matrix. The column space has is a subset of R^3 spanned by the second and fourth columns of A. The left null space of A the subset of R^3 that is spanned by any vector that is perpendicular to both of those columns. The row space of A is a subset of R^5 that is spanned by the first two rows of A, and the null space of A is the subset of R^5 that is spanned by (0, 0, 1, 0, 0)', (-2, 0, 0, 1, 0)', and (-4, -2, 0, 0, 1). I got that basis from the free variables you gave in your example.