What gets sent to zero
The null space of A (also called its kernel) is the set of all vectors x with A*x = 0 — everything the matrix collapses onto the zero vector. The zero vector is always in there; the real question is whether anything else is. If only x = 0 solves A*x = 0, the columns are independent; if other x exist, the columns are dependent.
What it says about solutions
The null space controls the whole solution story for a system of equations A*x = b. If you find one solution p (a particular solution), then every solution looks like p plus something from the null space: x = p + n. So if the null space is just {0}, the solution is unique; if the null space is bigger, there are infinitely many solutions, all parallel shifts of one another.
- Reduce A to echelon form; pivot columns are independent, the rest are free.
- Each free column gives one free variable, and one independent null-space vector.
- No free variables -> null space is {0} -> A*x = b has at most one solution.
Rank + nullity = columns
Now the accounting law. Every column is either a pivot column (counted by the rank) or a free column (counted by the nullity). They cannot be both and nothing is left out, so they must add up. This is the rank-nullity theorem: rank + nullity = number of columns.
A is 3 x 4, rank(A) = 2 columns = 4 nullity = columns - rank = 4 - 2 = 2 -> 2 pivot columns, 2 free columns -> A*x = 0 has a 2-dimensional null space -> A*x = b (if solvable) has infinitely many solutions