How many of solutions are there for a nullspace based on rank of a matrix?

Question

This always confuses me. How to know the number of solutions for a matrix for a NULL space :

1. Which is full row rank but not full col rank ?
2. Which is full col rank but not full row rank ?
3. Intersection of #1 and #2 above
4. Something which is none of the above i.e. not full row rank and not full col rank either

I do know the answer, I just do not know how to intuitively figure it out ?

Answer 1

The rank-nullity theorem states that $rank(A)+null(A) = n$ where $n$ is the number of columns of $A$.

1. Full row rank means that the dimension of the row space is $n$ and thus $null(A) = n - rank(A) = n-n=0$.
2. Not full row rank means that the dimension of the row space is $ 0$.
3. I assume you mean full row rank and full column rank. This is the same as case 1 above.
4. This is the same as case 2 above.