Rank of matrix and the number of solutions

For a  m by n matrix A with rank of r, the number of solutions for linear system of Ax = b is 

1. r < m and r < n
   The number of solutions is 0 or infinite.
2. r = m < n
   The number of solutions is 1 or infinite. (The solution EXISTs.)
3. r = n < m
   The number of solution is 0 or 1. (The solution is  UNIQUE if exists.)
4. r = m = n
   The number of solutions always 1. (INVERTIBLE case.) 

Gilbert Strang

I’ve been a big fan of Gilbert Strang, MIT professor, after buying his math book, Linear algebra and its applications.

You can see his video lectures on elementrary linear algebra course on MIT’s open courseware site.
http://ocw.mit.edu/OcwWeb/Mathematics/18-06Spring-2005/CourseHome/