1. Codimension.

If a system of equations has nonempty solution set,
then the rank of the augmented matrix of the system
is the ***codimension*** of the solution set,
not dimension. dimension + codimension = # variables.
For example, today, two of our three systems
had nonempty solution sets, each with five variables.
One system's augmented matrix had rank 2 and 
its solution set had dimension 5 - 2 = 3. 
The other system's augmented matrix had rank 3 
and its solution set had dimension 5 - 3 = 2.
If a system of equations had 100 variables and
an augmented matrix of rank 87, then the
dimension of the solution set would be 13
if nonempty.
(A zero-dimensional solution set is a single point;
the empty set has undefined dimension.)

2. Links.

2a. Today's in-class matrix computations:

http://nbviewer.jupyter.org/urls/dl.dropbox.com/s/emp83e26kpepg7c/03-rref.ipynb

2b. Computer-assisted explicit sequence of 
(corrected) row operations for Gauss-Jordan elimination
of the big matrix from Day 2:

http://nbviewer.jupyter.org/urls/dl.dropbox.com/s/j0pmptjt3jkelzc/02-rref.ipynb

2c. Optional: try copying some of my commands from above links to:

http://live.sympy.org

Then try making up your own commands.