When you have the system
and you are solving for
.
The matrix you end up with is actually:
Now if you done the row-reductions (assuming you were correct!) we get:
You can bring the bottom three rows to:
However this matrix is still now in row-reduced echelon form.
We will add the (3rd row
2) to 2nd row and 1st row.
Finally, add (2nd row
) to 1st row:
This tells you that:
.
Thus,
(notice that
is "free").
This means if
then
solves
, and conversely.
Thus, the nullspace is
We see the dimension is one, and so the nullity is one.