I will try not to use so many pictures again, it was just a pain in the butt to do. So instead I will combine them all into one picture and label them as steps and explain them by the step number, unless someone know of how to put this directly onto the computer.
I also ask that you only open the image once while reading the entire post, I have very limited bandwidth.
A Matrix is in Echelon Form if it has the following three properties:
1. All nonzero rows are above any rows of all zeros.
2. Each leading entry of a row is in a column to the right of the leading entry of the row above it.
3. All entries in a column below a leading entry are zeros.
This is shown in examples one and two of the picture.
If the matrix also meets the following requirements it is in Reduced Echelon Form:
4. The leading entry in each nonzero row is 1
5. each leading 1 is the only nonzero entry in its column (except for the last column if it is an augmented matrix.
This is shown in examples three and four of the picture. The fifth example is a general form where * depicts any number and the arrows show where there must be zeros.
Pivot Positions and columns
A pivot position is the position of the first nonzero value of a matrix in echelon form. A pivot column is a column with a pivot postion. Each variable that does not have a pivot position is a free variable, one that the answer to is unknown.
Example six is of an augmented matrix first forwards row reduced to create echelon form to show pivot positions and columns, then backwards row reduced to obtain the answer.