1.4 Pivots and their Influence on Solution Spaces
By now, we have seen many examples illustrating how the reduced row echelon matrix provides a convenient description of the solution space to a system of linear equations. In this section, we will use this understanding to make some general observations about how certain features of the reduced row echelon matrix reflect the nature of the solution space.
Reading
Read Chapter 1 Section 4 of Understanding Linear Algebra by David Austin and try out the preview exercise.
In the following videos you will see additional examples of finding the solutions of a system based on an augmented matrix in reduced row echelon form:
Theorem: In the reduced row echelon form of an augmented matrix, every pivot variable can be expressed as a function of the free variables.
Theorem: If there is a pivot in the augment column of a matrix in reduced row echelon form, then one of the equations in our system is 0 = 1. There will be no solutions.
Theorem: If there is a pivot in every row of the reduced row echelon form of a coefficient matrix, then no matter the augmented column, the system will always be consistent.