Page 46 - 35Linear Algebra

46 Systems of Linear Equations

Here is a verbal description of the preceeding example of the standard ap-
proach. We say that x, y, and z are pivot variables because they appeared
with a pivot coefficient in RREF. Since w never appears with a pivot co-
efficient, it is not a pivot variable. In the second line we put all the pivot
variables on one side and all the non-pivot variables on the other side and
added the trivial equation w = w to obtain a system that allowed us to easily
read off solutions.

The Standard Approach To Solution Sets

1. Write the augmented matrix.

2. Perform EROs to reach RREF.

3. Express the pivot variables in terms of the non-pivot variables.

There are always exactly enough non-pivot variables to index your solutions.
In any approach, the variables which are not expressed in terms of the other
variables are called free variables. The standard approach is to use the non-
pivot variables as free variables.
Non-standard approach: solve for w in terms of z and substitute into the
other equations. You now have an expression for each component in terms
of z. But why pick z instead of y or x? (or x + y?) The standard approach
not only feels natural, but is canonical, meaning that everyone will get the
same RREF and hence choose the same variables to be free. However, it is
important to remember that so long as their set of solutions is the same, any
two choices of free variables is fine. (You might think of this as the difference
between using Google Maps TM or Mapquest TM ; although their maps may
look different, the place hhome sici they are describing is the same!)
When you see an RREF augmented matrix with two columns that have
no pivot, you know there will be two free variables.

Example 20 (Standard approach, multiple free variables)

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