06-240/Classnotes For Tuesday November 21
More about the Wongpak Matrices
So let us assume row reduction leads us to the systems or . What does it tell us about the solutions? Let us start from the second system:
Well, quite clearly if this system has no solutions, but if it has solutions no matter what , and are. Finally, for any given values of , and we can choose the values of and (the variables corresponding the columns containing no pivots) as we please, and then get solutions by setting the "pivotal variables" in terms of the non-pivotal ones as follows: , and .
What about the system corresponding to ? It is
Here too we have solutions iff , and if , we have the freedom to choose the non-pivotal variables and as we please. But now the formulas for fixing the pivotal variables , and in terms of the non-pivotal ones are a bit harder.