Linear Algebra, 2020a

60 Chapter One. Linear Systems
and to C’s.
c 1,1 x 1 + c 1,2 x 2 + ···+ c 1,n x n = 0
c 2,1 x 1 + c 2,2 x 2 + ···+ c 2,n x n = 0
.
c m,1 x 1 + c m,2 x 2 + ···+ c m,n x n = 0
(∗∗∗)
With B and C different only in column n, suppose that they differ in row i.
Subtract row i of (∗∗∗) from row i of (∗∗) to get the equation (b i,n −c i,n )·x n = 0.
We’ve assumed that b i,n ≠ c i,n and so we get x n = 0. Thus x n is not a free
variable and so in (∗∗) and (∗∗∗) the n-th column contains the leading entry of
some row, since in an echelon form matrix any column that does not contain a
leading entry is associated with a free variable.
But now, with B and C equal on the first n − 1 columns, by the definition of
reduced echeleon form their leading entries in the n-th column are in the same
row. And, both leading entries would have to be 1, and would have to be the
only nonzero entries in that column. Therefore B = C.
QED
We have asked whether any two echelon form versions of a linear system have
the same number of free variables, and if so are they exactly the same variables?
With the prior result we can answer both questions “yes.” There is no linear
system such that, say, we could apply Gauss’s Method one way and get y and z
free but apply it another way and get y and w free.
Before the proof, recall the distinction between free variables and parameters.
This system
x + y = 1
y + z = 2
has one free variable, z, because it is the only variable not leading a row. We
have the habit of parametrizing using the free variable y = 2 − z, x =−1 + z, but
we could also parametrize using another variable, such as z = 2 − y, x = 1 − y.
So the set of parameters is not unique, it is the set of free variables that is
unique.
2.7 Corollary If from a starting linear systems we derive by Gauss’s Method two
different echelon form systems, then the two have the same free variables.
Proof The prior result says that the reduced echelon form is unique. We get
from any echelon form version to the reduced echelon form by eliminating up, so
any echelon form version of a system has the same free variables as the reduced
echelon form version.
QED
We close with a recap. In Gauss’s Method we start with a matrix and then
derive a sequence of other matrices. We defined two matrices to be related if we