1.2 ROW ECHELON FORM
KIAM HEONG KWA
p. 13 A matrix is said to be in
row echelon form
if
(1) the ﬁrst nonzero entry in each nonzero row is
1
;
(2) if the
k
th row does not consist entirely of zeros, the number of
leading zeros
1
in the
(
k
+ 1)
th row is greater than the number
of leading zeros in the
k
th row;
(3) all zero rows are below all nonzero rows.
pp. 1617 A matrix is said to be in
reduced echelon form
if, in addition to
being in row echelon form, the ﬁrst nonzero entry in each nonzero
row is the only nonzero entry in its column.
The process of using elementary row operations to transform the
augmented matrix of a linear system into one in row echelon form
and into one in reduced row echelon form are called
Gaussian elim
ination
and
GaussJordan reduction
respectively.
Example 1
(Exercise 1.2.2(a) in the text)
.
Consider the augmented
matrix
x
1
x
2
=
rhs
1
2
4
0
1
3
0
0
1
in row echelon form. The corresponding system is inconsistent be
cause there is no ordered pair
(
x
1
,x
2
)
such that
0
x
1
+ 0
x
2
= 1
.
Example 2
(Exercise 1.2.2(c) in the text)
.
Consider the augmented
matrix
x
1
x
2
x
3
=
rhs
1

2
4
1
0
0
1
3
0
0
0
0
Date
: June 18, 2011.
1
The
leading zeros
of a row vector are the collection of all zero entries preceding all
nonzero entries. For instances, the row vectors
(1
,
2
,
3
,
4
,
5)
,
(0
,
1
,
2
,
0
,
3)
and
(0
,
0
,
0
,
1
,
0)
have no leading zeros, one leading zero, and three leading zeros respectively.
1
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KIAM HEONG KWA
in row echelon form. The variables in the associated linear system
corresponding to the ﬁrst nonzero elements in each row of the aug
mented matrix are called the
lead variables
; the remaining vari
ables are called the
free variables
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 Summer '08
 KIM
 Linear Algebra, Algebra, Gaussian Elimination, Row echelon form, augmented matrix, Leading zero

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