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Wednesday
January 7
−
Lecture 2 :
The Gauss-Jordan elimination algorithm and
RREF.
(Refers to sections 2.1 and 2.2)
Expectations:
1.
Define a system of
m
linear equations in
n
unknowns.
2.
Define coefficient matrix, augmented coefficient matrix.
3.
Define Row echelon form (REF),"Reduced row echelon form" (RREF), "leading-
1's" and "pivots" in a matrix.
4.
Solve a system of linear equations by using the Gauss-Jordan elimination
algorithm.
5.
Recognize
basic unknowns
and
free unknowns
.
6.
Define the rank of a coefficient matrix.
2.1
Introduction
−
Linear equations are equations of the form
a
11
x
1
+
a
12
x
2
+ .
.. +
a
1
n
x
n
=
b
1
. Solving simultaneously many linear equations means finding the set of all variables
that satisfy all of the given linear equations. In this lecture we discuss a method for
solving these which involves only the coefficients and the constants.
2.1.1
Example
−
We give an example where we solve a system of 3
linear
equations
in 3 unknowns by using the mathematical principles above. (By
linear
equations we
mean that the exponents of all variables are 0 or 1.) The solution of this system
involves the application of the following principles:
a = b
c = d
then
a + c = b + d
ac = bd
.
Please illustrate how these 2 principles are invoked over and over the solution of this
system:
4
x
−
8
y
+
−
4
z
= 36
2
x
−
y
+
z
= 6
3
x
−
2
y
+
−
2
z
= 2.
The system has solution (
x
,
y
,
z
) = (
−
2,
−
7, 3).
See details of solution
.

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2.1.2
Geometric interpretation.
•
The 3 equations described above, when plotted in 3-space, form planes.
•
The solution indicates the only point that belongs to all 3 planes, i.e., the 3
planes intersect at that point.
2.2
Definition
−
System of linear equations. A
system of m linear equations with n
unknowns
has the form :
a
11
x
1
+
a
12
x
2
+ .
.. +
a
1
n
x
n
=
b
1
.
a
21
x
1
+
a
22
x
2
+ .
.. +
a
2
n
x
n
=
b
2
.
...
a
m
1
x
1
+
a
m
2
x
2
+ .
.. +
a
mn
x
n
= b
m
.
where
b
1
,
b
2
, .
..,
b
m
, are given numbers and
x
1
,
x
2
, .
..,
x
n
are unknowns. The
a
ij
's, 1
≤
i
≤
m
and 1
≤
j
≤
n
,
are numbers which we call
coefficients
.
2.2.1

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