Math 220
Robert Creese
September 3, 2019
Robert Creese
September 3, 2019
1 / 62
1.1 Systems of linear equations
Robert Creese
September 3, 2019
2 / 62
1.1 Linear Equation
A
Linear Equation
in variables
x
1
,
x
2
,
· · ·
,
x
n
is an equation that can be
written as
a
1
x
2
+
a
2
x
2
+
· · ·
a
n
x
n
=
b
.
(1)
Here
a
1
,
a
2
,
· · ·
a
n
are called coefficients. The coefficients and
b
are
normally known and are real or complex numbers.
For example 4
x
1
+ 5
x
2
=

1 +
x
2
is a linear equation since it can be
written in the form in (1).
However
x
1
x
2
+ 3
x
1

4
x
2
= 3 is not a linear equation. Note that we
cannot multiply variables with other variables or use exponents.
Robert Creese
September 3, 2019
3 / 62
1.1 Systems of linear equations
A
System of Linear Equations
in variables
x
1
,
x
2
,
· · ·
,
x
n
is a set of one
or more linear equations in variables
x
1
,
x
2
,
· · ·
,
x
n
.
An example of a system of linear equations is
x
1
+2
x
2

x
3
= 0
2
x
1
+4
x
3
= 6
(2)
Note that the number of variables and number of equations may not
necessarily match.
Robert Creese
September 3, 2019
4 / 62
1.1 Systems of linear equations
A
solution of a system
, is a set of numbers (
s
1
,
s
2
,
· · ·
s
n
) where each
equation is true if
x
1
=
s
1
,
· · ·
x
n
=
s
n
.
For example (1
,
0
,
1) is a solution the system
x
1
+2
x
2

x
3
= 0
2
x
1
+4
x
3
= 6
(3)
The
solution set of a system
is the set of all possible solutions to that
system. .
Two linear systems are called
equivalent
if they share the same solution
set.
Robert Creese
September 3, 2019
5 / 62
1.1 Systems of linear equations
A system of linear equations will always have either
1
No solution
2
One solution
3
Infinitely many solutions
This can be illustrated in the case where have system of two equations
with two variables. In this case each equations represents a line and each
solution is a point where the lines intersect. So there are three cases
1
Parallel Lines (never intersect so no solution) Ex.
x
2
=
x
1
,
x
2
= 1 +
x
1
2
Intersecting lines (Intersect at one point so one solution)
Ex.
x
2
=
x
1
,
x
2
= 2
x
1
3
Coinciding lines (lines are the same so every point on the line is a
solution, infinite solutions) Ex.
x
2
=
x
1
,
2
x
2
= 2
x
1
Robert Creese
September 3, 2019
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1.1 Systems of linear equations
A system of linear equations is
consistent
if it has at least one solution.
A system of linear equations is
inconsistent
if it has no solution.
Systems of linear equations will often be referred to as a system frequently
for the purpose of brevity.
Robert Creese
September 3, 2019
7 / 62
1.1 Matrix Notation
Here a
matrix
is a rectangular array of numbers. The shape of a matrix is
important.
If a matrix
A
has
m
rows and
n
columns, then
A
has size
m
×
n
and is a
m
×
n
matrix.
For example the following linear system
x
1
+4
x
2
+3
x
3
= 2
2
x
2

x
3
=

10

2
x
1
+
x
2
=

7
(4)
has a
coefficient matrix
(matrix including the coefficients of a linear
system)
1
4
3
0
2

1

2
1
0
There is a 0 in the second row, first column since there is no
x
1
in the
second equation (
x
1
has coefficient 0).
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