Math 568
Systems of Linear Equations and Matrices
In these notes, we define a linear system and their associated matrices. We also indicate the algebra which
can be preformed on these objects.
1.
Definitions and Notation
A
linear equation in
n
variables
is an equation of the form:
a
1
x
1
+
a
2
x
2
+
· · ·
+
a
n
x
n
=
b
and a
system of
m
linear equations in
n
variables
is a collection of linear equations in the same
n
variables:
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
A
solution
to a system of linear equations in
n
variables is an vector [
s
1
, s
2
, . . . , s
n
] such that the components
satisfy
all
of the equations in the system when we set
x
i
=
s
i
.
We say that a system of linear equations is
consistent
if it has at least one solution; otherwise we say that it is
inconsistent
. A system of linear equations
may have more than one solution (we will see later that it must have infinitely many solutions in this case) and
the collection of all solutions of a linear system is called its
solution set
.
Consider the following two linear systems:
x
1
+
x
2
= 2
x
1

x
2
= 0
and
x
1
= 1
x
2
= 1
Notice that they have exactly the same solution set, namely
{
[1
,
1]
}
.
We say that these two systems are
equivalent. More generally, two systems of linear equations (in the same variables) are said to be
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
This is the end of the preview.
Sign up
to
access the rest of the document.
 Winter '08
 White
 Linear Algebra, Algebra, Linear Equations, Equations, Matrices, Systems Of Linear Equations, augmented matrix, Linear Equations and Matrices

Click to edit the document details