§
1.1 and
§
1.2
1.1
Chapter 1
Matrices and Systems of Linear Equations
§
1.1: Introduction to Matrices and Systems of Linear Equations
§
1.2: Echelon Form and GaussJordan Elimination
Lecture
Linear Algebra  Math 2568M
on Friday, January 11, 2013
Oguz Kurt
MW 605
[email protected]
2929659
Off. Hrs:
MWF 10:2011:20
The Ohio State University
§
1.1 and
§
1.2
1.2
Outline
1
§
1.1 and
§
1.2
§
1.1 and
§
1.2
Linear Equations
Definition
§
1.1 and
§
1.2
1.4
Systems of Linear Equations (SLEs)
Definition
A
system of linear equations
is a finite set of linear equations, each
with the same variables. A
solution
of system of linear equations is a
vector that is
simultaneously
a solution of each equation in the
system. The
solution set
of a system of linear equations is the set of
all
solutions of the system.
Example
The system
2
x

3
y
=
7
3
x
+
y
=
5
has
[
2
,

1
]
as a solution.
§
1.1 and
§
1.2
1.5
Consistency and type of solutions
Note:
A system of linear equations is called
consistent
if it has at
least one solution. A system with no solutions is called
inconsistent
.
A system of linear equations with real coefficients has either
1
a unique solution (a consistent system) or
2
infinitely many solutions (a consistent system) or
3
no solutions (an inconsistent system).
§
1.1 and
§
1.2
1.6
Solving a System of Linear Equations
Example:
Solve the system
x

y

z
=
4
2
y
+
z
=
5
3
z
=
9
Note: To solve this system, we usually use
back substitution
.
3
z
=
9
→
z
=
3
z
=
3
,
2
y
+
z
=
5
→
2
y
=
2
→
y
=
1
z
=
3
,
y
=
1
,
x

y

z
=
4
→
x
=
8
[
8
,
1
,
3
]
is the unique solution for this SLEs.
§1.1 and§1.21.7Using Augmented MatrixAssume we have the following SLEs withmequations andnunknowns:a1,1x1+a1,2x2...a1,nxn=b1a2,1x1+a2,2x2...a2,nxn=b2...am,1x1+an,2x2...am,nxn=bmWe can represent it as follows:x1x2......xnEq. 1a1,1a1,2......a1,nb1Eq. 2a2,1a2,2......a2,nb2...............Eq. mam,1am,2......am,nbm
§
1.1 and
§
1.2
1.8
Augmented Matrix
The following matrix is called the
augmented matrix
of this SLEs:
a
1
,
1
a
1
,
2
...
...
a
1
,
n
b
1
a
2
,
1
a
2
,
2
...
...
a
2
,
n
b
2
.
.
.
.
.
.
.
.
.
.
.
.
a
m
,
1
a
m
,
2
...
...
a
m
,
n
b
m
Augmented Matrix can be used to solve SLEs.