1.1 Introduction to Systems of Linear Equations
Definitions
A linear equation in n variables
123
,,,...
n
xxx x
has the form
11 2 2 3 3 ... .
nn
ax a x ax a x b++++=
The coefficients
123
,,,...
n
aaa a
are real numbers, and the constant term
b
is a real number. The number
1
a
is the leading coefficient
and
1
x
is the leading variable.
A system of m linear equations in n variables is a set of equations, each of which is linear in the same n variables:
11 11 12 12 13 13 1 1 1
21 21 22 22 23 23 2 2 2
11 2 2 3 3
...
...
...
nn
nn
mm mm m m mnmn m
ax ax ax ax b
ax ax ax ax b
ax ax ax ax b
++++=
ļ¬
ļÆ++++ =
ļÆ
ļ
ļÆ
ļÆ++++=
ļ®
A solution of a linear system is sequence of numbers
123
,,,...
n
sss s
that is a solution of each equation in the system.
Definition
A system of linear equations is consistent when it has at least one solution and inconsistent when it has no solution.
For a system of linear equations, exactly one of the statements below is true.
1. The system has exactly one solution (consistent system).
2. The system has infinitely many solutions (consistent system).
3. The system has no solution (inconsistent system).
A system is in row-echelon form if it has a āstair-stepā pattern with leading coefficients of 1.
Two systems of linear equations are equivalent when they have the same solution set. To solve a system that is not in
row-echelon form, first rewrite it as an equivalent system that is in row-echelon form using the operations listed below.
1. Interchange two equations.
2. Multiply an equation by a nonzero constant.
3. Add a multiple of an equation to another equation.
Rewriting a system of linear equations in row-echelon form usually involves a chain of equivalent systems, using one of
the three basic operations to obtain each system. This process is called Gaussian elimination.
Ex 1 Use elimination to rewrite a system in row-echelon form (Gaussian elimination)
12 3
13
12 3
234
4210
2313 8
xx x
xx
xx x
+ā =
ļ¬
ļÆ+=
ļ
ļÆā+ ā =ā
ļ®
