MATH1850U/2050U:
Chapter 1
1
LINEAR SYSTEMS
Introduction to Systems of Linear Equations (1.1; pg.2)
Definition:
A
linear equation
in the
n
variables,
n
x
x
x
,
,
2
1
is defined to be an
equation that can be written in the form:
b
x
a
x
a
x
a
n
n
2
2
1
1
where
n
a
a
a
,
,
2
1
and
b
are real constants.
The
i
a
are called the
coefficients
, and the
variables
i
x
are sometimes called the
unknowns
.
If it cannot be written in this form, it is
called a
nonlinear equation
.
Examples:
4
3
2
1
8
10
6
5
x
x
x
x
4
2
2
3
1
6
3
9
x
x
x
x
9
7
3
2
1
x
x
x
Application:
Suppose that $100 is invested in 3 stocks.
If A , B, and C, denote the
number of shares of each stock that are purchased and they have units costs $5, $1.5, and
$3 respectively, write the linear equation describing this scenario.
Definition:
A
solution of a linear equation
b
x
a
x
a
x
a
n
n
2
2
1
1
is a sequence
(or
n
tuple) of
n
numbers
n
s
s
s
,
,
2
1
such that the equation is satisfied when we
substitute
n
n
s
x
s
x
s
x
,
,
2
2
1
1
in the equation.
The set of ALL solutions of the
equation is called the
solution set
(or sometimes the
general solution
) of the equation.
Example:
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Chapter 1
2
Definition:
A finite set of linear equations in
n
variables,
n
x
x
x
,
,
2
1
, is called a
system
of linear equations
or a
linear system
.
A sequence of
n
numbers
n
s
s
s
,
,
2
1
is a
solution of the system
of linear equations if
n
n
s
x
s
x
s
x
,
,
2
2
1
1
is a solution of
every equation in the system.
Example:
Verify that
1
,
1
,
1
z
y
x
is a solution of the linear system:
1
2
4
3
2
3
2
z
y
y
x
z
y
x
Question:
Does every system of equations have a solution?
Definition:
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 Spring '11
 PaulaTu
 Linear Equations, Equations, Gaussian Elimination, Linear Systems, Systems Of Linear Equations, elementary row operations

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