04.05.1
Chapter 04.05
System of Equations
After reading this chapter, you should be able to:
1.
setup simultaneous linear equations in matrix form and viceversa,
2.
understand the concept of the inverse of a matrix,
3.
know the difference between a consistent and inconsistent system of linear equations,
and
4.
learn that a system of linear equations can have a unique solution, no solution or
infinite solutions.
Matrix algebra is used for solving systems of equations.
Can you illustrate this
concept?
Matrix algebra is used to solve a system of simultaneous linear equations.
In fact, for many
mathematical procedures such as the solution to a set of nonlinear equations, interpolation,
integration, and differential equations, the solutions reduce to a set of simultaneous linear
equations.
Let us illustrate with an example for interpolation.
Example 1
The upward velocity of a rocket is given at three different times on the following table.
Table 5.1.
Velocity vs. time data for a rocket
Time,
t
Velocity,
v
(s)
(m/s)
5
106.8
8
177.2
12
279.2
The velocity data is approximated by a polynomial as
12.
t
5
,
2
c
bt
at
t
v
Set up the equations in matrix form to find the coefficients
c
b
a
,
,
of the velocity profile.
Solution
The polynomial is going through three data points
3
3
2
2
1
1
,
t
and
,
,
,
,
v
v
t
v
t
where from
table 5.1.
8
.
106
,
5
1
1
v
t
2
.
177
,
8
2
2
v
t
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
04.05.2
Chapter 04.05
2
.
279
,
12
3
3
v
t
Requiring that
c
bt
at
t
v
2
passes through the three data points gives
c
bt
at
v
t
v
1
2
1
1
1
c
bt
at
v
t
v
2
2
2
2
2
c
bt
at
v
t
v
3
2
3
3
3
Substituting the data
3
3
2
2
1
1
,
and
,
,
,
,
v
t
v
t
v
t
gives
8
.
106
5
5
2
c
b
a
2
.
177
8
8
2
c
b
a
2
.
279
12
12
2
c
b
a
or
8
.
106
5
25
c
b
a
2
.
177
8
64
c
b
a
2
.
279
12
144
c
b
a
This set of equations can be rewritten in the matrix form as
2
.
279
2
.
177
8
.
106
12
144
8
64
5
25
c
b
a
c
b
a
c
b
a
The above equation can be written as a linear combination as follows
2
.
279
2
.
177
8
.
106
1
1
1
12
8
5
144
64
25
c
b
a
and further using matrix multiplication gives
2
.
279
2
.
177
8
.
106
1
12
144
1
8
64
1
5
25
c
b
a
The above is an illustration of why matrix algebra is needed. The complete solution to the set
of equations is given later in this chapter.
This is the end of the preview.
Sign up
to
access the rest of the document.
 Spring '08
 Kaw,A
 Linear Algebra, Vector Space, Invertible matrix, Infinite Solutions

Click to edit the document details