Lecture 9
Introduction to Linear Systems
How linear systems occur
Linear systems of equations naturally occur in many places in engineering, such as structural anal-
ysis, dynamics and electric circuits.
Computers have made it possible to quickly and accurately
solve larger and larger systems of equations. Not only has this allowed engineers to handle more
and more complex problems where linear systems naturally occur, but has also prompted engineers
to use linear systems to solve problems where they do not naturally occur such as thermodynamics,
internal stress-strain analysis, fluids and chemical processes.
It has become standard practice in
many areas to analyze a problem by transforming it into a linear systems of equations and then
solving those equation by computer. In this way, computers have made linear systems of equations
the most frequently used tool in modern engineering.
In Figure 9.1 we show a truss with equilateral triangles.
In Statics you may use the “method of
joints” to write equations for each node of the truss
1
. This set of equations is an example of a linear
system. Making the approximation
√
3
/
2
≈
.
8660, the equations for this truss are:
.
5
T
1
+
T
2
=
R
1
=
f
1
.
866
T
1
=
−
R
2
=
−
.
433
f
1
−
.
5
f
2
−
.
5
T
1
+
.
5
T
3
+
T
4
=
−
f
1
.
866
T
1
+
.
866
T
3
= 0
−
T
2
−
.
5
T
3
+
.
5
T
5
+
T
6
= 0
.
866
T
3
+
.
866
T
5
=
f
2
−
T
4
−
.
5
T
5
+
.
5
T
7
= 0
,
(9.1)
where
T
i
represents the tension in the
i
-th member of the truss.
You could solve this system by hand with a little time and patience; systematically eliminating
variables and substituting. Obviously, it would be a lot better to put the equations on a computer
and let the computer solve it.
In the next few lectures we will learn how to use a computer
effectively to solve linear systems. The first key to dealing with linear systems is to realize that they

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