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Week 3
Week 3
Linear Algebraic Equations
Week 3
Outline
• What are linear algebraic equations
• Matrix algebra overview
• Solving linear algebraic equations
 graphics methods
 direct methods:
Gauss elimination
LU decomposition
Thomas methods
Week 3
Linear Algebraic Equations
• Background
 Singlecomponent systems result in a single
equation that can be solved using rootlocation
techniques as discussed in week 2
 Multicomponent systems result in a coupled set
of mathematical equations that must be solved
simultaneously. The equations are coupled
because the individual parts of the system are
influenced by other parts
Week 3
Engineering Systems
• Mechanical (Springmass system)
⎥
⎥
⎥
⎥
⎦
⎤
⎢
⎢
⎢
⎢
⎣
⎡
=
⎥
⎥
⎥
⎥
⎦
⎤
⎢
⎢
⎢
⎢
⎣
⎡
⎥
⎥
⎥
⎥
⎦
⎤
⎢
⎢
⎢
⎢
⎣
⎡
−
−
+
−
−
+
−
−
+
F
x
x
x
x
k
k
k
k
k
k
k
k
k
k
k
k
k
0
0
0
0
0
0
0
0
0
4
3
2
1
4
4
4
4
3
3
3
3
2
2
2
2
1
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Week 3
Engineering Systems
• Electrical (circuit)
Week 3
Engineering Systems
• Civil/Environmental
Week 3
Linear Algebraic Equations
• General form
m
m
nm
n
n
m
m
m
m
b
x
a
x
a
x
a
b
x
a
x
a
x
a
b
x
a
x
a
x
a
=
+
+
+
=
+
+
+
=
+
+
+
L
M
M
M
M
M
L
L
2
2
1
1
2
2
2
22
1
21
1
1
2
12
1
11
the
a
’s are constant coefficients, the
b
’s are constants,
the
x
’s are unknows, and
n
is the number of equations
n
,
,
,
i
b
x
a
m
j
i
j
ij
L
2
1
1
=
∑
=
=
Week 3
Linear Algebraic Equations
• Matrix notation
matrix
A
, vector
x
and vector
b
n = m
, square matrix
A
⎥
⎥
⎥
⎥
⎥
⎦
⎤
⎢
⎢
⎢
⎢
⎢
⎣
⎡
=
⎟
⎟
⎟
⎟
⎟
⎠
⎞
⎜
⎜
⎜
⎜
⎜
⎝
⎛
⎥
⎥
⎥
⎥
⎥
⎦
⎤
⎢
⎢
⎢
⎢
⎢
⎣
⎡
m
m
nm
n
n
m
m
b
b
b
x
x
x
a
a
a
a
a
a
a
a
a
M
M
L
M
O
M
M
L
L
2
1
2
1
2
1
2
22
21
1
12
11
A
n
×
m
x
b
3
Week 3
Linear Algebraic Equations
• Simple example: 2
×
2 System of Linear
equations (SL)
4
3
10
2
4
2
1
2
1
=
+
−
=
+
x
x
x
x
⎥
⎦
⎤
⎢
⎣
⎡
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
⎥
⎦
⎤
⎢
⎣
⎡
=
−
4
10
3
1
2
4
2
1
x
x
notation
matrix
The solution to the above SL is x
1
= 1.57142854 and x
2
= 1.85714281
or the vector
x
,
⎟
⎠
⎞
⎜
⎝
⎛
=
85714281
1
57142854
1
.
.
x
Week 3
Matrix Algebra
• Square matrix
⎥
⎥
⎥
⎥
⎦
⎤
⎢
⎢
⎢
⎢
⎣
⎡
=
nn
n
n
n
a
a
a
an
a
a
a
a
a
A
L
M
O
M
M
L
L
2
1
22
21
1
12
11
• Diagonal matrix
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
=
≠
≠
=
⎥
⎥
⎥
⎥
⎦
⎤
⎢
⎢
⎢
⎢
⎣
⎡
=
j
i
for
a
j
i
for
a
ij
ij
nxn
nn
a
a
a
D
0
0
0
0
0
0
0
0
22
11
L
M
O
M
M
L
L
4
x
4
5
.
0
0
0
0
0
1
0
0
0
0
1
.
3
0
0
0
0
5
.
2
⎥
⎥
⎥
⎥
⎥
⎥
⎦
⎤
⎢
⎢
⎢
⎢
⎢
⎢
⎣
⎡
Week 3
An important property of identity matrix is that any matrix multiplied by an
identity matrix will yield the same matrix.
I A
=
A
Matrix Algebra
• Identity matrix
• Triangular matrix
4
x
4
1
0
0
0
0
1
0
0
0
0
1
0
0
0
0
1
⎥
⎥
⎥
⎥
⎥
⎥
⎦
⎤
⎢
⎢
⎢
⎢
⎢
⎢
⎣
⎡
4
4
4
0
0
0
4
2
1
0
0
0
7
2
5
1
0
1
5
4
2
3
x
U
⎥
⎥
⎥
⎥
⎦
⎤
⎢
⎢
⎢
⎢
⎣
⎡
=
.
.
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This note was uploaded on 10/27/2009 for the course MECH 201 taught by Professor Weihuali during the Three '09 term at University of Wollongong, Australia.
 Three '09
 WeihuaLi

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