Engr/Math/Physics 25
Chp8 Linear
Algebraic Eqns-1
Bruce Mayer, PE
Registered Electrical & Mechanical Engineer
BMayer@ChabotCollege.edu
Engineering/Math/Physics 25: Computational Methods
1
Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-25_Linear_Equations-1.ppt
Learning Goals
 Define Linear Algebraic Equations
 Solve Systems of Linear Equations
by Hand using
• Gaussian Elimination (Elem. Row Ops)
• Cramer’s Method
 Distinguish between Equation System
Conditions: Exactly Determined,
OverDetermined, UnderDetermined
 Use MATLAB to Solve Systems of Eqns
Engineering/Math/Physics 25: Computational Methods
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-25_Linear_Equations-1.ppt
Linear Equations  Example
 In Many Engineering Analyses
(e.g. ENGR36 & ENGR43) The
Engineer Must Solve Several
Equations in Several Unknowns; e.g.:
6x

3y

4z

41
12 x

5y

7z

 26
 5x

2y

6z

14
 Contains 3 Unknowns (x,y,z)
in the 3 Equations
Engineering/Math/Physics 25: Computational Methods
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-25_Linear_Equations-1.ppt
1
2
3
Linear Systems - Characteristics
 Examine the System
of Equations
6 x  3 y  4 z  41
12 x  5 y  7 z  26
 5 x  2 y  6 z  14
 We notice These
Characteristics that
DEFINE Linear
Systems
Engineering/Math/Physics 25: Computational Methods
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 ALL the Variables
are Raised
EXACTLY to the
Power of ONE (1)
 COEFFICIENTS of
the Variables are all
REAL Numbers
 The Eqns Contain
No Transcendental
Functions (e.g. ln,
cos, ew)
Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-25_Linear_Equations-1.ppt
Gaussian Elimination – ERO’s
 A “Well Conditioned” System of Eqns
can be Solved by Elementary Row
Operations (ERO):
• Interchanges: The vertical position of
two rows can be changed
• Scaling: Multiplying a row by a
nonzero constant
• Replacement: The row can be replaced by
the sum of that row and a
nonzero multiple of any other row
Engineering/Math/Physics 25: Computational Methods
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-25_Linear_Equations-1.ppt
ERO Example - 1
 Let’s Solve The
System of Eqns
1
2
3
6 x  3 y  4 z  41
12 x  5 y  7 z  26
 5 x  2 y  6 z  14
 INTERCHANGE, or
Swap, positions of
Eqns (1) & (2)
Engineering/Math/Physics 25: Computational Methods
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1
2
3
12 x  5 y  7 z  26
6 x  3 y  4 z  41
 5 x  2 y  6 z  14
 Next SCALE by
using Eqn (1) as the
PIVOT To Multiply
• (2) by 12/6
• (3) by 12/[−5]
Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-25_Linear_Equations-1.ppt
ERO Example - 2
 The Scaling
Operation
1
12 x  5 y  7 z  26
12
2
6 x  3 y  4 z  41
6
12
3
 5 x  2 y  6 z  14
5
1
2
3
12 x  5 y  7 z  26
12 x  6 y  8 z  82
12 x  4.8 y  14.4 z  33.6
Engineering/Math/Physics 25: Computational Methods
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 Note that the 1st
Coeffiecient in the
Pivot Eqn is Called
the Pivot Value
• The Pivot is used to
SCALE the Eqns
Below it
 Next Apply
REPLACEMENT by
Subtracting Eqs
• (2) – (1)
• (3) – (1)
Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-25_Linear_Equations-1.ppt
ERO Example - 3
 The Replacement
Operation Yields
1
2
3
12 x  5 y  7 z  26
0 x  11y  15 z  108
0 x  9.8 y  7.4 z  7.6
Or
1
2
3
12 x  5 y  7 z  26
 11y  15 z  108
 9.8 y  7.4 z  7.6
Engineering/Math/Physics 25: Computational Methods
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 Note that the
x-variable has been
ELIMINATED below
the Pivot Row
• Next Eliminate in
the “y” Column
 We can use for the
y-Pivot either of −11
or −9.8
• For the best numerical
accuracy choose the
LARGEST pivot
Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-25_Linear_Equations-1.ppt
ERO Example - 4
 Our Reduced Sys
1
2
3
12 x  5 y  7 z  26
 11y  15 z  108
 9.8 y  7.4 z  7.6
 Since |−11| > |−9.8|
we do NOT need to
interchange (2)↔(3)
 Scale by Pivot
against Eqn-(3)
Engineering/Math/Physics 25: Computational Methods
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1
2
12 x  5 y  7 z  26
 11y  15 z  108
 11
3
 9.8 y  7.4 z  7.6
 9.8
Or
1
2
3
12 x  5 y  7 z
 26
 11y  15 z
 108
 11y  8.306 z  8.531
Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-25_Linear_Equations-1.ppt
ERO Example - 5
 Perform
Replacement by
Subtracting (3) – (2)
1
2
3
12 x  5 y  7 z
 26
 11y  15 z
 108
 23.306 z  116.531
 Now Easily Find
the Value of z from
Eqn (3)
z  116.531 23.306  5
Engineering/Math/Physics 25: Computational Methods
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 The Hard Part is
DONE
 Find y & x by BACK
SUBSTITUTION
 From Eqn (2)
108  15 z 108  75
y

 11
 11
y  33  11  3
Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-25_Linear_Equations-1.ppt
ERO Example - 6
 BackSub into (1)
1
2
3
6 x  3 y  4 z  41
12 x  5 y  7 z  26
 5 x  2 y  6 z  14
12 x  5 y  7 z  26
7 z  5 y  26
x
12
35  15  26 24
x

2
12
12
 x=2
 Thus the Solution
Set for Our Linear
System
Engineering/Math/Physics 25: Computational Methods
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 y = −3
 z=5
Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-25_Linear_Equations-1.ppt
Importance of Pivoting
 Computers use finite-precision arithmetic
 A small error is introduced in each arithmetic
operation, AND… error propagates
 When the pivot element is very small, then the
multipliers will be even smaller
 Adding numbers of widely differing magnitude
can lead to a loss of significance.
 To reduce error, row interchanges are
made to maximize the magnitude of
the pivot element
Engineering/Math/Physics 25: Computational Methods
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-25_Linear_Equations-1.ppt
Gaussian Elimination Summary
 INTERCHANGE Eqns Such that
the PIVOT Value has the
Greatest Magnitude
 SCALE the Eqns below the Pivot Eqn
using the Pivot Value ratio’ed against
the Corresponding Value below
 REPLACE Eqns Below the Pivot by
Subtraction to leave ZERO Coefficients
Below the Pivot Value
Engineering/Math/Physics 25: Computational Methods
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-25_Linear_Equations-1.ppt
Poorly Conditioned Systems
 For Certain Systems Guassian
Elimination Can Fail by
• NO Solution → Singular System
• Numerically Inaccurate Results →
ILL-Conditioned System
 In a SINGULAR SYSTEM Two or More
Eqns are Scalar Multiples of each other
 In ILL-Conditioned Systems 2+ Eqns are
NEARLY Scalar Multiples of each other
Engineering/Math/Physics 25: Computational Methods
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-25_Linear_Equations-1.ppt
A Singular (Inconsistent) Sys
 Consider 2-Eqns in
2-Unknowns
1
2
x  2y  4
2x  4 y  5
 Perform Elimination
by
• Swapping Eqns
• Mult (2) by 2/1
• Subtract (2) – (1)
Engineering/Math/Physics 25: Computational Methods
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1
2
1
2
1
2
2x  4 y  5
0x  0 y  3
2
0  3 ????
2x  4 y  5
x  2y  4
2x  4 y  5
2x  4 y  8
Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-25_Linear_Equations-1.ppt
Singular System - Geometry
 Plot This System on
the XY Plane
x  2y  4
2x  4 y  5
 The Lines do NOT
CROSS to Define a
A Solution Point
y
1
2
 Singular Systems
Have at least Two
“PARALLEL” Eqns
Engineering/Math/Physics 25: Computational Methods
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-25_Linear_Equations-1.ppt
ILL-Conditioned Systems
 A small deviation in one or more of the
CoEfficients causes a LARGE
DEVİATİON in the SOLUTİON.
1x

2y

3
x 1

0.48 x  0.99 y  1.47
y 1
1x

2y

3
x3

0.49 x  0.99 y  1.47
y0
Engineering/Math/Physics 25: Computational Methods
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-25_Linear_Equations-1.ppt
ILL-Conditioned Systems - 2
 Systems in Which a
Small Change in a
CoEfficient Produces
Large Changes in
the Solution are
said to be STIFF
Tilt Region
• Essentially the Lines
Have very nearly
Equal SLOPES
• “Tilting” The Equations just a bit Dramatically Shifts
the Solution (Crossing Point)
Engineering/Math/Physics 25: Computational Methods
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-25_Linear_Equations-1.ppt
Matrix Methods for LinSys - 1
 Consider the
Electrical Ckt
Shown at Right
 The Operation of
this Ckt May be
Described in
Terms of the
• Mesh Currents, I1-I4
• Sources: 4 mA, 12 V
• Resistors: 1 & 2 kΩ
Engineering/Math/Physics 25: Computational Methods
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 Notice Mesh
Currents I1 & I2 are
Defined by
SOURCES
Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-25_Linear_Equations-1.ppt
Matrix Methods for LinSys - 3
 Using Techniques
from ENGR43 find
 0
 I2
 I2
 I2
I1
I1
0
0
 0
 I3
 3I 3
 I3
 0
 0
 2I 4
 2I 4
 Recall Matrix
Multiplication to
Write the
Equation system
in Matrix Form
Engineering/Math/Physics 25: Computational Methods
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
4mA

0
 8mA
  12mA
0   I1   4 
1 0 0
1 1  1 0   I   0 

 2   

0 1 3  2   I 3   8 

  

0  1  1 2   I 4   12
A
x b
Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-25_Linear_Equations-1.ppt
Matrix Methods for LinSys - 3
 Thus The (linear)
Ckt Can be
Described by
Ax  b
 Where
• A  Coefficient
Matrix
– m-Rows x n-Colunms
• b  Constraint Vector
• x  Solution Vector
Engineering/Math/Physics 25: Computational Methods
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 This Can Be Written
in Std Math Notation
 a11
 

A   ai1

 
am1
 x1 

 
x   xi 
 

 xm 
 a1i
 
 aii

 ani
 a1n 
 
 ain 

  
 amn 
 b1 

 
b   bi 
 

bm 
Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-25_Linear_Equations-1.ppt
Determinants - 1
 If we Solve a LinSys
by Elimination we
may do a Lot of
work Before
Discovering that the
system is Singular
or Very-Stiff
 Determinants Can
Alert us ahead of
time to these
Difficulties
Engineering/Math/Physics 25: Computational Methods
22
 Determinants are
Defined only for
SQUARE Arrays
 The 2x2 Definition
a11 a12
D2 
a11a 22 a 21a12
a 21 a 22
 D2 is Sometimes
called the
“Basic Minor”
Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-25_Linear_Equations-1.ppt
Determinants - 2
 Calculating Larger-Dimension DETs
becomes very-Tedious very-Quickly
• Consider a 3x3 Det
a11
a12
D3  a21 a22
a31 a32
a13
a23   a11 detMinor 1   a12 detMinor 2   a13 detMinor 3 
a33
D3  a11 a22a33  a32a23   a12 a21a33  a31a23   a13 a21a32  a31a22 
• Example
4
9
6
  465   22  935  6  677   39
D3ex  7 13  2
 348  261  696  87
 3 11 5
Engineering/Math/Physics 25: Computational Methods
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-25_Linear_Equations-1.ppt
Determinants - 3
 A Determinant, no matter what its size,
Returns a SINGLE Value
 Matrix vs. Determinant
• For Square Matrix A the Notation
A  det A  A
 MATLAB vs det
• The det Calc is quite
Painful, but
MATLAB’s “det” Fcn
Makes it Easy
Engineering/Math/Physics 25: Computational Methods
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 For the D3ex
>> A = [-4,9,6; 7,13,-2;
-3,11,5];
>> D3ex = det(A)
D3ex =
87
Bruce Mayer, PE

BMayer@ChabotCollege.edu • ENGR-25_Linear_Equations-1.ppt
Determinant Indicator - 1
 The LARGER the Magnitude of the
Determinant relative to the Coefficients,
The LESS-Stiff the System
 If det=0, then the System is SINGULAR
1
2
x  2y  4
 det  0  SINGULAR
2x  4 y  5
det  0.03
1x

2y

3

STIFF
0.48 x  0.99 y  1.47
1x
0.49 x


2y
0.99 y
Engineering/Math/Physics 25: Computational Methods
25


3

1.47
det  0.01
STIFF
Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-25_Linear_Equations-1.ppt
Determinant Indicator - 2
 Consider this 5 x
System
7x


2y
2y
5
 Check the “Stiffness” D 
7


13
23
2
 24
2
 Thus The system appears NON-Stiff
 Find Solution by Elimination as
x3
Engineering/Math/Physics 25: Computational Methods
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y 1
Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-25_Linear_Equations-1.ppt
MATLAB Left Division
 MATLAB has a very  The Syntax is Quite
nice Utility for
Simple
Solving Well• the hassle is
Conditioned Linear
entering the Matrix-A
and Vector-b
Systems of the Form
Ax  b
 Well Conditioned →
 x = A\b
• Square System →
No. of Eqns &
Unknwns are Equal
• det  0
Engineering/Math/Physics 25: Computational Methods
27
Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-25_Linear_Equations-1.ppt
Left-Div Example - 1
 Consider a 750 kg
Crate suspended by
3 Ropes or Cables
In MATRIX form
 Using Force
 
AT  w
Mechanics from
ENGR36 Find 3 Eqns
in 3 Unknowns
 0.48TAB  0TAC  0.5195TAD  0
0.8TAB  0.8824TAC  0.7792TAD  750 * 9.81
 0.36TAB  0.4706TAC  0.3506TAD  0
Engineering/Math/Physics 25: Computational Methods
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-25_Linear_Equations-1.ppt
Left-Div Example - 2
 The MATLAB
Command Window
Session
>> A = [-0.48, 0,
0.5195;...
0.8, 0.8824, 0.7792;...
-0.36, 0.4706, -.3506];
>> w = [0; 9.81*750; 0]
>> T = A\w
T =
1.0e+003 *
2.6254
3.8157
2.4258
Engineering/Math/Physics 25: Computational Methods
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 Or
• TAB = 2.625 kN
• TAC = 3.816 kN
• TAD = 2.426 kN
Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-25_Linear_Equations-1.ppt
Matrix Inverse - 1
 Recall The Matrix
Formulation for nEqns in n-Unknowns
Ax  b
 In Matrix Land
xb A
 To Isolate x, employ
the Matrix Inverse
A-1 as Defined by
Engineering/Math/Physics 25: Computational Methods
30
1
A AI
 Note that the
IDENTITY Matrix , I,
Has Property
Ix  x
 Use A-1 in Matrix Eqn
1
1
A Αx  A b
1
or Ix  A b
1
or x  A b
Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-25_Linear_Equations-1.ppt
Matrix Inverse - 2
 Thus the Matrix
Shorthand for the
Solution
Ax  b 
1
xA b
 Determining the
Inverse is NOT
Trivial (Ask your
MTH6 Instructor)
Engineering/Math/Physics 25: Computational Methods
31
 In addition A-1 is, in
general, Less
Numerically
Accurate Than
Pivoted Elimination
 However
1
xA b
is Symbolically
Elegant and
Will be Useful
in that regard
Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-25_Linear_Equations-1.ppt
Compare MatInv & LeftDiv
% Bruce Mayer, PE
% ENGR25 * 21Oct09
% file = Compare_MatInv_LeftDiv_0910
%
A = [3 -7 8; 7 6 -5; -9 0 2]
b = [13; -29; 37]
Ainv = inv(A)
xinv = Ainv*b
xleft = A\b
%
% CHECK Both by b = A*x
CHKinv = A*xinv
CHKleft = A*xleft
Engineering/Math/Physics 25: Computational Methods
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-25_Linear_Equations-1.ppt
All Done for Today
Matrix
Inversion
by Adjoint
Engineering/Math/Physics 25: Computational Methods
33
 Given A, Find A-1
Adj ( A)
A 
|A|
1
The “Adjoint” of a matrix
is the transpose of the
matrix made up of the
“CoFactors” of the
original matrix.
Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-25_Linear_Equations-1.ppt
Engr/Math/Physics 25
Appendix
f x   2 x  7 x  9 x  6
3
2
Bruce Mayer, PE
Licensed Electrical & Mechanical Engineer
BMayer@ChabotCollege.edu
Engineering/Math/Physics 25: Computational Methods
34
Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-25_Linear_Equations-1.ppt