class 1 - slides
DESCRIPTION
not anyTRANSCRIPT
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Engineering 112Engineering 112Engineering Methods IIEngineering Methods II
Bob LeMasterBob LeMasterCollege of EngineeringCollege of Engineering
University of Tennessee MartinUniversity of Tennessee Martin
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Course FormatCourse Format
Two two-hour instruction/laboratory sessions per week.
Session One: Introduction to Linear Algebra and Computer Programming - Matlab
Instructors: Bob LeMaster (2 Sections)Robert Benton ( 1 Section)
Session Two: Introduction to Computer Aided Design Software - AutoCAD
Instructor: Ed Wheeler (all sections)
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How will my course grade be How will my course grade be determined?determined?
A grade will be assigned for each session by the instructor.The grade for each session will be combined into an overall course grade - equally weighted.You must pass both sessions to pass the course.
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What is Linear Algebra?What is Linear Algebra?A set of mathematical rules which are used to
manipulate arrays of numbers.
The theoretical foundations were developed by mathematicians studying the properties of linear equations.
1.95.1x2.6x1.6x15.01.4x9.1x3x
11.23x4.5x2x
321
321
321
=++
=++
=++
=
1.915.011.2
xxx
5.12.61.61.49.13.03.04.52.0
3
2
1
BAx =
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What is MATLAB?What is MATLAB?
A computer programming language developed to perform numeral computations encountered in
science and engineering operations.
MATrix LABoratory
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Why do I need to know linear Why do I need to know linear algebra?algebra?
Systems of linear and nonlinear equations are encountered in every engineering discipline.
Controls
Circuits
Structures
Kinematics
Stress
Power Systems
IndustrialProcessSimulation Fluids
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Finite Element Stress AnalysisFinite Element Stress Analysis(SSME Turbopump Example)(SSME Turbopump Example)
Deflection and stress analysis of the Space Shuttle Main Engine (SSME) Alternate Turbopump Development (ATD) High Pressure Oxidizer Turbopump (HPOTP) required the simultaneous solution of approximately 750,000 nonlinear equations using methods that you will be exposed to in this class.
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Computational Fluid DynamicsComputational Fluid Dynamics(Space Shuttle Example)(Space Shuttle Example)
Calculation of temperature distributions required the simultaneous solution of thousands of nonlinear equations using techniques that you will be exposed in this course.
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Why do I need to learn to Why do I need to learn to program in MATLAB?program in MATLAB?Computers dominate the industrial workplace.
The fundamental programming concepts learned using MATLAB will enable you to adapt to other programming environments.
You will be required to use MATLAB in future courses.
MATLAB is widely used in industry.
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What will I be able to do at the What will I be able to do at the end of the course?end of the course?
Represent systems of equations using matrix notationCompute the Euclidean norm of column or row matricesPerform basic linear algebra operations symbolically, manually, and in MATLAB Solve systems of equations using Cramers ruleSolve systems of equations using MATLABFind the eigenvalues and eigenvectors of matricesDevelop computational software that employs structured programming control methodologyDisplay computed results in tabular and graphical formats
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What will be expected of me?What will be expected of me?
Attend classAttendance will be takenIt is professional to be prompt and to have good attendance
Your future employers will expect you to have acquired these work habits
Attendance record will be taken into account for borderline grade situations
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What will be expected of me?What will be expected of me?(Continued)(Continued)
Do homeworkYou will not be able to do well on exams without being proficient in solving problemsThere is no other way to become proficient at problem solving than to solve problemsKeep a notebook of all homework
Will aid in studying for examsWill aid review for EIT prior to graduationFacilitates grading
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What will be expected of me?What will be expected of me?(Continued)(Continued)
Pass four examsEach exam will be worth 15% of session final grade.
Out of class assignments will be worth 20% of session final grade.
Pass Comprehensive FinalWill be worth 20 % of session one final grade.
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Resource MaterialsResource Materials(Text)(Text)
Etter, D.M., D.C. Kuncicky, Introduction to MATLAB, Prentice-Hall.
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Resource MaterialsResource Materials(Reference)(Reference)
Getting Started html document onMathworks web site
http:/www.mathworks.com/access/helpdesk/help/helpdesk.shtml
Click on Getting Started
Click on This manual in PDF
Open or save to disk
Print
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Resource MaterialsResource Materials(Linear Algebra Reference Texts)(Linear Algebra Reference Texts)
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Linear EquationsLinear Equations(Two Dimensions)(Two Dimensions)
y
x
y=mx+b
b
m
1
bmxy +=
One independent variable (x)One dependent variable (y)y-intercept (b)slope (m)
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Linear EquationsLinear Equations(Three Dimensions)(Three Dimensions)
=
=
=
=++
nbintercept x
mb interceptx
lbintercept x
bnxmxlx
3
2
1
321
1x
2x
3x
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Matrix NotationMatrix Notation
=
1.915.011.2
xxx
5.12.61.61.49.13.03.04.52.0
3
2
1
[ ]{ } { }BxA =
[ ]
{ } { }
=
=
=
9.10.152.11
,
1.56.26.14.11.90.30.35.40.2
B xxx
x
A
3
2
1
Symbolic FormatSymbolic Format
Matrix FormatMatrix Format
1.95.1x2.6x1.6x15.01.4x9.1x3x
11.23x4.5x2x
321
321
321
=++
=++
=++
Algebraic FormatAlgebraic Format
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Matrix NotationMatrix Notation(Symbols)(Symbols)
[ ] Two dimensional array of numbers{ } Column array of numbers
Row array of numbers
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Matrix NotationMatrix Notation(Size Designation)(Size Designation)
[ ] [ ] nm,or nm, Two dimensional array of numbershaving m rows and n columns{ } { }mor m Column array of m numbers
nor n Row array containing n numbers
m is used to designate the number of rowsm is used to designate the number of rowsn is used to designate the number of columnsn is used to designate the number of columns
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Matrix NotationMatrix Notation(Text Books)(Text Books)
In many text books, it is standard practice to represent matrices with bold fonts.
[ ]{ } { }BxA =Is equivalent to
Ax=B
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Matrix NotationMatrix Notation(Element Addressing)(Element Addressing)
[ ]
=
3,23,1
2,22,1
1,21,1
3,2
AAAAAA
A
column-j and row-i theat located matrix A ofElement A ji,
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Matrix NotationMatrix Notation(Example)(Example)
[ ]
{ } { }
=
=
=
9.10.152.11
,
1.56.26.14.11.90.30.35.40.2
B xxx
x
A
3
2
1 15.0Bxx
1.6A9.1A
2
22
3,1
2,2
=
=
=
=
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Reasons for Matrix NotationReasons for Matrix Notation
1) Enables many numbers to be represented as a single object2) Represents the object by a single symbol3) Allows mathematical operations to be performed with the symbols instead of arrays of numbers
Matrix Notation is a Mathematical Shortcut
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Relationship Between Vectors Relationship Between Vectors and Matricesand Matrices
2x
1x
3x
Vectors are mathematical entities used to represent physical parameters.
Vectors have three characteristics: 1) magnitude, 2) orientation, and 3) sense.
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Relationship Between Vectors Relationship Between Vectors and Matricesand Matrices
(Components)(Components)
1x
3x
kAjAiAA321 xxx
++=
ij
k
2x
The components of a vector may be written in matrix format.
321
3
2
1
xxx
x
x
x
AAAor AAA
Not all row or column matrices are vectors.
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Matrix AdditionMatrix Addition
[ ] [ ] [ ]
ji,ji,ji, BAC
BAC
+=
+=
Note that the number of rows and columns in [A] and [B] must be equal.
n1,...,jm1,...,i
=
=
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Matrix AdditionMatrix Addition(Example)(Example)
[ ]
[ ]
=
=
1.86.65.17.64.52.24.13.05.0
B
6.83.62.14.68.51.23.11.02.0
A
[ ]
=
6.82.102.72.120.134.32.70.40.7
C
ji,ji,ji, BAC +=
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Properties of Matrix AdditionProperties of Matrix Addition
[ ] [ ] [ ] [ ][ ] [ ]( ) [ ] [ ] [ ] [ ]( )[ ] [ ] [ ][ ] [ ] 0A-A
A0ACBACBA
ABBA
=+
=+
++=++
+=+
[ ][ ] [ ]Ain valueof negative iselement each A-
zero are elements all0
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Transpose of a MatrixTranspose of a Matrix[ ] [ ]
[ ].Ain columns and rows theinginterchangby obtainedmatrix (nxm) theis
Amatrix (mxn)an of A transposeThe T
[ ]
=
826243
A [ ]
=
822463
A T
[ ]
=
241
B [ ] 241B T =
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Symmetric MatrixSymmetric Matrix
[ ] [ ]AA T =A matrix having real components is symmetric if
[ ]
=
413152321
A [ ]
=
413152321
A T
Note that symmetric matrices must be square(i.e. same number of rows as columns).
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Diagonal MatrixDiagonal Matrix
A square matrix whose elements above and below the principal diagonal are all zero is called a diagonal matrix.
[ ]
=
300050001
A
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Identity MatrixIdentity Matrix
A diagonal matrix whose elements on the principal diagonal are all 1 is called an identity matrix or unit matrix.
[ ]
=
100010001
I
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AssignmentAssignment1. Download a copy of
Getting Started from The Math Works website.
[ ] [ ]
[ ] [ ]
[ ] [ ][ ] [ ]
[ ] [ ] [ ] [ ]( )symmetric? matrices theofany Are g)
BA&BAf)
B4Ae)2A&Cd)
[D]c)[C][D]b)[C]
[A][B] & [B][A] a)Find
.351
421D ,
403210
C
,1225
B ,4032
ALet .2
TTT
TT
++
+
++
=
=
=
=
Problem 2 should be done on engineering pad paper and turned in next class.