Elementary Linear Algebra

549 Pages·6.766 MB·English

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Elementary Linear Algebra
Kenneth Kuttler
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Kenneth Kuttler
Elementary Linear Algebra
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Elementary Linear Algebra
© 2012 Kenneth Kuttler & Ventus Publishing ApS
ISBN 978-87-403-0018-5
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Elementary Linear Algebra Contents
Contents
Preface 10
1 Some Prerequisite Topics 11
1.1 Sets And Set Notation 11
1.2 Functions 13
1.3 Graphs Of Functions 14
1.4 The Complex Numbers 17
1.5 Polar Form Of Complex Numbers 20
1.6 Roots Of Complex Numbers 21
1.7 The Quadratic Formula 23
1.8 Exercises 24
2 Fn 27
2.1 Algebra in Fn 29
2.2 Geometric Meaning Of Vectors 30
2.3 Geometric Meaning Of Vector Addition 31
2.4 Distance Between Points In Rn Length Of A Vector 33
2.5 Geometric Meaning Of Scalar Multiplication 37
2.6 Exercises 38
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Elementary Linear Algebra Contents
2.7 Vectors And Physics 39
2.8 Exercises 45
3 Vector Products 47
3.1 The Dot Product 47
3.2 The Geometric Significance Of The Dot Product 49
3.3 Exercises 57
3.4 The Cross Product 59
3.5 The Vector Identity Machine 67
3.6 Exercises 69
4 Systems Of Equations 73
4.1 Systems Of Equations, Geometry 73
4.2 Systems Of Equations, Algebraic Procedures 76
4.3 Exercises 91
5 Matrices 97
5.1 Matrix Arithmetic 97
5.2 Exercises 117
6 Determinants 125
6.1 Basic Techniques And Properties 125
6.2 Applications 134
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Elementary Linear Algebra Contents
6.2.1 A Formula For The Inverse 134
6.2.2 Cramer’s Rule 138
6.3 Exercises 141
7 The Mathematical Theory Of Determinants 149
7.1 The Function sgnn 149
7.2 The Determinant 151
7.3 The Cayley Hamilton Theorem 162
8 Rank Of A Matrix 165
8.1 Elementary Matrices 165
8.2 THE Row Reduced Echelon Form Of A Matrix 173
8.3 The Rank Of A Matrix 178
8.5 Fredholm Alternative 197
8.6 Exercises 202
9 Linear Transformations 211
9.1 Linear Transformations 211
9.2 Constructing The Matrix Of A Linear Transformation 214
9.3 Exercises 225
10 The LU Factorization 231
10.1 Definition Of An LU factorization 231
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Elementary Linear Algebra Contents
10.2 Finding An LU Factorization By Inspection 232
10.3 Using Multipliers To Find An LU Factorization 233
10.4 Solving Systems Using The LU Factorization 234
10.5 Justification For The Multiplier Method 236
10.6 The PLU Factorization 238
10.7 The QR Factorization 240
10.8 Exercises 244
11 Linear Programming 249
11.1 Simple Geometric Considerations 249
11.2 The Simplex Tableau 250
11.3 The Simplex Algorithm 256
11.4 Finding A Basic Feasible Solution 268
11.5 Duality 270
11.6 Exercises 276
12 Spectral Theory 277
12.1 Eigenvalues And Eigenvectors Of A Matrix 277
12.2 Some Applications Of Eigenvalues And Eigenvectors 300
12.3 The Estimation Of Eigenvalues 307
12.4 Exercises 308
13 Matrices And The Inner Product 317
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Elementary Linear Algebra Contents
13.1 Symmetric And Orthogonal Matrices 317
13.2 Fundamental Theory And Generalizations 331
13.3 Least Square Approximation 341
13.4 The Right Polar Factorization 346
13.5 The Singular Value Decomposition 350
13.6 Approximation In The Frobenius Norm 351
13.7 Moore Penrose Inverse 354
13.8 Exercises 354
14 Numerical Methods For Solving Linear Systems 367
14.1 Iterative Methods For Linear Systems 367
14.2 The Operator Norm 377
14.3 The Condition Number 379
14.4 Exercises 381
15 Numerical Methods For Solving The Eigenvalue Problem 385
15.1 The Power Method For Eigenvalues 385
15.2 The Shifted Inverse Power Method 388
15.3 The Rayleigh Quotient 399
15.4 The QR Algorithm 403
15.5 Exercises 414
16 Vector Spaces 419
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Elementary Linear Algebra Contents
16.1 Algebraic Considerations 419
16.2 Exercises 420
16.3 Vector Spaces And Fields 428
16.4 Exercises 444
16.5 Inner Product Spaces 451
16.6 Exercises 468
17 Linear Transformations 475
17.1 Matrix Multiplication As A Linear Transformation 475
17.2 L(V,W) As A Vector Space 476
17.3 Eigenvalues And Eigenvectors Of Linear Transformations 478
17.4 Block Diagonal Matrices 483
17.5 The Matrix Of A Linear Transformation 489
17.6 Exercises 500
A The Jordan Canonical Form* 505
B The Fundamental Theorem Of Algebra 515
C Answers To Selected Exercises 519
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Elementary Linear Algebra Preface
Preface
This is an introduction to linear algebra. The main part of the book features row operations and
everythingisdoneintermsoftherowreducedechelonformandspecificalgorithms. Attheend,the
more abstract notions of vector spaces and linear transformations on vector spaces are presented.
However, this is intended to be a first course in linear algebra for students who are sophomores
or juniors who have had a course in one variable calculus and a reasonable background in college
algebra. I have given complete proofs of all the fundamental ideas, but some topics such as Markov
matrices are not complete in this book but receive a plausible introduction. The book contains a
complete treatment of determinants and a simple proof of the Cayley Hamilton theorem although
these are optional topics. The Jordan form is presented as an appendix. I see this theorem as the
beginningofmoreadvancedtopicsinlinearalgebraandnotreallypartofabeginninglinearalgebra
course. There are extensions of many of the topics of this book in my on line book [11]. I have also
not emphasized that linear algebra can be carried out with any field although there is an optional
section on this topic, most of the book being devoted to either the real numbers or the complex
numbers. It seems to me this is a reasonable specialization for a first course in linear algebra.
Linear algebra is a wonderful interesting subject. It is a shame when it degenerates into nothing
more than a challenge to do the arithmetic correctly. It seems to me that the use of a computer
algebra system can be a great help in avoiding this sort of tedium. I don’t want to over emphasize
the use of technology, which is easy to do if you are not careful, but there are certain standard
things which are best done by the computer. Some of these include the row reduced echelon form,
PLU factorization, and QR factorization. It is much more fun to let the machine do the tedious
calculationsthantosufferwiththemyourself. However,itisnotgoodwhentheuseofthecomputer
algebra system degenerates into simply asking it for the answer without understanding what the
oracular software is doing. With this in mind, there are a few interactive links which explain how
to use a computer algebra system to accomplish some of these more tedious standard tasks. These
are obtained by clicking on the symbol (cid:31). I have included how to do it using maple and scientific
notebook because these are the two systems I am familiar with and have on my computer. Other
systemscouldbefeaturedaswell. Itisexpectedthatpeoplewillusesuchcomputeralgebrasystems
to do the exercises in this book whenever it would be helpful to do so, rather than wasting huge
amounts of time doing computations by hand. However, this is not a book on numerical analysis so
no effort is made to consider many important numerical analysis issues.
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