applied linear algebra and differential machas/applied-linear-algebra-and... applied linear algebra

Download Applied Linear Algebra and Differential machas/applied-linear-algebra-and... Applied Linear Algebra

Post on 22-Feb-2020

22 views

Category:

Documents

13 download

Embed Size (px)

TRANSCRIPT

  • Applied Linear Algebra and Differential Equations

    Lecture notes for MATH 2350

    Jeffrey R. Chasnov

  • The Hong Kong University of Science and Technology Department of Mathematics Clear Water Bay, Kowloon

    Hong Kong

    Copyright c○ 2017-2019 by Jeffrey Robert Chasnov

    This work is licensed under the Creative Commons Attribution 3.0 Hong Kong License. To view a copy of this license, visit http://creativecommons.org/licenses/by/3.0/hk/ or send a letter to Creative Commons, 171 Second Street, Suite 300, San Francisco, California, 94105, USA.

  • Preface What follows are my lecture notes for a mathematics course offered to second-year engineering students at the the Hong Kong University of Science and Technology. Material from our usual courses on linear algebra and differential equations have been combined into a single course (essentially, two half-semester courses) at the request of our Engineering School. I have tried my best to select the most essential and interesting topics from both courses, and to show how knowledge of linear algebra can improve students’ understanding of differential equations.

    All web surfers are welcome to download these notes and to use the notes and videos freely for teaching and learning.

    I also have some online courses on Coursera. You can click on the links below to explore these courses.

    If you want to learn differential equations, have a look at

    Differential Equations for Engineers

    If your interests are matrices and elementary linear algebra, try

    Matrix Algebra for Engineers

    If you want to learn vector calculus (also known as multivariable calculus, or calcu- lus three), you can sign up for

    Vector Calculus for Engineers

    And if you simply want to enjoy mathematics, my very first online course is still available:

    Fibonacci Numbers and the Golden Ratio

    Jeffrey R. Chasnov Hong Kong

    January 2020

    iii

    https://www.coursera.org/learn/differential-equations-engineers/ https://www.coursera.org/learn/matrix-algebra-engineers https://www.coursera.org/learn/vector-calculus-engineers https://www.coursera.org/learn/fibonacci

  • Contents 0 A short mathematical review 1

    0.1 The trigonometric functions . . . . . . . . . . . . . . . . . . . . . . . . . 1 0.2 The exponential function and the natural logarithm . . . . . . . . . . . 1 0.3 Definition of the derivative . . . . . . . . . . . . . . . . . . . . . . . . . 2 0.4 Differentiating a combination of functions . . . . . . . . . . . . . . . . 2

    0.4.1 The sum or difference rule . . . . . . . . . . . . . . . . . . . . . 2 0.4.2 The product rule . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 0.4.3 The quotient rule . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 0.4.4 The chain rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

    0.5 Differentiating elementary functions . . . . . . . . . . . . . . . . . . . . 3 0.5.1 The power rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 0.5.2 Trigonometric functions . . . . . . . . . . . . . . . . . . . . . . . 3 0.5.3 Exponential and natural logarithm functions . . . . . . . . . . . 3

    0.6 Definition of the integral . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 0.7 The fundamental theorem of calculus . . . . . . . . . . . . . . . . . . . 4 0.8 Definite and indefinite integrals . . . . . . . . . . . . . . . . . . . . . . . 5 0.9 Indefinite integrals of elementary functions . . . . . . . . . . . . . . . . 5 0.10 Substitution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 0.11 Integration by parts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 0.12 Taylor series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 0.13 Functions of several variables . . . . . . . . . . . . . . . . . . . . . . . . 7 0.14 Complex numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

    I Linear algebra 13

    1 Matrices 17 1.1 Definition of a matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 1.2 Addition and multiplication of matrices . . . . . . . . . . . . . . . . . . 17 1.3 The identity matrix and the zero matrix . . . . . . . . . . . . . . . . . . 19 1.4 General notation, transposes, and inverses . . . . . . . . . . . . . . . . 19 1.5 Rotation matrices and orthogonal matrices . . . . . . . . . . . . . . . . 23 1.6 Matrix representation of complex numbers . . . . . . . . . . . . . . . . 24 1.7 Permutation matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 1.8 Projection matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

    2 Systems of linear equations 27 2.1 Gaussian Elimination . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 2.2 When there is no unique solution . . . . . . . . . . . . . . . . . . . . . . 28 2.3 Reduced row echelon form . . . . . . . . . . . . . . . . . . . . . . . . . 29 2.4 Computing inverses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 2.5 LU decomposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

    v

  • CONTENTS

    3 Vector spaces 35 3.1 Vector spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 3.2 Linear independence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 3.3 Span, basis and dimension . . . . . . . . . . . . . . . . . . . . . . . . . . 37 3.4 Inner product spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 3.5 Vector spaces of a matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

    3.5.1 Null space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 3.5.2 Application of the null space . . . . . . . . . . . . . . . . . . . . 41 3.5.3 Column space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 3.5.4 Row space, left null space and rank . . . . . . . . . . . . . . . . 43

    3.6 Gram-Schmidt process . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 3.7 Orthogonal projections . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 3.8 QR factorization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 3.9 The least-squares problem . . . . . . . . . . . . . . . . . . . . . . . . . . 47 3.10 Solution of the least-squares problem . . . . . . . . . . . . . . . . . . . 49

    4 Determinants 51 4.1 Two-by-two and three-by-three determinants . . . . . . . . . . . . . . . 51 4.2 Laplace expansion and Leibniz formula . . . . . . . . . . . . . . . . . . 52 4.3 Properties of the determinant . . . . . . . . . . . . . . . . . . . . . . . . 53 4.4 Use of determinants in Vector Calculus . . . . . . . . . . . . . . . . . . 58

    5 Eigenvalues and eigenvectors 63 5.1 The eigenvalue problem . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 5.2 Matrix diagonalization . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66 5.3 Symmetric and Hermitian matrices . . . . . . . . . . . . . . . . . . . . . 68

    II Differential equations 69

    6 Introduction to odes 73 6.1 The simplest type of differential equation . . . . . . . . . . . . . . . . . 73

    7 First-order odes 75 7.1 The Euler method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 7.2 Separable equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76 7.3 Linear equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 7.4 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

    7.4.1 Compound interest . . . . . . . . . . . . . . . . . . . . . . . . . . 82 7.4.2 Chemical reactions . . . . . . . . . . . . . . . . . . . . . . . . . . 83 7.4.3 Terminal velocity . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 7.4.4 Escape velocity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86 7.4.5 RC circuit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87 7.4.6 The logistic equation . . . . . . . . . . . . . . . . . . . . . . . . . 89

    8 Second-order odes, constant coefficients 91 8.1 The Euler method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 8.2 The principle of superposition . . . . . . . . . . . . . . . . . . . . . . . 92 8.3 The Wronskian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92 8.4 Homogeneous odes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

    8.4.1 Distinct real roots . . . . . . . . . . . . . . . . . . . . . . . . . . . 94 8.4.2 Distinct complex-conjugate roots . . . . . . . . . . . . . . . . . . 96

    vi CONTENTS

  • CONTENTS

    8.4.3 Degenerate roots . . . . . . . . . . . . . . . . . . . . . . . . . . . 98 8.5 Difference equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99 8.6 Inhomogeneous odes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100 8.7 Resonance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104 8.8 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107

    8.8.1 RLC circuit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107 8.8.2 Mass on a spring . . . . . . . . . . . . . . . . . . . . . . . . . . . 109 8.8.3 Pendulum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110

    8.9 Damped resonance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111

    9 Series solutions 113 9.1 Ordinary points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113

    10 Systems of linear differential equations 119 10.1 Distinct real eigenvalues . . . . . . . . . . . . . . . . . . . . . . . . . . . 119 10.2 Solution by diagonalization . . . . . . . . . . . . . . . . . . . . . . . . . 121 10.3 Solution by the matrix

Recommended

View more >