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notes152-ch10

notes152-ch10 - MATH 152 Spring 2004-05 Applied Linear...

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MATH 152 Spring 2004-05 Applied Linear Algebra & Differential Equations Lecture Notes Dr. Tony Yee Department of Mathematics The Hong Kong University of Science and Technology April 28, 2005

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Contents Table of Contents iii 1 Introduction 3 1.1 What are Differential Equations? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.2 Classification of Differential Equations . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.3 Solutions of Ordinary Differential Equations . . . . . . . . . . . . . . . . . . . . . . . 5 2 First-Order Differential Equations 11 2.1 First-Order Linear Differential Equations . . . . . . . . . . . . . . . . . . . . . . . . 11 2.2 Separable Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 2.3 Exact Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 2.4 The Existence and Uniqueness Theorem . . . . . . . . . . . . . . . . . . . . . . . . . 25 2.5 Modeling with First-Order Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 2.6 Euler and Runge–Kutta Numerical Methods . . . . . . . . . . . . . . . . . . . . . . . 35 3 Second-Order Linear Equations 43 3.1 Homogeneous Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 3.1.1 Homogeneous Equations with Real Constant Coefficients . . . . . . . . . . . 46 3.1.2 Method of Reduction of Order . . . . . . . . . . . . . . . . . . . . . . . . . . 52 3.2 Nonhomogeneous Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 3.2.1 Method of Undetermined Coefficients . . . . . . . . . . . . . . . . . . . . . . 55 3.2.2 Method of Variation of Parameters . . . . . . . . . . . . . . . . . . . . . . . . 60 3.3 Mechanical Vibrations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 4 Laplace Transform 81 4.1 Introduction – Examples and Properties . . . . . . . . . . . . . . . . . . . . . . . . . 81 4.2 Initial Value Problems and Inverse Laplace Transform . . . . . . . . . . . . . . . . . 86 4.3 Discontinuous Forcing Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 4.4 Impulse Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96 4.5 Convolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99 5 Matrix 103 5.1 Introduction and Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 5.2 Addition and Scalar Multiplication . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 5.3 Multiplication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106 5.4 Inverse . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110 5.5 Transpose . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112 5.6 Partitioned Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114 iii

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CONTENTS 6 Systems of Linear Equations 117 6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117 6.2 Solve Systems of Linear Algebraic Equations . . . . . . . . . . . . . . . . . . . . . . 119 6.2.1 Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119 6.2.2 Gaussian Elimination . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120 6.3 Theory of Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125 6.3.1 Row Echelon Form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125 6.3.2 Reduced Row Echelon Form . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126 6.3.3 Existence and Uniqueness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128 6.4 Rank . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133 6.5 Find Inverse of Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135 7 Euclidean Vector 139 7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139 7.1.1 Euclidean Vector and Euclidean Space . . . . . . . . . . . . . . . . . . . . . . 139 7.1.2 Addition and Scalar Multiplication . . . . . . . . . . . . . . . . . . . . . . . . 140 7.1.3 Column and Row Vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141 7.2 General Solution in Vector Form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142 7.3 Span . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147 7.3.1 Span of One or Two Vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . 148 7.3.2 Determine Span . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149 7.4 Linear Independence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153 7.4.1 Independence of One or Two Vectors . . . . . . . . . . . . . . . . . . . . . . . 153 7.4.2 Linear Dependence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154 7.4.3 Determine Independence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155 7.5 Basis of R n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158 8 Eigenvalue and Eigenvector 163 8.1 Diagonalization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163 8.2 Eigenvalue, Eigenvector, Eigenspace . . . . . . . . . . . . . . . . . . . . . . . . . . . 165 8.3 Find Eigenvalue and Eigenvector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170 8.3.1 2 × 2 Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170 8.3.2 3 × 3 Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171 8.3.3 Complex Eigenvalue . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174 9 Systems of Differential Equations 179 9.1 Homogeneous Linear Differential Systems . . . . . . . . . . . . . . . . . . . . . . . . 179 9.2 Nonhomogeneous Differential Systems . . . . . . . . . . . . . . . . . . . . . . . . . . 184 10 Orthogonality 187 10.1 Dot Product . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187 10.2 Orthogonality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190 10.3 Orthogonal Projection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195 10.4 Orthogonalization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 198 iv
Chapter 10 Orthogonality 10.1 Dot Product Many real world problems involve geometrical quantities such as length, angle, area, volume, etc. Some other problems can be interpreted into geometrical quantities. The theory of linear algebra we have learned so far is not sufficient for treating these types of problems. We need to introduce more operations. The most basic of such operations is the dot product of two vectors in R n : ( x 1 , x 2 , · · · , x n ) · ( y 1 , y 2 , · · · , y n ) = x 1 y 1 + x 2 y 2 + · · · + x n y n .

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