Notes for Math 152: Linear Systems Spring, 2013 Richard Froese and Brian Wetton © 2010 Richard Froese and Brian Wetton. Permission is granted to make and distribute copies of this document provided the copyright notice and this permission notice are preserved on all copies. Contents 1 Introduction 5 1.1 Course Goals . 5 1.2 About the Subject . 6 1.2.1 Connection to Geometry . 6 1.2.2 Linear Systems . 6 1.2.3 Eigen-analysis . 7 1.3 About These Notes . 8 1.4 About The Computer Labs . 9 2 Vectors and Geometry 10 2.1 Chapter Introduction . 10 2.2 Vectors . 10 2.2.1 Multiplication by a number and vector addition . 11 2.2.2 Co-ordinates . 12 2.2.3 Properties of vector addition and scalar multiplication . 16 2.2.4 MATLAB: basic scalar and vector operations . 17 2.2.5 Problems . 18 2.3 Geometrical Aspects of Vectors . 19 2.3.1 Length of a vector . 19 2.3.2 The dot product . 19 2.3.3 Projections . 22 2.3.4 MATLAB: norm and dot commands . 23 2.3.5 Problems . 23 2.4 Determinants and the Cross Product . 25 2.4.1 The determinant in two and three dimensions . 25 2.4.2 The cross product . 27 2.4.3 The triple product and the determinant in three dimensions 30 2.4.4 MATLAB: assigning matrices and det and cross commands 31 2.4.5 MATLAB: generating scripts with the MATLAB editor . 31 2.4.6 MATLAB: floating point representation of real numbers . 32 2.4.7 Problems . 33 2.5 Lines and Planes . 34 2.5.1 Describing linear sets . 34 2.5.2 Lines in two dimensions: Parametric form . 34 1 CONTENTS CONTENTS 2.5.3 Lines in two dimensions: Equation form . 35 2.5.4 Lines in three dimensions: Parametric form . 35 2.5.5 Lines in three dimensions: Equation form . 36 2.5.6 Planes in three dimensions: Parametric form . 37 2.5.7 Planes in three dimensions: Equation form . 37 2.5.8 Problems . 37 2.6 Introduction to Linear Systems . 40 2.6.1 Description of points and the geometry of solutions to systems of equations . 40 2.6.2 Describing the whole plane in two dimensions and all of space in three dimensions . 42 2.6.3 Linear dependence and independence . 43 2.6.4 Problems . 44 2.7 Additional Topics . 45 2.7.1 Application: rotational motion . 45 2.7.2 Application: 3-D graphics . 46 2.8 Solutions to Chapter Problems . 49 3 Solving Linear Systems 63 3.1 Linear Systems . 63 3.1.1 General Form of Linear Systems . 63 3.1.2 Solving Linear Systems by Substitution . 63 3.1.3 Elementary row (equation) operations . 64 3.1.4 Augmented Matrices . 65 3.1.5 Problems . 66 3.2 Gaussian Elimination . 67 3.2.1 Using MATLAB for row reductions . 73 3.2.2 Problems . 74 3.3 Homogeneous Equations . 76 3.3.1 Properties of solutions of homogeneous systems. 77 3.3.2 Connection of solutions to homogeneous and inhomoge- neous systems. 77 3.3.3 Problems . 79 3.4 Geometric Applications . 80 3.4.1 Problems . 81 3.5 Resistor Networks . 82 3.5.1 Elements of Basic Circuits . 82 3.5.2 Two Simple Examples Made Complicated . 84 3.5.3 Loop Currents . 86 3.5.4 Alternate Presentation of Resistor Networks . 90 3.5.5 Problems . 93 3.6 Additional Topics . 96 3.6.1 Quadratic Functions . 96 3.6.2 Least squares fit . 98 3.6.3 Equilibrium configuration of hanging weights and springs 99 3.6.4 Problems . 101 2 CONTENTS CONTENTS 3.7 Solutions to Chapter Problems . 102 4 Matrices and Determinants 117 4.1 Matrix operations . 117 4.1.1 MATLAB . 120 4.1.2 Problems . 120 4.2 Linear Transformations and Matrices . 121 4.2.1 Linear Transformations . 121 4.2.2 Rotations in two dimensions . 122 4.2.3 Projections in two dimensions . 123 4.2.4 Reflections in two dimensions . 124 4.2.5 Every linear transformation is multiplication by a matrix 125 4.2.6 Composition of linear transformations and matrix product 127 4.2.7 Problems . 127 4.3 Application: random walks . 129 4.3.1 Problems . 134 4.4 The Transpose . 136 4.4.1 MATLAB . 138 4.4.2 Problems . 138 4.5 Matrix Inverses . 138 4.5.1 Computing the inverse . 141 4.5.2 Inverses of Products . 144 4.5.3 MATLAB . 145 4.5.4 Problems . 145 4.6 Return to Resistor Networks . 146 4.7 Determinants . 150 4.7.1 Definition of Determinants . 150 4.7.2 Determinants of Triangular matrices . 151 4.7.3 Summary of determinant calculation rules . 152 4.7.4 Calculation of determinant using row operations . 153 4.7.5 More expansion formulae . 153 4.7.6 MATLAB . 155 4.7.7 Problems . 155 4.8 Additional Topics . 157 4.8.1 Application: General Least Squares . 157 4.8.2 Least squares solutions . 158 4.8.3 Problems . 159 4.8.4 Elementary matrices . 159 4.8.5 Problems . 162 4.8.6 Exchanging two rows changes the sign of the determinant 163 4.8.7 The determinant is linear in each row separately . 164 4.8.8 Adding a multiple of one row to another doesn't change the determinant . 165 4.8.9 The determinant of QA . 166 4.8.10 The determinant of A is zero exactly when A is not invertible167 4.8.11 The product formula: det(AB) = det(A) det(B) . 167 3 CONTENTS CONTENTS 4.8.12 The determinant of the transpose . 168 4.8.13 An impractical formula for the inverse . 169 4.8.14 Cramer's rule, an impractical way to solve systems . 170 4.8.15 Problems . 170 4.9 Solutions to Chapter Problems . 170 5 Complex numbers 193 5.1 Complex arithmetic . 193 5.2 Complex exponential . 194 5.3 Polar representation of a complex number . 196 5.4 MATLAB . 196 5.5 Problems . 197 5.6 Solutions to Chapter Problems . 197 6 Eigen-analysis 200 6.1 Eigenvalues and eigenvectors . 200 6.1.1 Computing the eigenvalues and eigenvectors . 201 6.1.2 Complex eigenvalues and eigenvectors . 207 6.1.3 MATLAB . 209 6.1.4 Problems . 210 6.2 Eigenanalysis simplifies matrix powers . 211 6.2.1 Problems . 215 6.3 Systems of linear differential equations . 215 6.3.1 Problems . 221 6.4 LCR circuits . 222 6.4.1 Capacitors and inductors . 222 6.4.2 Differential equations for LCR circuits . 223 6.4.3 Alternate description of LCR circuits . 225 6.4.4 Problems . 227 6.5 Additional Topics . 227 6.5.1 Diagonalization . 227 6.5.2 Computing high powers of a matrix . 229 6.5.3 Another formula for the determinant . 230 6.5.4 The matrix exponential and differential equations . ..