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Exam (elaborations)

Introduction to Applied Linear Algebra Vectors, Matrices, and Least Squares

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I Vectors 1 1 Vectors 3 1.1 Vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.2 Vector addition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 1.3 Scalar-vector multiplication . . . . . . . . . . . . . . . . . . . . . . . . 15 1.4 Inner product . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 1.5 Complexity of vector computations . . . . . . . . . . . . . . . . . . . . 22 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 2 Linear functions 29 2.1 Linear functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 2.2 Taylor approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 2.3 Regression model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 3 Norm and distance 45 3.1 Norm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 3.2 Distance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 3.3 Standard deviation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 3.4 Angle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 3.5 Complexity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 4 Clustering 69 4.1 Clustering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 4.2 A clustering objective . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 4.3 The k-means algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . 74 4.4 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 4.5 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87 viii Contents 5 Linear independence 89 5.1 Linear dependence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89 5.2 Basis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 5.3 Orthonormal vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 5.4 Gram{Schmidt algorithm . . . . . . . . . . . . . . . . . . . . . . . . . 97 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 II Matrices 105 6 Matrices 107 6.1 Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107 6.2 Zero and identity matrices . . . . . . . . . . . . . . . . . . . . . . . . 113 6.3 Transpose, addition, and norm . . . . . . . . . . . . . . . . . . . . . . 115 6.4 Matrix-vector multiplication . . . . . . . . . . . . . . . . . . . . . . . . 118 6.5 Complexity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124 7 Matrix examples 129 7.1 Geometric transformations . . . . . . . . . . . . . . . . . . . . . . . . 129 7.2 Selectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131 7.3 Incidence matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132 7.4 Convolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144 8 Linear equations 147 8.1 Linear and ane functions . . . . . . . . . . . . . . . . . . . . . . . . 147 8.2 Linear function models . . . . . . . . . . . . . . . . . . . . . . . . . . 150 8.3 Systems of linear equations . . . . . . . . . . . . . . . . . . . . . . . . 152 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159 9 Linear dynamical systems 163 9.1 Linear dynamical systems . . . . . . . . . . . . . . . . . . . . . . . . . 163 9.2 Population dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . 164 9.3 Epidemic dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168 9.4 Motion of a mass . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169 9.5 Supply chain dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . 171 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174 10 Matrix multiplication 177 10.1 Matrix-matrix multiplication . . . . . . . . . . . . . . . . . . . . . . . 177 10.2 Composition of linear functions . . . . . . . . . . . . . . . . . . . . . . 183 10.3 Matrix power . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186 10.4 QR factorization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191 Contents ix 11 Matrix inverses 199 11.1 Left and right inverses . . . . . . . . . . . . . . . . . . . . . . . . . . . 199 11.2 Inverse . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202 11.3 Solving linear equations . . . . . . . . . . . . . . . . . . . . . . . . . . 207 11.4 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 210 11.5 Pseudo-inverse . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217 III Least squares 223 12 Least squares 225 12.1 Least squares problem . . . . . . . . . . . . . . . . . . . . . . . . . . . 225 12.2 Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227 12.3 Solving least squares problems . . . . . . . . . . . . . . . . . . . . . . 231 12.4 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 234 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 239 13 Least squares data tting 245 13.1 Least squares data tting . . . . . . . . . . . . . . . . . . . . . . . . . 245 13.2 Validation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 260 13.3 Feature engineering . . . . . . . . . . . . . . . . . . . . . . . . . . . . 269 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 279 14 Least squares classi cation 285 14.1 Classi cation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 285 14.2 Least squares classi er . . . . . . . . . . . . . . . . . . . . . . . . . . . 288 14.3 Multi-class classi ers . . . . . . . . . . . . . . . . . . . . . . . . . . . 297 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 305 15 Multi-objective least squares 309 15.1 Multi-objective least squares . . . . . . . . . . . . . . . . . . . . . . . 309 15.2 Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 314 15.3 Estimation and inversion . . . . . . . . . . . . . . . . . . . . . . . . . 316 15.4 Regularized data tting . . . . . . . . . . . . . . . . . . . . . . . . . . 325 15.5 Complexity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 330 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 334 16 Constrained least squares 339 16.1 Constrained least squares problem . . . . . . . . . . . . . . . . . . . . 339 16.2 Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 344 16.3 Solving constrained least squares problems . . . . . . . . . . . . . . . . 347 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 352

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Introduction to
Applied Linear Algebra
Vectors, Matrices, and Least Squares



Stephen Boyd
Department of Electrical Engineering
Stanford University

Lieven Vandenberghe
Department of Electrical and Computer Engineering
University of California, Los Angeles

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Information on this title: www.cambridge.org/9781316518960
DOI: 10.1017/9781108583664
© Cambridge University Press 2018
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First published 2018
Printed in the United Kingdom by Clays, St Ives plc, 2018
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ISBN 978-1-316-51896-0 Hardback
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