Availability: Linear Algebra | Technische Universität München

Linear Algebra: theory, intuition, code

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Bibliographic Details
Main Author:	Cohen, Mike X. 1979- (Author)
Format:	Book
Language:	English
Published:	[Erscheinungsort nicht ermittelbar] sincXpress [2021]
Subjects:	Lineare Algebra
Links:	http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=032957710&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA
Physical Description:	582 Seiten Illustrationen, Tabellen
ISBN:	9789083136608

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MARC


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Record in the Search Index

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adam_text	Contents 0.1 Front matter...................................................................... 0.2 Dedication......................................................................... 0.3 Forward........................................................................... 1 2 2 2 Introduction 11 1.1 What is linear algebra andwhy learn it?.................... 12 1.2 About this book............................................................... 12 1.3 Prerequisites..................................................................... 15 1.4 Exercises and code challenges...................................... 17 1.5 Online and other resources............................................ 18 2 Vectors 21 2.1 Scalars.............................................................................. 22 2.2 Vectors: geometry and algebra...................................... 23 2.3 Transpose operation......................................................... 30 2.4 Vector addition and subtraction................................... 31 2.5 Vector-scalar multiplication ......................................... 33 2.6 Exercises........................................................................... 37 2.7 Answers........................................................................... 39 2.8 ֊ Code challenges............................................................... 41 2.9 Code solutions.................................................................. 42 3 Vector multiplication 43 3.1 Vector dot product: Algebra......................................... 44 3.2 Dot product properties.................................................. 46 3.3 Vector dot product: Geometry...................................... 50 3.4 Algebra and geometry .................................................. 52 3.5 Linear weighted combination......................................... 57 3.6 The outer product............................................................ 59 3.7 Hadamard multiplication............................................... 62 3.8 Cross product.................................................................. 64 3.9 Unit vectors..................................................................... 65 3.10 3.11 3.12 3.13 Exercises.......................................................................... Answers ......................................................................... Code challenges.................................... Code solutions................................................................ 68 70 72 73 4 Vector spaces 4.1 Dimensions and fields.................................................... 75 76 4.2 4.3 4.4 4.5 4.6 4.7 4.8 4.9 5 Matrices 107 5.1 Interpretations and uses of matrices...............................108 5.2 5.3 5.4 5.5 5.6 5.7 5.8 5.9 5.10 5.11 5.12 5.13 6 Vector spaces ................................................................ 78 Subspaces and ambient spaces..................................... 78 Subsets............................................................................ 85 Span............................................................................... 86 Linear independence .................................................... 90 Basis............................................................................... 97 Exercises............................................................................ 102 Answers............................................................................ 105 Matrix terms and notation.............................................. 109 Matrix dimensionalities..................... 110 The transpose operation.................................................110 Matrix zoology ................................................................ 112 Matrix addition and subtraction.....................................124 Scalar-matrix mult..............................................................126 Shifting a matrix.......................................................... 126 Diagonal and trace.......................................................... 129 Exercises.............................................................................131 Answers.............................................................................134 Code challenges................................................................ 136 Code solutions................................................................... 137 Matrix multiplication 6.1 6.2 6.3 6.4 6.5 6.6 6.7 6.8 139 Standard multiplication................................................. 140 Multiplication and eqns......................................................148 Multiplication with diagonals ........................................150 LIVE EVIL .......................................................................152 Matrix-vector multiplication........................................... 155 Creating symmetric matrices...........................................157 Multiply symmetric matrices...........................................160 Hadamard multiplication................................................. 161 6.9 6.10 6.11 6.12 6.13 6.14 6.15 7 Rank 179 7.1 Six things about matrixrank.......................................... 180 7.2 Interpretations of matrix rank..................................... 181 7.3 Computing matrix rank.................................................. 183 7.4 Rank and scalar multiplication..................................... 185 7.5 Rank of added matrices.................................................. 186 7.6 Rank of multiplied matrices.........................................188 7.7 Rank of A, AT, ATA, and AAT............................... 190 7.8 7.9 7.10 7.11 7.12 7.13 7.14 7.15 8 Frobenius dot product .............................................. 163 Matrix norms..................................................................166 What about matrix division?.........................................169 Exercises........................................................................... 176 Answers........................................................................... 173 Code challenges...............................................................175 Code solutions..................................................................176 Rank of random matrices...............................................193 Boosting rank by shifting ............................................194 Rank difficulties.............................................................. 196 Rank and span..................................................................198 Exercises...........................................................................200 Answers ...........................................................................201 Code challenges.............................................................. 202 Code solutions..................................................................203 Matrix spaces 205 8.1 Column space of a matrix ...............................................206 8.2 Column space: A and AAT........................................... 209 8.3 8.4 8.5 8.6 8.7 8.8 8.9 8.10 8.11 8.12 8.13 Determining whether v Є C(A)..................................... 210 Row space of a matrix..................................................... 212 Row spaces of A and ATA...............................................214 Null space of a matrix ..................................................... 214 Geometry of the null space...............................................219 Orthogonal subspaces........................................................ 221 Matrix space orthogonalities............................................223 Dimensionalities of matrix spaces.................................. 227 More on Ax = b and Ay= 0..........................................230 Exercises..............................................................................232 Answers..............................................................................234 8.14 Code challenges.................................................................236 8.15 Code solutions....................................................................237 9 Complex numbers 239 9.1 Complex numbers and C.................................................. 240 9.2 What are complexnumbers?............................................ 241 9.3 The complex conjugate................................................. 244 9.4 Complex arithmetic....................................................... 246 9.5 Complex dot product.................................................... 249 9.6 Special complex matrices.............................................. 261 9.7 Exercises..........................................................................253 9.8 Answers..........................................................................254 9.9 Code challenges............................................................. 255 9.10 Code solutions................................................................ 256 10 Systems of equations 259 10.1 Algebra and geometry of eqns........................................260 10.2 From systems to matrices.............................................. 264 10.3 Row reduction................................................................... 268 10.4 Gaussian elimination....................................................... 278 10.5 Row-reduced echelon form...........................................281 10.6 Gauss-Jordan elimination.................................................284 10.7 Possibilities for solutions................................................. 285 10.8 Matrix spaces, row reduction........................................288 10.9 Exercises............................................................................ 291 10.10 Answers............................................................................ 292 10.11 Coding challenges............................................................. 293 10.12 Code solutions................................................................... 294 11 Determinant 295 11.1 Features of determinants.................................................296 11.2 Determinant of a 2 x 2 matrix........................................297 11.3 The characteristic polynomial........................................299 11.4 3 x 3 matrix determinant.................................................302 11.5 The full procedure.............................................................305 11.6 Δ of triangles...................................................................307 11.7 Determinant and row reduction.....................................309 11.8 Δ and scalar multiplication..............................................313 11.9 Theory vs practice .................. 314 11.10 Exercises............................................................................316 / 11.11 Answers.............................................................................. 317 11.12 Code challenges..................................................................318 11.13 Code solutions.....................................................................319 12 Matrix inverse 321 12.1 Concepts and applications...............................................322 12.2 Inverse of a diagonal matrix............................................327 12.3 Inverse of a 2 x 2 matrix..................................................328 12.4 The MCA algorithm ........................................................ 330 12.5 Inverse via row reduction..................................................336 12.6 Left inverse ........................................................................339 12.7 Right inverse........................................................................342 12.8 The pseudoinverse, part 1 345 12.9 Exercises..............................................................................348 12.10 Answers ..............................................................................350 12.11 Code challenges................................................................. 352 12.12 Code solutions.................................................................... 353 13 Projections 357 13.1 Projections in R2.............................................................. 358 13.2 Projections in MN.............................................................. 361 13.3 Orth and par vect comps..................................................365 13.4 Orthogonal matrices ........................................................ 370 13.5 Orthogonalization via GS..................................................374 13.6 QR decomposition.............................................................. 377 13.7 Inverse via QR.................................................................... 381 13.8 Exercises...........................................................................383 13.9 Answers ...........................................................................384 13.10 Code challenges................................................................. 385 13.11 Code solutions.....................................................................386 14 Least-squares 389 14.1 Introduction........................................................................390 14.2 5 steps of model-fitting..................................................... 391 14.3 Terminology........................................................................394 14.4 Least-squares via left inverse............................................395 14.5 Least-squares via projection........................................... 397 14.6 Least-squares via row-reduction..................................... 399 14.7 Predictions and residuals..................................................401 14.8 Least-squares example .....................................................402 14.9 Code challenges.............................................................. 409 14.10 Code solutions................................................................. 410 15 Eigendecomposiťion 415 15.1 Eigenwhatnow?.............................................................. 416 15.2 Finding eigenvalues.......................................................... 421 15.3 Finding eigenvectors ........................................................426 15.4 Diagonalization.................................................................430 15.5 Conditions for diagonalization........................................433 15.6 Distinct, repeated eigenvalues........................................435 15.7 Complex solutions............................................................. 440 15.8 Symmetric matrices........................................................442 15.9 Eigenvalues singular matrices ..................................... 445 15.10 Eigenlayers of a matrix.................................................... 448 15.11 Matrix powers and inverse.............................................. 451 15.12 Generalized eigendecomposition.....................................455 15.13 Exercises..........................................................................458 15.14 Answers ..........................................................................459 15.15 Code challenges................................................................ 460 15.16 Code solutions................................................................... 461 16 The SVD 467 16.1 Singular value decomposition........................................ 468 16.2 Computing the SVD .................................................... 470 16.3 Singular values and eigenvalues ..................................474 16.4 SVD of a symmetric matrix........................................478 16.5 SVD and the four subspaces........................................ 479 16.6 SVD and matrix rank.................................................... 482 16.7 SVD spectral theory .......................................................484 16.8 Low-rank approximations.................................................488 16.9 Normalizing singular values ...........................................491 16.10 Condition number of a matrix........................................493 16.11 SVD and the matrix inverse...........................................495 16.12 MP Pseudoinverse, part 2 ..............................................496 16.13 Code challenges................................................................499 16.14 Code solutions................................................................... 502 17 Quadratic form 517 17.1 Algebraic perspective.......................................................518 17.2 Geometric perspective ....................................................523 17.3 17.4 17.5 17.6 17.7 17.8 17.9 The normalized quadratic form ..................................... 526 Evecs and the qf surface..................................................530 Matrix definiteness........................................................... 532 The definiteness of ATA.................................................. 534 Λ and definiteness.............................................................. 535 Code challenges..................................................................539 Code solutions.....................................................................540 18 Covariance matrices 545 18.1 Correlation........................................................................546 18.2 Variance and standard deviation..................................... 548 18.3 Covariance........................................................................549 18.4 Correlation coefficient........................................................ 551 18.5 Covariance matrices........................................................ 553 18.6 Correlation to covariance..................................................554 18.7 Code challenges.................. 555 18.8 Code solutions.....................................................................556 19 PCA 557 19.1 PCA: interps and apps.....................................................558 19.2 How to perform a PCA..................................................... 561 19.3 The algebra of PCA........................................................... 563 19.4 Regularization.....................................................................565 19.5 Is PCA always the best?..................................................568 19.6 Code challenges................................................................. 570 19.7 Code solutions.....................................................................571 20 The end. 575 20.1 The end... of the beginning!............................................576 20.2 Thanks! ՝.............................................................................. 577
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spellingShingle	Cohen, Mike X. 1979- Linear Algebra theory, intuition, code Lineare Algebra (DE-588)4035811-2 gnd
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title	Linear Algebra theory, intuition, code
title_auth	Linear Algebra theory, intuition, code
title_exact_search	Linear Algebra theory, intuition, code
title_full	Linear Algebra theory, intuition, code Dr. Mike X Cohen
title_fullStr	Linear Algebra theory, intuition, code Dr. Mike X Cohen
title_full_unstemmed	Linear Algebra theory, intuition, code Dr. Mike X Cohen
title_short	Linear Algebra
title_sort	linear algebra theory intuition code
title_sub	theory, intuition, code
topic	Lineare Algebra (DE-588)4035811-2 gnd
topic_facet	Lineare Algebra
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