2 edition
简介
Linear algebra is the foundation of science and engineering. Knowledge of linear algebra is a prerequisite for studying statistics, machine learning, computer graphics, signal processing, chemistry, economics, quantum mechanics, and countless other applications. Indeed, linear algebra offers a powerful toolbox for modelling the real world.
The NO BULLSHIT GUIDE TO LINEAR ALGEBRA shows the connections between the computational techniques of linear algebra, their geometric interpretations, and the theoretical foundations. This university-level textbook contains lessons on linear algebra written in a style that is precise and concise. Each concept is illustrated through definitions, formulas, diagrams, explanations, and examples of real-world applications. Readers build their math superpowers by solving practice problems and learning to use the computer algebra system SymPy to speed up tedious matrix arithmetic tasks.
目录
1 Math fundamentals 9
1.1 Solving equations..................... 10
1.2 Numbers.......................... 12
1.3 Variables ......................... 16
1.4 Functions and their inverses............... 18
1.5 Basic rules of algebra................... 21
1.6 Solving quadratic equations ............... 25
1.7 The Cartesian plane ................... 29
1.8 Functions ......................... 32
1.9 Function reference .................... 38
1.10Polynomials........................ 54
1.11Trigonometry....................... 58
1.12Trigonometric identities ................. 63
1.13Geometry ......................... 65
1.14Circle ........................... 67
1.15 Solving systems of linear equations . . . . . . . . . . . 69
1.16 Set notation........................ 73
1.17 Math problems ...................... 79
2 Vectors 89
2.1 Vectors .......................... 90
2.2 Basis............................ 98
2.3 Vector products...................... 99
2.4 Complex numbers .................... 102
2.5 Vectors problems ..................... 107
3 Intro to linear algebra 111
3.1 Introduction........................ 111
3.2 Review of vector operations ............... 117
3.3 Matrix operations .................... 121
3.4 Linearity.......................... 126
3.5 Overview of linearalgebra................ 131
3.6 Introductory problems .................. 135
4 Computational linear algebra 137
4.1 Reduced rowe chelonform................ 138
4.2 Matrix equations ..................... 150
4.3 Matrix multiplication................... 154
4.4 Determinants ....................... 158
4.5 Matrix inverse ...................... 169
4.6 Computational problems................. 176
5 Geometrical aspects of linear algebra 181
5.1 Lines and planes ..................... 181
5.2 Projections ........................ 189
5.3 Coordinate projections.................. 194
5.4 Vector spaces ....................... 199
5.5 Vector space techniques ................. 210
5.6 Geometric alproblems .................. 220
6 Linear transformations 223
6.1 Linear transformations.................. 223
6.2 Finding matrix representations . . . . . . . . . . . . . 234
6.3 Change of basis for matrices............... 245
6.4 Invertible matrix theorem ................ 249
6.5 Linear transformations problems . . . . . . . . . . . . 256
7 Theoretical linear algebra 257
7.1 Eigenvalues and eigenvectors .............. 258
7.2 Special types of matrices................. 271
7.3 Abstract vector spaces.................. 277
7.4 Abstract inner product spaces. . . . . . . . . . . . . . 281
7.5 Gram–Schmidt orthogonalization . . . . . . . . . . . . 288
7.6 Matrix decompositions.................. 292
7.7 Linear algebra with complex numbers . . . . . . . . . 298
7.8 Theory problems ..................... 313
8 Applications 317
8.1 Balancing chemical equations .............. 318
8.2 Input–output models in economics . . . . . . . . . . . 320
8.3 Electric circuits...................... 321
8.4 Graphs........................... 327
8.5 Fibonacci sequence.................... 330
8.6 Linear programming ................... 332
8.7 Least squares approximate solutions . . . . . . . . . . 333
8.8 Computer graphics.................... 342
8.9 Cryptography....................... 354
8.10 Error correcting codes .................. 366
8.11 Fourier analysis...................... 375
8.12Applications problems .................. 389
9 Probability theory 391
9.1 Probability distributions................. 391
9.2 Markov chains ...................... 398
9.3 Google’sPage Rank algorithm.............. 404
9.4 Probability problems................... 410
10 Quantum mechanics 411
10.1 Introduction........................ 412
10.2 Polarizing lenses experiment............... 418
10.3 Dirac notation for vectors ................ 425
10.4 Quantum in formation processing . . . . . . . . . . . . 431
10.5 Postulates of quantum mechanics . . . . . . . . . . . . 434
10.6 Polarizing lenses experiment revisited . . . . . . . . . 448
10.7 Quantum physics is not that weird . . . . . . . . . . . 452
10.8 Quantum mechanics applications . . . . . . . . . . . . 457
10.9 Quantum mechanics problems.............. 473
End matter 475
Conclusion ........................... 475
Social stuff ........................... 477
Acknowledgements ....................... 477
General linear algebra links .................. 477
A Answers and solutions 479
B Notation 499
Mathnotation ......................... 499
Set notation........................... 500
Vectors notation ........................ 500
Complex numbers notation .................. 501
Vector space notation ..................... 501
Notation for matrices and matrix operations . . . . . . . . . 502
Notation for linear transformations . . . . . . . . . . . . . . 503
Matrix decompositions..................... 503
Abstract vector space notation ................ 504