LAFF started as a Massive Open Online Course (MOOC) funded in part by the University of Texas System and the National Science Foundation (grant ACI-1148125), created by Prof. Robert van de Geijn and Dr. Maggie Myers at The University of Texas at Austin, and launched on the edX platform . The materials continue to be available with edX through at least Summer 2014.
The "Notes to LAFF With" are a PDF book that becomes the "hub" through which the other LAFF material (e.g., the videos) can be accessed. It goes beyond the notes that were released as part of the edX MOOC by also providing an index into the materials and incorporating extensive solutions for the homework exercises.
From the MOOC description:
Linear Algebra: Foundations to Frontiers (LAFF) is packed full of challenging, rewarding material that is essential for mathematicians, engineers, scientists, and anyone working with large datasets. Students appreciate our unique approach to teaching linear algebra because: It’s visual. It connects hand calculations, mathematical abstractions, and computer programming. It illustrates the development of mathematical theory. It’s applicable. In this course, you will learn all the standard topics that are taught in typical undergraduate linear algebra courses all over the world, but using our unique method, you'll also get more! LAFF was developed following the syllabus of an introductory linear algebra course at The University of Texas at Austin taught by Professor Robert van de Geijn, an expert on high performance linear algebra libraries. Through short videos, exercises, visualizations, and programming assignments, you will study Vector and Matrix Operations, Linear Transformations, Solving Systems of Equations, Vector Spaces, Linear Least-Squares, and Eigenvalues and Eigenvectors. In addition, you will get a glimpse of cutting edge research on the development of linear algebra libraries, which are used throughout computational science.
Download it for Free: http://www.ulaff.net/
Author(s): Robert van de Geijn, Maggie Myers
Year: 2014
Language: English
Pages: 905
Take Off......Page 17
Outline Week 1......Page 18
What You Will Learn......Page 20
Notation......Page 21
Unit Basis Vectors......Page 24
Equality (=), Assignment (:=), and Copy......Page 25
Vector Addition (add)......Page 26
Scaling (scal)......Page 29
Vector Subtraction......Page 31
Scaled Vector Addition (axpy)......Page 33
Linear Combinations of Vectors......Page 35
Dot or Inner Product (dot)......Page 37
Vector Length (norm2)......Page 40
Vector Functions......Page 42
Vector Functions that Map a Vector to a Vector......Page 45
Starting the Package......Page 49
A Routine that Scales a Vector (scal)......Page 50
A Vector Length Routine (norm2)......Page 51
2. Linear Transformations and Matrices (Answers)......Page 52
Coding with Slicing and Redicing: Dot Product......Page 53
Algorithms with Slicing and Redicing: axpy......Page 54
Coding with Slicing and Redicing: axpy......Page 55
Other Norms......Page 56
Overflow and Underflow......Page 60
Homework......Page 61
Summary of Vector Operations......Page 62
Summary of the Properties of Vector Operations......Page 63
Summary of the Routines for Vector Operations......Page 64
Rotating in 2D......Page 65
Outline......Page 68
What You Will Learn......Page 69
What is a Linear Transformation?......Page 70
Of Linear Transformations and Linear Combinations......Page 74
What is the Principle of Mathematical Induction?......Page 76
Examples......Page 77
From Linear Transformation to Matrix-Vector Multiplication......Page 80
Practice with Matrix-Vector Multiplication......Page 84
It Goes Both Ways......Page 87
Rotations and Reflections, Revisited......Page 89
The Importance of the Principle of Mathematical Induction for Programming......Page 93
Puzzles and Paradoxes in Mathematical Induction......Page 94
Summary......Page 95
Timmy Two Space......Page 99
Outline Week 3......Page 100
What You Will Learn......Page 101
The Zero Matrix......Page 102
The Identity Matrix......Page 104
Diagonal Matrices......Page 108
Triangular Matrices......Page 110
Transpose Matrix......Page 114
Symmetric Matrices......Page 117
Scaling a Matrix......Page 120
Adding Matrices......Page 124
Via Dot Products......Page 128
Via axpy Operations......Page 132
Compare and Contrast......Page 135
Homework......Page 137
Summary......Page 138
Predicting the Weather......Page 143
Outline......Page 149
What You Will Learn......Page 150
Partitioned Matrix-Vector Multiplication......Page 151
Transposing a Partitioned Matrix......Page 154
Matrix-Vector Multiplication, Again......Page 159
Transpose Matrix-Vector Multiplication......Page 164
Triangular Matrix-Vector Multiplication......Page 166
Symmetric Matrix-Vector Multiplication......Page 177
Motivation......Page 182
From Composing Linear Transformations to Matrix-Matrix Multiplication......Page 183
Computing the Matrix-Matrix Product......Page 184
Special Shapes......Page 188
Hidden Markov Processes......Page 196
Homework......Page 197
Summary......Page 199
Composing Rotations......Page 203
Outline......Page 204
What You Will Learn......Page 205
Partitioned Matrix-Matrix Multiplication......Page 206
Properties......Page 208
Matrix-Matrix Multiplication with Special Matrices......Page 209
Lots of Loops......Page 216
Matrix-Matrix Multiplication by Columns......Page 218
Matrix-Matrix Multiplication by Rows......Page 220
Matrix-Matrix Multiplication with Rank-1 Updates......Page 223
Slicing and Dicing for Performance......Page 226
How It is Really Done......Page 231
Homework......Page 233
Summary......Page 238
Solving Linear Systems......Page 243
Outline......Page 244
What You Will Learn......Page 245
Reducing a System of Linear Equations to an Upper Triangular System......Page 246
Appended Matrices......Page 249
Gauss Transforms......Page 253
Computing Separately with the Matrix and Right-Hand Side (Forward Substitution)......Page 257
Towards an Algorithm......Page 258
LU factorization (Gaussian elimination)......Page 262
Solving L z = b (Forward substitution)......Page 266
Solving U x = b (Back substitution)......Page 269
Putting it all together to solve A x = b......Page 271
Cost......Page 272
Blocked LU Factorization......Page 278
Summary......Page 284
Introduction......Page 291
Outline......Page 292
What You Will Learn......Page 293
When Gaussian Elimination Works......Page 294
The Problem......Page 299
Permutations......Page 301
Gaussian Elimination with Row Swapping (LU Factorization with Partial Pivoting)......Page 306
When Gaussian Elimination Fails Altogether......Page 312
Back to Linear Transformations......Page 313
Simple Examples......Page 315
More Advanced (but Still Simple) Examples......Page 321
Properties......Page 325
Library Routines for LU with Partial Pivoting......Page 326
Summary......Page 328
When LU Factorization with Row Pivoting Fails......Page 335
Outline......Page 339
What You Will Learn......Page 340
Solving A x = b via Gauss-Jordan Elimination......Page 341
Solving A x = b via Gauss-Jordan Elimination: Gauss Transforms......Page 343
Solving A x = b via Gauss-Jordan Elimination: Multiple Right-Hand Sides......Page 347
Computing A-1 via Gauss-Jordan Elimination......Page 352
Computing A-1 via Gauss-Jordan Elimination, Alternative......Page 357
Cost of Matrix Inversion......Page 361
Solving A x = b......Page 365
But.........Page 366
Symmetric Positive Definite Matrices......Page 367
Solving A x = b when A is Symmetric Positive Definite......Page 368
Welcome to the Frontier......Page 372
Summary......Page 375
M.2 Sample Midterm......Page 377
M.3 Midterm......Page 383
Solvable or not solvable, that's the question......Page 391
Outline......Page 397
What you will learn......Page 398
When Solutions Are Not Unique......Page 399
When Linear Systems Have No Solutions......Page 400
When Linear Systems Have Many Solutions......Page 402
What is Going On?......Page 404
Toward a Systematic Approach to Finding All Solutions......Page 406
Definition and Notation......Page 409
Examples......Page 410
Operations with Sets......Page 411
Subspaces......Page 414
The Column Space......Page 417
The Null Space......Page 420
Span......Page 422
Linear Independence......Page 424
Bases for Subspaces......Page 429
The Dimension of a Subspace......Page 431
Typesetting algrithms with the FLAME notation......Page 432
Summary......Page 433
Visualizing Planes, Lines, and Solutions......Page 437
Outline......Page 445
What You Will Learn......Page 446
The Important Attributes of a Linear System......Page 447
Orthogonal Vectors......Page 454
Orthogonal Spaces......Page 456
Fundamental Spaces......Page 457
A Motivating Example......Page 461
Finding the Best Solution......Page 464
Why It is Called Linear Least-Squares......Page 469
Solving the Normal Equations......Page 470
Summary......Page 471
Low Rank Approximation......Page 475
Outline......Page 476
What You Will Learn......Page 477
Component in the Direction of .........Page 478
An Application: Rank-1 Approximation......Page 482
Projection onto a Subspace......Page 486
An Application: Rank-2 Approximation......Page 488
An Application: Rank-k Approximation......Page 490
The Unit Basis Vectors, Again......Page 492
Orthonormal Vectors......Page 493
Orthogonal Bases......Page 497
Orthogonal Bases (Alternative Explanation)......Page 499
The QR Factorization......Page 503
Solving the Linear Least-Squares Problem via QR Factorization......Page 505
The QR Factorization )Again)......Page 506
Change of Basis......Page 509
The Best Low Rank Approximation......Page 512
The Problem with Computing the QR Factorization......Page 516
Summary......Page 517
Predicting the Weather, Again......Page 523
Outline......Page 526
What You Will Learn......Page 527
The Algebraic Eigenvalue Problem......Page 528
Simple Examples......Page 529
Diagonalizing......Page 541
Eigenvalues and Eigenvectors of 3 3 Matrices......Page 543
Eigenvalues and Eigenvectors of n n matrices: Special Cases......Page 549
Eigenvalues of n n Matrices......Page 551
Diagonalizing, Again......Page 553
Properties of Eigenvalues and Eigenvectors......Page 556
Predicting the Weather, One Last Time......Page 558
The Power Method......Page 561
The Inverse Power Method......Page 565
More Advanced Techniques......Page 570
Summary......Page 571
F.2 Sample Final......Page 577
F.3 Final......Page 580
Answers......Page 589
1. Vectors in Linear Algebra (Answers)......Page 590
3. Matrix-Vector Operations (Answers)......Page 630
4. From Matrix-Vector Multiplication to Matrix-Matrix Multiplication (Answers)......Page 651
5. Matrix-Matrix Multiplication (Answers)......Page 679
6. Gaussian Elimination (Answers)......Page 706
7. More Gaussian Elimination and Matrix Inversion (Answers)......Page 715
8. More on Matrix Inversion (Answers)......Page 749
Midterm (Answers)......Page 763
9. Vector Spaces (Answers)......Page 781
10. Vector Spaces, Orthogonality, and Linear Least Squares (Answers)......Page 802
11. Orthogonal Projection, Low Rank Approximation, and Orthogonal Bases (Answers)......Page 821
12. Eigenvalues, Eigenvectors, and Diagonalization (Answers)......Page 839
Final (Answers)......Page 861
LAFF Routines (Python)......Page 885
``What You Will Learn'' Check List......Page 887
Index......Page 901
