جبر خطی: از بنیاد تا مرزها

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/ 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