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Advanced Topics in Linear Algebra and Geometric Transformations

This course covers essential topics in linear algebra and geometric transformations, including matrix operations, Gaussian elimination, and the study of hyperplanes and convex hulls. Key algorithms for nearest neighbor problems, null spaces, and QR decomposition are explored. The lessons extend into spectral geometry, eigenvalue problems, and practical applications in computer graphics and geometric transformations. Students will engage with theoretical concepts while solving computational problems through lectures, quizzes, and hands-on exercises.

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Advanced Topics in Linear Algebra and Geometric Transformations

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Presentation Transcript

  1. CS232

  2. Schedule • 1. Introduction • 2. Points vs vector (distance, balls, sphere) • Chapter 1 • 3. Divide and Conquer: Algorithms for Near Neighbor Problem • Handout (section)

  3. 4. HyperplanesChapter 2 • Ray intersections • Lines • By linear equations • By two points • When does a line passing the origin • Intersection of two lines • Matrix and algebraic approach (two variables and two equations)

  4. 3D • Ray and mirrors • Planes in three dimensions • By linear equations • By three points • When does a plane passing the origin

  5. Hyperplanes • Intersection of three planes • Matrix and algebraic approach (three variables and equations) • Hypereplanes in n-dimensions • By linear equations • By n points • When does a hyperplane passing through the origin • Intersection of n hyperplanes in n dimensions

  6. Matrix Form • What is a matrix? • Matrix vector multiplication • (inner product after all) • Matrix form of intersection of n hyperplanes --- system of linear equations?

  7. Column Picture: combination of vectors • Find proper linear combinations of vectors • Visualize hyperplane is hard, so you might eventually like the column pictures.

  8. Repeated the questions • Row pictures: n hyperplanes meets at a single points • Column pictures: combines n vectors to produce another vector

  9. Gaussian Elimination • Gaussian Elimination in 2 dimensions • example • Pictures • Pivots • Multipliers • Upper triangular matrix • Back substitution

  10. Two dimensions • Unique solution • No solution • Infinitely many solutions • What if the pivot is 0!!!

  11. 3D • Gaussian Elimination in 3 dimensions • example • Pictures • Pivots • Multipliers • Upper triangular matrix • Back substitution • Can be extended to any dimensions

  12. 5. Gaussian Elimination(General form) • Matrix Algebra • Matrix addition • Scalar times a matrix • Matrix multiplication • (dimensions have to agree) • Associative law • Non commutative law

  13. Gaussian Elimination(General form) • Identity matrix • Elimination matrix

  14. Permutation Matrix

  15. Matrix algebra(General form) • All the laws (page 58 – 59)

  16. Complexity of Matrix Multiplication • cube

  17. Block Multiplication

  18. Strassen’s Fast Matrix Mulplication • Divide and conquer

  19. 6. Inverse Matrix7 Quiz 18 LU factorization • Rest of chapter 2

  20. 9. Two dimensional convex Hull • From the handout • Convex combination

  21. 10. Algorithms for Null space • 3.1 – 3.3

  22. 11. Complete Linear Solver • 3.4 – 3.6

  23. 12. No class13 Geometric Projection • 4.1 – 4.2

  24. 14. Midterm15 Least Square Algorithm

  25. 16. QR Decomposition

  26. 17-18 no classes spring break

  27. 19. Hubs and AuthorityTheory for WebsHand out • Understanding webs • How Google works

  28. 20. Simplex and its Volume • Chapter 5

  29. 21. Determinants: Matrix Representation of volume

  30. 22. Eivenvalue problem and Spectral Geometry

  31. 23. Quiz 2

  32. 24. Diagonalization

  33. 25. Quadratic Shapes • Positive Definite matrices

  34. 26. Dimensional Reduction • Singular value Decomposition

  35. 27. Application: Computer Graphics

  36. 28. Spherical Geometry • Points on sphere • Caps • Stereographic Transformation

  37. 29. Geometric Transformation • Chapter 7

  38. 30 Geometric Transformation

  39. 31. Triangulations and Voronoi Diagram

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