CS232

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

4. Hyperplanes Chapter 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)

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

Hyperplanes Hypereplanes in n-dimensions 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

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

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

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

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

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

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

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

Gaussian Elimination (General form) Identity matrix Elimination matrix

Permutation Matrix

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

Complexity of Matrix Multiplication cube

Block Multiplication

Strassen’s Fast Matrix Mulplication Divide and conquer

6. Inverse Matrix 7 Quiz 1 8 LU factorization Rest of chapter 2

9. Two dimensional convex Hull From the handout Convex combination

10. Algorithms for Null space 3.1 – 3.3

11. Complete Linear Solver 3.4 – 3.6

12. No class 13 Geometric Projection 4.1 – 4.2

14. Midterm 15 Least Square Algorithm

16. QR Decomposition

17-18 no classes spring break

19. Hubs and Authority Theory for Webs Hand out Understanding webs How Google works

20. Simplex and its Volume Chapter 5

21. Determinants: Matrix Representation of volume

22. Eivenvalue problem and Spectral Geometry

23. Quiz 2

24. Diagonalization

25. Quadratic Shapes Positive Definite matrices

26. Dimensional Reduction Singular value Decomposition

27. Application: Computer Graphics

28. Spherical Geometry Points on sphere Caps Stereographic Transformation

29. Geometric Transformation Chapter 7

30 Geometric Transformation

31. Triangulations and Voronoi Diagram