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. 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)

3D

• Ray and mirrors

• Planes in three dimensions– By linear equations– By three points– When does a plane passing the origin

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

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 Matrix7 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 class13 Geometric Projection

• 4.1 – 4.2

14. Midterm15 Least Square Algorithm

16. QR Decomposition

17-18 no classes spring break

19. Hubs and AuthorityTheory 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