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Presentation on theme: "CS232."— 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. 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)

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

5 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

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

17 Block Multiplication

18 Strassen’s Fast Matrix Mulplication
Divide and conquer

19 6. Inverse Matrix 7 Quiz 1 8 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 class 13 Geometric Projection
4.1 – 4.2

24 14. Midterm 15 Least Square Algorithm

25 16. QR Decomposition

26 17-18 no classes spring break

27 19. Hubs and Authority Theory for Webs Hand 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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