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CS232

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**Schedule 1. Introduction 2. Points vs vector (distance, balls, sphere)**

Chapter 1 3. Divide and Conquer: Algorithms for Near Neighbor Problem Handout (section)

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**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)

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**3D Ray and mirrors Planes in three dimensions By linear equations**

By three points When does a plane passing the origin

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**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

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**Matrix Form What is a matrix? Matrix vector multiplication**

(inner product after all) Matrix form of intersection of n hyperplanes --- system of linear equations?

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**Column Picture: combination of vectors**

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

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**Repeated the questions**

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

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**Gaussian Elimination Gaussian Elimination in 2 dimensions example**

Pictures Pivots Multipliers Upper triangular matrix Back substitution

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**Two dimensions Unique solution No solution Infinitely many solutions**

What if the pivot is 0!!!

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**3D Gaussian Elimination in 3 dimensions**

example Pictures Pivots Multipliers Upper triangular matrix Back substitution Can be extended to any dimensions

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**5. Gaussian Elimination (General form)**

Matrix Algebra Matrix addition Scalar times a matrix Matrix multiplication (dimensions have to agree) Associative law Non commutative law

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**Gaussian Elimination (General form)**

Identity matrix Elimination matrix

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Permutation Matrix

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**Matrix algebra (General form)**

All the laws (page 58 – 59)

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**Complexity of Matrix Multiplication**

cube

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Block Multiplication

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**Strassen’s Fast Matrix Mulplication**

Divide and conquer

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**6. Inverse Matrix 7 Quiz 1 8 LU factorization**

Rest of chapter 2

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**9. Two dimensional convex Hull**

From the handout Convex combination

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**10. Algorithms for Null space**

3.1 – 3.3

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**11. Complete Linear Solver**

3.4 – 3.6

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**12. No class 13 Geometric Projection**

4.1 – 4.2

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**14. Midterm 15 Least Square Algorithm**

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16. QR Decomposition

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**17-18 no classes spring break**

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**19. Hubs and Authority Theory for Webs Hand out**

Understanding webs How Google works

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20. Simplex and its Volume Chapter 5

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**21. Determinants: Matrix Representation of volume**

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**22. Eivenvalue problem and Spectral Geometry**

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23. Quiz 2

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24. Diagonalization

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25. Quadratic Shapes Positive Definite matrices

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**26. Dimensional Reduction**

Singular value Decomposition

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**27. Application: Computer Graphics**

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**28. Spherical Geometry Points on sphere Caps**

Stereographic Transformation

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**29. Geometric Transformation**

Chapter 7

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**30 Geometric Transformation**

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**31. Triangulations and Voronoi Diagram**

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