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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)
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