2 Schedule 1. Introduction 2. Points vs vector (distance, balls, sphere) Chapter 13. Divide and Conquer: Algorithms for Near Neighbor ProblemHandout (section)
3 4. Hyperplanes Chapter 2 Ray intersections Lines By linear equations By two pointsWhen does a line passing the originIntersection of two linesMatrix and algebraic approach (two variables and two equations)
4 3D Ray and mirrors Planes in three dimensions By linear equations By three pointsWhen 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-dimensionsBy linear equationsBy n pointsWhen does a hyperplane passing through the originIntersection 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 vectorsVisualize hyperplane is hard, so you might eventually like the column pictures.
8 Repeated the questions Row pictures: n hyperplanes meets at a single pointsColumn pictures: combines n vectors to produce another vector
9 Gaussian Elimination Gaussian Elimination in 2 dimensions example PicturesPivotsMultipliersUpper triangular matrixBack substitution
10 Two dimensions Unique solution No solution Infinitely many solutions What if the pivot is 0!!!
11 3D Gaussian Elimination in 3 dimensions examplePicturesPivotsMultipliersUpper triangular matrixBack substitutionCan be extended to any dimensions
12 5. Gaussian Elimination (General form) Matrix AlgebraMatrix additionScalar times a matrixMatrix multiplication(dimensions have to agree)Associative lawNon commutative law