Reflections and projective linear algebra | Universal Hyperbolic Geometry 15 | NJ Wildberger
Reflections are the fundamental symmetries in hyperbolic geometry. The reflection in a point interchanges any two null points on any line through the point. Using the projective parametrization of the circle, we associate to the reflecting point a 2x2 projective matrix. So we need to develop some basics about projective linear algebra: where we consider vectors and matrices but only up to scalars. CONTENT SUMMARY: pg 1: @00:10 Reflection in a is determined by its action on null points pg 2: @04:01 Reflections on null points; null point, join of null points (red, green, blue bilinear forms), a lies on L; null point reflection formula (the star formula) pg 3: @08:01 Linear algebra (in 2 dim's) in a nutshell; projective rather than affine linear algebra; pg 4: @15:22 Projective linear algebra (in 2 dim's) pg 5: @19:56 (the star formula rewrite); The projective matrix of the point a; trace and determinant; a is a null point when determinant of its projective matrix is zero; trace zero matrix pg 6: @26:37 Reflection matrix theorem; example pg 7: @30:35 Point/matrix correspondence; sl(2) Lie algebra; null point zero determinant exercise 15.1a pg 8: @33:36 exercise 15.2; Reflection matrix conjugation theorem; pg 9: 36:32 example of Reflection matrix conjugation theorem pg 10: @41:43 proof of Reflection matrix conjugation theorem pg 11: @47:20 2 Corollaries (THANKS to EmptySpaceEnterprise) ************ Research Gate page: https://www.researchgate.net/profile/... Blog: http://njwildberger.com/ Online courses at openlearning.com (currently Algebraic Calculus One): https://www.openlearning.com/courses/... Please join us for an exciting new approach to one of mathematics' most important subjects! Patreon: / njwildberger Your support would be much appreciated. Wild Egg Maths YT channel: https://www.youtube.com/channel/UCriF... Insights into Mathematics Playlists: • The Algebra of Boole, Logic and Circuit An... (31 videos) • Box Arithmetic: a new framework for Mathem... (18 videos) • Hypergroups and Diffusion Symmetry: an int... (6 videos) • Rational Trigonometry for maths, physics a... (4 videos) • Sociology and Pure Maths (44 videos) • Old Babylonian mathematics and Plimpton 322 (8 videos) • Math Foundations (226 videos) • Math Seminars N J Wildberger (26 videos) • Math History (ancient to modern) (45 videos) • Geometric Linear Algebra (43 videos) • Algebraic Topology (40 videos) • Universal Hyperbolic Geometry (55 videos) • Differential Geometry (34 videos) • Elementary Probability and Statistics (8 videos) • Math Terminology for Incoming Uni Students (9 videos) • Famous Math Problems ( 46 videos) • Elementary Mathematics Explained (K-6) (40 videos) • Ancient Mathematics and insights of Howard... (7 videos) • Wild West Banking: A mathematician goes We... (7 videos) • Playing Go: the ancient oriental board game (19 videos) • Maths and Music (21 videos) • Year 9 Mathematics (review fractions, deci... (10 videos) • Wild Trig: An introduction to Rational Tri... (94 videos) Wild Egg Maths Playlists: • Intro to Algebraic Calculus with Box Arith... (4 videos) • Classical to Quantum (for Members) (64 videos) • Solving Polynomial Equations and the Geode... (45 videos) • De Casteljau Bezier curves and associated ... (20 videos) • Exploring q-series (for Members) (8 videos) • Six: A mathematical exploration (9 videos) • Algebraic Calculus One: a new foundation f... (52 videos) • Advice to prospective research mathematici... (9 videos)
