Reflections and Projective Linear Algebra - Universal Hyperbolic Geometry

Overview

Explore the fundamental symmetries of hyperbolic geometry in this 50-minute lecture on reflections and projective linear algebra. Delve into the concept of reflection in a point and its effect on null points. Learn about the projective parametrization of the circle and its association with 2x2 projective matrices. Develop a foundation in projective linear algebra, focusing on vectors and matrices up to scalars. Examine the null point reflection formula, the projective matrix of a point, and the reflection matrix theorem. Investigate the point/matrix correspondence, the sl(2) Lie algebra, and tackle exercises on null points and reflection matrix conjugation. Gain insights into this advanced mathematical topic through examples, proofs, and corollaries presented by Professor N.J. Wildberger.

Syllabus

CONTENT SUMMARY: pg 1: @ Reflection in a is determined by its action on null points
pg 2: @ 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: @ Linear algebra in 2 dim's in a nutshell; projective rather than affine linear algebra;
pg 4: @ Projective linear algebra in 2 dim's
pg 5: @ 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: @ Reflection matrix theorem; example
pg 7: @ Point/matrix correspondence; sl2 Lie algebra; null point zero determinant exercise 15.1a
pg 8: @33:36 exercise 15.2; Reflection matrix conjugation theorem; pg 9: example of Reflection matrix conjugation theorem
pg 10: @ proof of Reflection matrix conjugation theorem
pg 11: @ 2 Corollaries THANKS to EmptySpaceEnterprise