Reflections and Projective Linear Algebra - Universal Hyperbolic Geometry

This lecture develops projective linear algebra to represent reflections in universal hyperbolic geometry. It covers null points, 2×2 projective matrices, trace and determinant, the sl(2) Lie algebra, and reflection matrix theorems.

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