### UnivHypGeom15: Reflections and projective linear algebra

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 pointspg 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 matrixpg 6: @26:37 Reflection matrix theorem; examplepg 7: @30:35 Point/matrix correspondence; sl(2) Lie algebra; null point zero determinant exercise 15.1apg 8: @33:36 exercise 15.2; Reflection matrix conjugation theorem; pg 9: 36:32 example of Reflection matrix conjugation theorempg 10: @41:43 proof of Reflection matrix conjugation theorempg? 11: @47:20 2 Corollaries (THANKS to EmptySpaceEnterprise)

Length:
50:29