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.