Setting the Stage -- Linear Algebra over the Complex Numbers -- Complex Scalar Product Spaces (a.k.a. Hilbert Spaces) -- Lie Groups and Lie Group Representations -- New Representations from Old -- Irreducible Representations and Invariant Integration -- Representations and the Hydrogen Atom -- The Algebra so(4) Symmetry of the Hydrogen Atom -- The Group SO(4) Symmetry of the Hydrogen Atom -- Projective Representations and Spin -- Independent Events and Tensor Products.This is a textbook for a senior-level undergraduate course for math, physics and chemistry majors. This one course can play two different but complimentary roles: it can serve as a capstone course for students finishing their education, and it can serve as motivating story for future study of mathematics. Some textbooks are like a vigorous regular physical training program, preparing people for a wide range of challenges by honing their basic skills thoroughly. Some are like a series of day hikes. This book is more like an intended trek to a particularly beautiful goal. We’ll take the easiest route to the top, and we’ll stop to appreciate local flora as well as distant peaks worthy of the vigorous training one would need to scale them. Advice to the Student: This book was written with many different readers in mind. Some will be mathematics students interested to see a beautiful and powerful application of a “pure” mathematical subject. Some will be students of physics and chemistry curious about the mathematics behind some tools they use, such as spherical harmonics. Because the readership is so varied, no single reader should be put off by occasional digressions aimed at certain other readers.