Linear Algebra

Linear Algebra Done Right

The title says it all, "Linear Algebra Done Right."  This is great for a second course in linear algebra but can be used as a first course for those interested in learning linear algebra "by proofs" from the start.

Finite Dimensional Vector Spaces

This is an awesome theoretical introduction to vector spaces!  Paul Halmos's classic is beyond repute.

Abstract Algebra

A Book of Abstract Algebra

"A book of Abstract Algebra" is a great, concise introductory book on abstract algebra.  It's great for self-learning (and even has some answers in the back).  NB: To get the most out of this book and hit the "standard" introductory topics, you should do the exercises since some significant "lecture" topics have been moved to the exercises.  

Algebra

This is a masterpiece reference tome.  Lang's "Algebra" is a classic reference for graduate level algebra.

Real Analysis

Understanding Analysis

This is a nice, user-friendly introduction to real analysis.  "Understanding Analysis" is a great Springer UTM (Undergraduate Texts in Mathematics) book. 

Introduction to Analysis

A less expensive but great alternative is this one from Dover publications.  The treatment of metric space topology is top notch for someone first learning the subject.

Physics

Classical Mechanics

This is a great introduction to classical mechanics!  Taylor takes the time to explain things clearly.  The treatment of the Lagrangian, in particular, is worth it.  This book is aimed at physics majors taking a mechanics course.

No-Nonsense Classical Mechanics

This is another great introduction to classical mechanics (from the author of the "Physics from Symmetry").  It focuses on key concepts and ideas and gives you a good overview of mechanics.  Everyone and their dog seems to love the treatment of Noether's Theorem.