What are the best texts on undergraduate linear algebra?

Question

I have recently finished a course in 'elementary linear algebra,' which entails basic systems of linear equations, in-depth study on matrices, the basics of vector space, inner product spaces, linear transformations, as well as a light discussion on eigenvectors and eigenvalues. The textbooks I referred to in this study were Larson's Elementary Linear Algebra and Anton's Elementary Linear Algebra.

I want to know basically what the next step is. I suppose a more advanced course in linear algebra or a move into abstract algebra would be logical, can anybody recommend any textbooks?

Answer 1

This book could be used at advanced undergraduate levels, and includes some abstract algebra (in a linear algebra context): http://link.springer.com/book/10.1007%2F978-94-007-2636-9

An older book, but also good: http://www.amazon.com/Matrices-Linear-Transformations-Edition-Mathematics/dp/0486663280

And maybe this one: http://www.amazon.com/Linear-Algebra-Edition-Stephen-Friedberg/dp/0130084514

Basically - I am studying linear algebra at honours level at the moment. In my final year of undergraduate the precribed book was Friedberg. For both honours levels courses Cullen was prescribed, but this year we are transitioning to a new textbook - the one by Golan - it approached the subject more from the abstract algebra side.

Answer 2

I recommend Alan Macdonald's introductory textbook Linear and Geometric Algebra, available from amazon.com. The author calls it sophomore level, and I would agree. This short, inexpensive little book contains a lot of material you'll have seen already, but it's presented from the non-traditional but increasingly popular perspective of geometric algebra. My experience is that geometric algebra provides a simple but amazingly powerful system for unifying many of the ideas of mathematical physics: scalars, vectors, k-vectors; outer (wedge) and inner (contraction) products; complex numbers; quaternionic, Pauli, and Dirac algebras; differential geometry; and much more. The topics of vector calculus and more are treated from a geometric algebra viewpoint in Macdonald's follow-up text, Vector and Geometric Calculus. LAGA and VAGC also introduce the reader to the free computer algebra system SymPy and its geometric algebra module GA.

The same viewpoint is taken in Geometric Algebra for Computer Science, by Dorst, Fontijne, & Mann. Although most of GACS is devoted to computer graphics, the initial chapters give a leisurely and well-written introduction to geometric algebra and its relationship to traditional presentations of linear algebra. GACS shows how projective geometry and conformal geometry can be encoded within the geometric algebra. I have severe reservations about GACS's discussion of the calculus associated with geometric algebra, however; their discussion is not up to the level displayed when discussing the geometric algebra, and I think that many of the calculus formulas displayed are incorrect. Get the revised edition of GACS; the first edition acquired an embarrassingly lengthy list of errata.

Considerably more advanced (graduate level) is Geometric Algebra for Physicists, by Doran and Lasenby. The number of ideas introduced therein is staggering.

For a general course in modern algebra I have two recommendations. Easier to read by far is John B. Fraleigh's A First Course in Abstract Algebra, which has gone through multiple editions. The second recommendation, not easy but still well written, is Algebra (Third Edition, AMS Chelsea Publishing, 1993), by MacLane and Birkhoff (not to be confused with A Survey of Modern Algebra, a classic earlier textbook by the same authors).