Advanced Linear Algebra

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1. mathaftermathJul 2016
   My favorite LA books are Linear Algebra by Friedberg/Insel[0] which is a combination of Axler style book with more computation oriented one (Terry Tao has a set of lectures based off this book). Another one I like is Modern Intro To LA by Henry Ricardo[1] which implicitly introduced me to Replacement theorem which is really overlooked in a ton of LA books. Again, this book's a rigorous mixture of both theory and computation done very well. If you've never seen higher level math before, there's Linear Algebra: Gateway to Mathematics by Robert Messer[2]. It has tons of commentary about elementary set theory and proof techniques along the way. Whenever someone mentions Axler's book, someone else brings up Treil's book. But there's a third one in the same league/group which is Linear Algebra: An Introduction to Abstract Mathematics by Robert Valenza[3]. Other favorites are Coding the Matrix by Philip Klein[4] for Python aficionados and Linear Algebra Through Geometry by Banchoff/Wermer[5] for those who like geometry.

   If you are way beyond all this, you can still pick up new things from Advanced Linear Algebra by Steven Roman[6].

   [0] https://www.amazon.com/Linear-Algebra-Stephen-H-Friedberg-eb...

   [1] https://www.amazon.com/Modern-Introduction-Linear-Algebra/dp...

   [2] https://www.amazon.com/Linear-Algebra-Mathematics-Robert-Mes...

   [3] https://www.amazon.com/Linear-Algebra-Introduction-Mathemati...

   [4] https://www.amazon.com/Coding-Matrix-Algebra-Applications-Co...

   [5] https://www.amazon.com/Algebra-Through-Geometry-Undergraduat...

   [6] https://www.amazon.com/Advanced-Linear-Algebra-Graduate-Math...

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2. nppOct 2010
   Advanced Linear Algebra, Roman; Linear Algebra, Hoffman & Kunze; Matrix Analysis, Horn & Johnson; Principles of Mathematical Analysis, Rudin; Real Analysis, Royden.

   Miscellaneous comments:

   - Reading pure abstract algebra (e.g. Dummit & Foote) isn't a good use of time if you intend to go into statistics, since it only shows up in a few very special subareas. If you decide to go into one of these areas, you can learn this later.

   - More advanced books on linear algebra usually emphasize the abstract study of vector spaces and linear transformations. This is fine, but you also need to learn about matrix algebra (some of which is in that Horn & Johnson book) and basic matrix calculus, since in statistics, you'll frequently be manipulating matrix equations. The vector space stuff generally does not help with this, and this material isn't in standard linear algebra books. (Similarly, you should learn the basics of numerical linear algebra and optimization -- convex optimization in particular shows up a lot in statistics.)

   - People have different opinions on books like Rudin, but you need to learn to read material like this if you're going into an area like probability. It's also more or less a de facto standard, so it is worth reading partly for that reason as well. So read Rudin/Royden (or equivalent, there are a small handful of others), but supplement them with other books if you need (e.g. 'The Way of Analysis' is the complete opposite of Rudin in writing style). It helps to read a few different books on the same topic simultaneously, anyway.

   - Two books on measure-theoretic probability theory that are more readable than many of the usual suspects are "Probability with Martingales" by Williams and "A User's Guide to Measure-Theoretic Probability" by Pollard. There is also a nice book called "Probability through Problems" that develops the theory through a series of exercises.

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