# Good article (or book) about coordinate-dependent linear algebra, for those already familiar with coordinate-free aspects.

I have a decent understanding of coordinate-free linear algebra.

For example: (not-necessarily-finite-dimensional) vector spaces, linear transforms, (possibly infinite) products of vector spaces, (possibly infinite) coproducts, biproducts, free vector spaces, the concepts of "basis" and dimension, subspaces, quotient spaces, multilinear maps, tensor products of vector spaces, the tensor-hom adjunction, canonical self-enrichment of $\mathbf{Vect}_K$, and dual spaces.

At the same time, there's a lot of holes in my knowledge:

• Matrix normal forms are a subject I know almost nothing about

• If you say "positive definite matrix" I will stare at you blankly

• If you say "special linear group" I will stare at you blankly

• Matrix similarity / congruence / equivalence, um what?

• I feel like I have no understanding of bilinear mappings into $\mathbb{R}$, despite that they're "the same" as matrices. (Given a matrix $A,$ we get a bilinear mapping into $\mathbb{R}$ given by $y,x \mapsto y^T Ax.$ This process is an isomorphism of vector spaces.)

These are mainly things that can be looked up on wikipedia, of course (except for the last dot point), but its tough to get the "big picture" and/or the geometric meaning without the help of a good article or book.

Question. Can anyone recommend an article or book that specifically deals with coordinate-dependent linear algebra, such as matrices, in a sophisticated, mathematically-mature way, and which preferably takes abstract linear algebra for granted, and even uses it to help to express and clarify the coordinate dependent stuff?

• Nothing comes to mind, but one thought is that you could pick up some computational/numerical linear algebra. I begrudgingly TA'd for such a class once and was surprised to learn how much linear algebra I hadn't really learned. There are many books on this subject, and I wouldn't be qualified to recommend one any better than Amazon reviews, though. – Callus Mar 23 '15 at 8:28
• @Callus, thanks for the tip. – goblin Mar 23 '15 at 8:28

From Georges Elencwajg's description, Shilov's book sounds pretty good. I have a suggestion that mmight sound somewhat unorthodox: read Artin's book Algebra. Though not specifically addressed to linear algebra, it answers all of your questions (and you don't even have to read it all,) as well as relating it to the very important issues of group theory and abstract algebra. If you work through it, it can serve as a really good springboard for further work.

• Thanks. I actually had Artin's Algebra as my first textbook in abstract algebra, and did not find it to my taste; but, I suspect that was caused by a lack of mathematical maturity on my part. I'll definitely check it out again; perhaps I will even love it this time :) – goblin Mar 25 '15 at 2:12
• I also had Artin's book as the first textboook in algebra, and even though I was taking classes with Artin himself, I also didn't enjoy the book very much. I always found that it wasn't very straightforward, as it tilts away from the standard math exposition style. However, looking back, I repeatedly conclude that this book is an authentic goldmine. It explains the intuition behind everything, and relates topics that in a more pedantic exposition aren't obviously related. I definitely encourage re-reading it. – Artur Araujo Mar 25 '15 at 11:11
• Also, though I have an obvious bias, I feel like these points you raised were best understood in the context of geometry. – Artur Araujo Mar 25 '15 at 11:15

I recommend the book Linear Algebra by the great Russian mathematician Georgiĭ Shilov.
It has the unique feature of combining very abstract concepts like spectra, jets and representations of simple algebras with extremely down to earth computations involving matrices.
In Chapter 1 already determinants are defined, then applied to Cramer's Rule and the very explicit Laplace development of a determinant through minors of arbitrary order is explained.
Later you will discover an algorithm for determining the rank of a matrix, tensors with all their upper and lower indices, elementary divisors, Gram determinants, the method of least squares and all the material which causes you to "stare blankly"...
Let's not forget the icing on the cake: many exercises are provided with hints or answers at the end of the book. And the book is dirt cheap (as always with Dover): around \$10 !
In conclusion: a book in the best Russian tradition that Shilov practically wrote for you !

Edit
Browsing the Web I found this fine page on a course by George Melvin at Berkeley based on Shilov's book.
You will find there many exercise sheets, some quite advanced, but solved in detail.

• Thanks, I'll check it out. – goblin Mar 24 '15 at 9:13
• Seconded! Forgot about this book. Really good one. – Callus Mar 24 '15 at 23:29