Modern Higher Algebra by A. Adrian Albert (1937, Dover/Cambridge). It covers both abstract algebra and linear algebra.
Most modern textbooks tend to approach linear algebra from geometric perspectives. Albert's text is one of the few that introduce the subject in a purely algebraic approach. With a solid algebraic foundation, the author was able to produce some elegant proofs or results that you don't often see in modern texts.
E.g. Albert's proof of Cayley-Hamilton theorem is essentially a one-liner. Some modern textbooks (such as Jim Hefferon's Linear Algebra) try to reproduce the same proof, but without setting up the proper algebraic framework, their proofs become much longer and much harder to understand. Readers of these modern textbooks may not realize that the theorem is simply a direct consequence of Factor Theorem for polynomials over non-commutative rings.
With only about 300 pages, the book's coverage is amazingly wide. When I first read the table of content, I was surprised to see that it not only covers undergraduate topics such as group, ring, field and Galois theory, but also advanced topics such as p-adic numbers. I haven't read the part on abstract algebra in details. However, if you want to re-learn linear algebra, this book may be an excellent choice.
For those who are reading this, let me stress that it is not that Prof. Hefferon's proof of Cayley-Hamilton theorem is bad (it is actually better than some really horrible proofs that appear in some well-received textbooks), but that Albert's treatment is superb --- it is far better than the treatments of the theorem in most modern textbooks, including Prof. Hefferon's. Also, I was certainly not commenting on the overall quality of Prof. Hefferon's book, and I thank him for offering his textbook for free.
Oh, forgive me, no offense taken. I should have put a smiley. I read your post with interest and shall check out the book.
(As you no doubt know, different books have different audiences. Before I wrote my Linear book, when I looked at the available textbooks I thought that there were low-level computational books that suited people with weak backgrounds, and high-level beautiful books that show the power of big, exciting, ideas. I had a room with students who were not ready for high. I wrote the book hoping that it could form part of an undergraduate program that deliberately worked at bringing students along to where they would be ready for such things. Naturally, with that mindset I read your post as meaning that the audience for the book you described is just different. Anyway, thanks again for the pointer.)
This is the piece I love about HN. A comment refers to an accomplished person and he/she happens to be right there. And either supplying a correction or taking the feedback positively.
Most modern textbooks tend to approach linear algebra from geometric perspectives. Albert's text is one of the few that introduce the subject in a purely algebraic approach. With a solid algebraic foundation, the author was able to produce some elegant proofs or results that you don't often see in modern texts.
E.g. Albert's proof of Cayley-Hamilton theorem is essentially a one-liner. Some modern textbooks (such as Jim Hefferon's Linear Algebra) try to reproduce the same proof, but without setting up the proper algebraic framework, their proofs become much longer and much harder to understand. Readers of these modern textbooks may not realize that the theorem is simply a direct consequence of Factor Theorem for polynomials over non-commutative rings.
With only about 300 pages, the book's coverage is amazingly wide. When I first read the table of content, I was surprised to see that it not only covers undergraduate topics such as group, ring, field and Galois theory, but also advanced topics such as p-adic numbers. I haven't read the part on abstract algebra in details. However, if you want to re-learn linear algebra, this book may be an excellent choice.