Linear Algebra Book Recommendation like Tao's Analysis

I am looking for a book that explains Linear Algebra, where it is build from axioms to higher level of Linear Algebra. It does not have to be a book on elementary level.

As example from other fields, Tao's Analysis would be perfect example.

Do you know if there is a book which is in similar manner as Tao's Analysis?

Thanks

Solution 1:

A few suggestions:

1) Linear Algebra, Friedberg, Insel and Spence. This was my textbook for advanced undergraduate. It starts with the vector space axioms, moves on to linear transformations and builds the rest of the theory from there. Probably the gentlest introduction to the subject in this list.

2) The Linear Algebra a Beginning Graduate Student Ought to Know by Golan. This was my textbook for postgrad, starts with some field theory (more as preliminaries) and the moves on to the vector space axioms and beyond. The treatment leans more towards abstract algebra.

3) Advanced Linear Algebra by Roman. Currently working through this. Has a detailed preliminary section with some abstract algebra. Starts with vector space axioms and moves on to linear transformations and beyond. This book includes modules over rings, etc. so even more understanding of abstract algebra required...but I like the way this is used to develop the treatment of canonical forms later on in the book.