Linear Algebra Done Right – Sheldon Axler

To be honest, I haven’t encountered quite this many math books with a title this pretentious. The author acknowledges this book is meant to be the second exposure to linear algebra, focused more on abstract vector spaces and linear mappings. The author explains in the preface that he chose the title because he pushes the notion of a determinant to the end of the book, whereas many linear algebra books introduce them early on and use them in proofs. He wanted to give proofs that are “more intuitive”. He does try to do so, attempting to make a readable text. However, he doesn’t motivate the proofs very well, simply diving right into them. Even if the student has been exposed to proofs before, it still helps to give some hint of why you started a proof (especially an algebra proof) the way you did. He does this occasionally, and sometimes offers side notes, which are helpful, but a brief discussion prior to starting rather than at the end is preferred. The exercises are decent, and I like that there are some that ask the reader to prove or disprove, rather than just prove. I do like the way he defines an eigenvalue, and it gives a more intuitive picture as to how they relate to invariant subspaces and linear independence before moving to the matrix way of treating eigenvalues. One thing I do not like is his treatment of the Jordan Canonical Form. It doesn’t really explain how to compute them, and the treatment is far too short. A student with a pretty strong introduction to matrix theory and linear algebra will do ok, but find the beginning of the text entirely too easy. A student for whom the beginning of the text is the appropriate difficulty will likely get lost in the later chapters. I don’t recommend this book for self-study, unless one has a strong mathematical background. Teaching out of this text will likely require a bit of supplementary material on the part of the instructor or careful study of the book prior to designing to course to gauge student levels. The book starts out reasonably readable, but then the later chapters, which would benefit the most from slowing down a bit, seem very rushed. I used this text as part of my graduate linear algebra course during my PhD studies, and found it simultaneously too easy (in the beginning chapters) and then wholly confusing (in the later chapters); I had to seek supplementary material when learning some of the concepts for the first time.

-Rachel Traylor, Ph.D. 

Review

Prerequisites

linear algebra, matrix theory, proofs


Topics Covered

  • vector spaces introduction
    • direct sums
  • spanning sets, linear independence, basis, and dimensions
  • linear maps
    • null spaces, ranges 
    • matrix of a linear map
    • invertibility
  • eigenvalues and eigenvectors
    • introduced via invariant subspaces
    • polynomials on operators
    • upper triangular matrices and invariant subspaces
  • inner product spaces, norms, and orthonormal bases
    • orthogonal projections
    • linear functionals and adjoints
  • self-anoint and normal operators 
  • the spectral theorem
  • isometries
  • singular-value decompositions
  • generalized eigenvectors, nilpotent operators, and the characteristic polynomial
  • minimal polynomial
  • Jordan basis
  • trace and determinant of a matrix

Attributes

Difficulty 2-5 (depending on the chapter)
Good for teaching? 3
Proof Quality 3
Quality of Exercises 3
Self-Study 2
Readability 3