Linear Algebra Done Right by Axler

[micromass]

• Author: Sheldon Axler
• Title: Linear Algebra Done Right
• Amazon link https://www.amazon.com/dp/0387982582/?tag=pfamazon01-20
• Prerequisities: A first linear algebra course, proofs
• Level: Undergrad

Table of Contents:

[LIST]
[*] Preface to the Instructor
[*] Preface to the Student
[*] Acknowledgments
[*] Vector Spaces
[LIST]
[*] Complex Numbers
[*] Definition of Vector Space
[*] Properties of Vector Spaces
[*] Subspaces
[*] Sums and Direct Sums
[*] Exercises
[/LIST]
[*] Finite-Dimensional Vector Spaces
[LIST]
[*] Span and Linear Independence
[*] Bases
[*] Dimension
[*] Exercises
[/LIST]
[*] Linear Maps
[LIST]
[*] Definitions and Examples
[*] Null Spaces and Ranges
[*] The Matrix of a Linear Map
[*] Invertibility
[*] Exercises
[/LIST]
[*] Polynomials
[LIST]
[*] Degree
[*] Complex Coefficients
[*] Real Coefficients
[*] Exercises
[/LIST]
[*] Eigenvalues and Eigenvectors
[LIST]
[*] Invariant Subspaces
[*] Polynomials Applied to Operators
[*] Upper-Triangular Matrices
[*] Diagonal Matrices
[*] Invariant Subspaces on Real Vector Spaces
[*] Exercises
[/LIST]
[*] Inner-Product Spaces
[LIST]
[*] Inner Products
[*] Norms
[*] Orthonormal Bases
[*] Orthogonal Projections and Minimization Problems
[*] Linear Functionals and Adjoints
[*] Exercises
[/LIST]
[*] Operators on Inner-Product Spaces
[LIST]
[*] Self-Adjoint and Normal Operators
[*] The Spectral Theorem
[*] Normal Operators on Real Inner-Product Spaces
[*] Positive Operators
[*] Isometries
[*] Polar and Singular-Value Decompositions
[*] Exercises
[/LIST]
[*] Operators on Complex Vector Spaces
[LIST]
[*] Generalized Eigenvectors
[*] The Characteristic Polynomial
[*] Decomposition of an Operator
[*] Square Roots
[*] The Minimal Polynomial
[*] Jordan Form
[*] Exercises
[/LIST]
[*] Operators on Real Vector Spaces
[LIST]
[*] Eigenvalues of Square Matrices
[*] Block Upper-Triangular Matrices
[*] The Characteristic Polynomial
[*] Exercises
[/LIST]
[*] Trace and Determinant
[LIST]
[*] Change of Basis
[*] Trace
[*] Determinant of an Operator
[*] Determinant of a Matrix
[*] Volume
[*] Exercises
[/LIST]
[*] Symbol Index
[*] Index
[/LIST]

 

[Sankaku]

Axler's book has a very strong style to it and so might not appeal to everyone. It is squarely aimed at being a second course (which he says at the beginning), and so he really explores the theoretical underpinnings of Linear Algebra. I did such a course using it and enjoyed it thoroughly. I did many of the excercises and found almost all of them at just about the right level. Very few were highly difficult, but I went in with more background than most of the other students.

Its drawbacks (?) are that it does not deal with applications or matrix manipulations. It also doesn't deal with determinants until close to the end, and it treads a very "narrow" path through the wide field of Linear Algebra. This allows Axler to stay focussed on the theory, and will give you the right mind-set to go on to things like Abstract Algebra or Functional Analysis.

 

[Fredrik]

The selection of topics is excellent for a physics student who wants to learn about complex vector spaces and linear operators for quantum mechanics. That "narrow path" it takes through linear algebra is an advantage for us.

Some books focus on systems of linear equations and problems in geometry in the early parts of the book. Axler starts with vector spaces and linear operators. It also has a very nice section on polynomials.

People (including the author) say that it's intended as a second course, but it explains everything you need, so you don't need the knowledge from another linear algebra course. You may however need a bit more mathematical maturity than people usually have when they start taking their first linear algebra course. That's what people say anyway. Honestly, I don't see it. I found this book very easy to read, but that could be because I only used it to refresh my memory after the second time I had forgotten all about linear algebra.

 

[mathwonk]

I had this book on hand when I taught my own summer course on linear algebra, math 4050, whose notes are on my webpage. I did not read this book in advance nor use it as a model, but the table of contents reminds me of my own organization. In fact I took my approach to normal operators on a real vector space from this book. I recall from what I did read that it was very intelligent and clear. I did not agree entirely with his insistence on not using determinants, as they are about the only elementary computational tool one has for working examples. (Of course I guess one could use row and column operations on matrices of polynomials, but this amounts to one tool for computing determinants, just without saying so.) I do agree that determinants are not always enlightening as a tool for theoretical developments.

As others have said, Axler is not a stand alone source for linear algebra for a beginner. It is a clear explanation of the theory, but that is not enough to understand a subject thoroughly, you have to be able to do examples too. So it is a second book for the usual case of a student whose first course was mostly computational.

Other books I had on hand included Shilov; Halmos; Insel ,Spence, and Friedberg, Dummitt and Foote, Chi Han Sah, and possibly Hoffman and Kunze, oh and perhaps Sergei Treil's free (and highly recommended) online book "Linear algebra done wrong". When I teach I normally consult 8-10 books and choose what I think is the clearest presentation of each topic, or just invent my own presentation when I can.

 

[jasonRF]

I think that Axler is a good book on theoretical linear algebra. In my opinion it is appropriate for a second exposure to the subject for the vast majority of us. If I had this thrown at me my soph. year I would have drowned! Perhaps an honors math student would be fine using it for a first course?

I self-studied this book in an attempt to improve my mathematical maturity and familiarity with higher level linear algebra. The motivation was work: I found myself re-organized into a group that was quite mathematical and I was out of place. I must say I thought it would be fun, too! My background was a standard soph. semester of linear algebra for engineers (in the US), and I am not a mathematician. I found Axler is a good book and engineer like me; I learned quite a bit about linear algebra, and improved my ability to write proofs. I also worked through "analysis with an introduction to proof" by Lay as part of this effort. This has actually helped me in my work and I can now do higher level work than I could before.

Engineers should not be afraid of this book - just note that it reads like a real math book, not like a typical engineering book. Also, there are NO applications. As a complement to Axler, I look to the standard EE reference "linear algebra and its applications" by Strang; it is matrix based with almost no abstraction, but has lots of applications.

By the way, as an EE who uses complex numbers/spaces for everything I was unable to motivate myself to work through chapter 9 on operators on real vector spaces (I was also running out of steam at that point). Anyone have an interesting example of where that material is used?

jasonRF

 

[fourier jr]

By the way, as an EE who uses complex numbers/spaces for everything I was unable to motivate myself to work through chapter 9 on operators on real vector spaces (I was also running out of steam at that point). Anyone have an interesting example of where that material is used?

jasonRF

the laplacian $$\Delta = \frac{\partial^2}{\partial x^2} + \frac{\partial^2}{\partial y^2}$$ is an operator on a real vector space, whose kernel is the set of harmonic functions. it doesn't really help you calculate the solutions, but it might help conceptualize what that chapter is about. multiplication by a matrix is another example, where the kernel is simply the nullspace of the matrix.

 

[jasonRF]

the laplacian $$\Delta = \frac{\partial^2}{\partial x^2} + \frac{\partial^2}{\partial y^2}$$ is an operator on a real vector space, whose kernel is the set of harmonic functions. it doesn't really help you calculate the solutions, but it might help conceptualize what that chapter is about. multiplication by a matrix is another example, where the kernel is simply the nullspace of the matrix.

I understand your point - many models are analyzed in the context of real vector spaces. Your particular example is somewhat special of course- we are usually free to define an inner product, and for a reasonable choice of inner product the Laplacian is self-adjoint for typical boundary conditions. So I see this example as more like chapter 7 material on the spectral theorem for self-adjoint operators. Of course, there are many other operators that are NOT self-adjoint!

Thanks,

jason