Linear Algebra by Friedberg, Insel and Spence

[micromass]

• Author: Stephen Friedberg, Arnold Insel, Lawrence Spence
• Title: Linear Algebra
• Amazon link https://www.amazon.com/dp/0130084514/?tag=pfamazon01-20
• Prerequisities: Being acquainted with proofs and rigorous mathematics. Knowing what matrices and determinants are, is also helpful.
• Level: Undergrad

Table of Contents:

[LIST]
[*] Preface
[*] Vector Spaces
[LIST]
[*] Introduction
[*] Vector Spaces
[*] Subspaces
[*] Linear Combinations and Systems of Linear Equations
[*] Linear Dependence and Linear Independence
[*] Bases and Dimension
[*] Maximal Linearly Independent Subsets
[/LIST]
[*] Linear Transformations and Matrices
[LIST]
[*] Linear Transformations, Null Spaces, and Ranges
[*] The Matrix Representation of a Linear Transformation
[*] Composition of Linear Transformations and Matrix Multiplication
[*] Invertibility and Isomorphisms
[*] The Change of Coordinate Matrix
[*] Dual Spaces
[*] Homogeneous Linear Differential Equations with Constant Coefficients
[/LIST]
[*] Elementary Matrix Operations and Systems of Linear Equations
[LIST]
[*] Elementary Matrix Operations and Elementary Matrices
[*] The Rank of a Matrix and Matrix Inverses
[*] Systems of Linear Equations - Theoretical Aspects
[*] Systems of Linear Equations - Computational Aspects
[/LIST]
[*] Determinants
[LIST]
[*] Determinants of Order [itex]2[/itex]
[*] Determinants of Order [itex]n[/itex]
[*] Properties of Determinants
[*] Summary - Important Facts about Determinants
[/LIST]
[*] Diagonalization
[LIST]
[*] Eigenvalues and Eigenvectors
[*] Diagonalizability
[*] Matrix Limits and Markov Chains
[*] Invariant Subspaces and the Cayley-Hamilton Theorem
[/LIST]
[*] Inner Product Spaces
[LIST]
[*] Inner Products and Norms
[*] The Gram-Schmidt Orthogonalization Process and Orthogonal Complements
[*] The Adjoint of a Linear Operator
[*] Normal and Self-Adjoint Operators
[*] Unitary and Orthogonal Operators and Their Matrices
[*] Orthogonal Projections and the Spectral Theorem
[*] Bilinear and Quadratic Forms
[*] Einstein's Special Theory of Relativity
[*] Conditioning and they Rayleigh Quotient
[*] The Geometry of Orthogonal Operators
[/LIST]
[*] Canonical Forms
[LIST]
[*] Generalized Eigenvectors
[*] Jordan Canonical Form
[*] The Minimal Polynomial
[*] Rational Canonical Form
[/LIST]
[*] Appendices
[LIST]
[*] Sets
[*] Functions
[*] Fields
[*] Complex Numbers
[*] Polynomials
[/LIST]
[*] Answers to Selected Exercises
[*] List of Frequently Used Symbols
[*] Index of Theorems
[*] Index
[/LIST]

 

[mathwonk]

I found this book very clear and used it for my upper level undergrad course. I have some differences of philosophy with them, but you can learn a lot here. I wish there were a recommendation stronger than loight and lighter than strong in this case. I.e. I give it a B not C.

 

[Sankaku]

I like this book, which I now use as my main linear algebra reference (with Hoffman & Kunze close behind). My second course in LA was from Axler. I like Axler's style, but he keeps to a very narrow (theoretical) path through Linear Algebra, which makes his book not so good for general reference.

Friedberg et al. give a well-organized and rigorous (at a basic level) summary of most of the linear algebra that undergrads need to know. Pretty much everything is here, but it will appeal more to mathematicians than, say, engineers.

The only drawback is that the hardback is stupidly expensive, and the paperback international (Indian) edition is printed on very low quality paper (beware Prentice Hall's "Eastern Economy Editions").

 

[mathwonk]

to be specific, i recall i was puzzled that they chose to ignore the powerful role of the minimal polynomial for much of this book in developing the structure of linear maps. I seem to recall that as a stated principle of theirs, that they chose to treat linear algebra without relying on facts about polynomials, for some reason.

In my own class notes which I wrote whole teaching the course, and posted free on my website, you will see how easily the concept of minimal polynomial leads to precise structure theorems more easily and naturally in my opinion than the approach used here.

I thought this book has, in contrast to the claims of some negative reviews at Amazon, many helpful numerical examples and problems, as well as clear explanations and proofs. I just felt that the proofs could be made conceptually easier by using the minimal polynomial more fully.