Linear Algebra Elementary Linear Algebra by Anton

[micromass]

• Author: Howard Anton
• Title: Elementary Linear Algebra
• Amazon Link: https://www.amazon.com/dp/0470458216/?tag=pfamazon01-20
• Prerequisities: High-school mathematics
• Level: Undergrad

Table of Contents:

[LIST]
[*] Systems of Linear Equations and Matrices
[LIST]
[*] Introduction to systems of Linear Equations
[*] Gaussian Elimination
[*] Matrices and Matrix Operations
[*] Inverses; Algebraic Properties of Matrices
[*] Elementary Matrices and a Method for Finding [itex]A^{-1}[/itex]
[*] More on Linear Systems and Invertible Matrices
[*] Diagonal, Triangular, and Symmetric Matrices
[*] Application: Applications of Linear Systems
[*] Application: Leontief Input-Output Models
[/LIST]
[*] Determinants
[LIST]
[*] Determinants by Cofactor Expansion
[*] Evaluating Determinants by Row Reduction
[*] Properties of Determinants; Cramer's Rule
[/LIST]
[*] Euclidean Vector Spaces
[LIST]
[*] Vectors in 2-Space, 3-Space, and n-Space
[*] Norm, Dot Product, and Distance in R^n
[*] Orthogonality
[*] The Geometry of Linear Systems
[*] Cross Product
[/LIST]
[*] General Vector Spaces
[LIST]
[*] Real Vector Spaces
[*] Subspaces
[*] Linear Independence
[*] Coordinates and Basis
[*] Dimension
[*] Change of Basis
[*] Row Space, Column Space, and Null Space
[*] Rank, Nullity, and the Fundamental Matrix Spaces
[*] Matrix Transformations from R^n to R^m
[*] Properties of Matrix Transformations
[*] Geometry of Matrix Operators in R^2
[*] Dynamical Systems and Markov Chains
[/LIST]
[*] Eigenvalues and Eigenvectors
[LIST]
[*] Eigenvalues and Eigenvectors
[*] Diagonalization
[*] Complex Vector Spaces
[*] Application: Differential Equations
[/LIST]
[*] Inner Product Spaces
[LIST]
[*] Inner Products
[*] Angle and Orthogonality in Inner Product Spaces
[*] Gram-Schmidt Process; QR-Decomposition
[*] Best Approximation; Least Squares
[*] Application: Least Squares Fitting to Data
[*] Application: Function Approximation; Fourier Series
[/LIST]
[*] Diagonalization and Quadratic Forms
[LIST]
[*] Orthogonal Matrices
[*] Orthogonal Diagonalization
[*] Quadratic forms
[*] Optimization Using Quadratic Forms
[*] Hermitian, Unitary, and Normal Matrices
[/LIST]
[*] Linear Transformations
[LIST]
[*] General Linear Transformations
[*] Isomorphism
[*] Compositions and Inverse Transformations
[*] Matrices for General Linear Transformations
[*] Similarity
[/LIST]
[*] Numerical Methods
[LIST]
[*] LU-Decompositions
[*] The Power Method
[*] Application: Internet Search Engines
[*] Comparison of Procedures for Solving Linear Systems
[*] Singular Value DEcomposition
[*] Application: Data Compression Using Singular Value Decomposition
[/LIST]
[*] Appendix: How to Read Theorems
[*] Appendix: Complex Numbers
[*] Answers to Exercises
[*] Index
[/LIST]

 

[Fredrik]

I have only read the 6th edition. I think it's an exceptionally well written book. Everything is explained clearly and the proofs are very easy to follow. However, it really bothers me that it doesn't introduce linear transformations until chapter 7, starting on page 295. Another problem is that it doesn't introduce complex vector spaces until chapter 10, starting on page 477. Because of these things, I can only "lightly" recommend it.

To a physics student, nothing in linear algebra is more important than linear operators (=transformations) on complex vector spaces. (In quantum mechanics, some of those operators represent measuring devices).