What Makes Jim Hefferon's Linear Algebra a Must-Read?

[bcrowell]

• Author: Jim Hefferon
• Title: Linear Algebra
• Download Link: http://joshua.smcvt.edu/linalg.html
• Prerequisities:

Table Of Contents:

[LIST]
[*] Linear Systems
[LIST]
[*] Solving Linear Systems
[LIST]
[*] Gauss’s Method
[*] Describing the Solution Set
[*] General=Particular+Homogeneous
[/LIST]
[*] Linear Geometry
[LIST]
[*] Vectors in Space
[*] Length and Angle Measures
[/LIST]
[*] Reduced Echelon Form
[LIST]
[*] Gauss-Jordan Reduction
[*] The Linear Combination Lemma
[/LIST]
[*] Computer Algebra Systems
[*] Input-Output Analysis
[*] Accuracy of Computations
[*] Analyzing Networks
[/LIST]
[*] Vector Spaces
[LIST]
[*] Definition of Vector Space
[LIST]
[*] Definition and Examples
[*] Subspaces and Spanning Sets
[/LIST]
[*] Linear Independence
[LIST]
[*] Definition and Examples
[/LIST]
[*] Basis and Dimension
[LIST]
[*] Basis
[*] Dimension
[*] Vector Spaces and Linear Systems
[*] Combining Subspaces
[/LIST]
[*] Fields
[*] Crystals
[*] Voting Paradoxes
[*] Dimensional Analysis
[/LIST]
[*] Maps Between Spaces
[LIST]
[*] Isomorphisms
[LIST]
[*] Definition and Examples
[*] Dimension Characterizes Isomorphism
[/LIST]
[*] Homomorphisms
[LIST]
[*] Definition
[*] Range space and Null space
[/LIST]
[*] Computing Linear Maps
[LIST]
[*] Representing Linear Maps with Matrices
[*] Any Matrix Represents a Linear Map
[/LIST]
[*] Matrix Operations
[LIST]
[*] Sums and Scalar Products
[*] Matrix Multiplication
[*] Mechanics of Matrix Multiplication
[*] Inverses
[/LIST]
[*] Change of Basis
[LIST]
[*] Changing Representations of Vectors
[*] Changing Map Representations
[/LIST]
[*] Projection
[LIST]
[*] Orthogonal Projection Into a Line
[*] Gram-Schmidt Orthogonalization
[*] Projection Into a Subspace
[/LIST]
[*] Line of Best Fit 
[*] Geometry of Linear Maps
[*] Magic Squares
[*] Markov Chains
[*] Orthonormal Matrices
[/LIST]
[*] Determinants
[LIST]
[*] Definition
[LIST]
[*] Exploration
[*] Properties of Determinants
[*] The Permutation Expansion
[*] Determinants Exist
[/LIST]
[*] Geometry of Determinants
[LIST]
[*] Determinants as Size Functions
[/LIST]
[*] Laplace’s Expansion
[LIST]
[*] Laplace’s Expansion Formula
[/LIST]
[*] Cramer’s Rule
[*] Speed of Calculating Determinants
[*] Chiò’s Method
[*] Projective Geometry
[/LIST]
[*] Similarity
[LIST]
[*] Complex Vector Space
[LIST]
[*] Review of Factoring and Complex Numbers
[*] Complex Representations
[/LIST]
[*] Similarity
[LIST]
[*] Definition and Examples
[*] Diagonalizability
[*] Eigenvalues and Eigenvectors
[/LIST]
[*] Nilpotence
[LIST]
[*] Self-Composition
[*] Strings
[/LIST]
[*] Jordan Form
[LIST]
[*] Polynomials of Maps and Matricess
[*] Jordan Canonical Form
[/LIST]
[*] Method of Powers
[*] Stable Populations
[*] Page Ranking
[*] Linear Recurrences
[/LIST]
[*] Appendix
[LIST]
[*] Propositions
[*] Quantifiers
[*] Techniques of Proof 
[*] Sets, Functions, and Relations
[/LIST]
[/LIST]

 

[bcrowell]

This is an excellent book. The fact that it's free is just an added bonus.

One thing that makes this book very different from the undergraduate math texts I used is the many interesting applications. Some of these are in separate sections, and some are interspersed throughout the text. The physics applications -- such as crystals, electrical networks, and dimensional analysis -- are excellent. Some seem like they might be a little on the difficult side for students with weaker preparation, but it is of course up to the instructor which ones to cover. It's a measure of the quality of the book that I was intrigued by the applications that were outside my specialty, such as voting paradoxes. When's the last time you found yourself getting interested in a textbook?

Many of the homework problems relate directly to these real-life applications. This is in welcome contrast to the usual, dreary set of "drill and kill" problems without any real context. The drudgery is also reduced by the explicit introduction of computer algebra systems in the first chapter. Many of the problems explicitly state that they are to be solved on a computer, and the assumption that the students will use computers has also allowed Hefferon to include many realistic problems that result in larger matrices than could be handled by hand.

In some ways, this book strikes me as more advanced than the ones used in my lower-division course. Says Hefferon,

The courses at the start of most mathematics programs work at having students correctly apply formulas and algorithms, and imitate examples. Later courses require some mathematical maturity: reasoning skills that are developed enough to follow different types of proofs, a familiarity with the themes that underly many mathematical investigations like elementary set and function facts, and an ability to do some independent reading and thinking, Where do we work on the transition?

Linear algebra is an ideal spot...

I was especially interested in the treatment of determinants, since I clearly recalled how the text used in my own undergraduate linear algebra course had introduced them abruptly and without motivation. I only really felt that I understood what a determinant was once I learned that it could be interpreted as the product of the eigenvalues. Hefferon takes an interesting in-between approach. He doesn't do eigenvalues until after determinants, but he doesn't just introduce determinants through a deus ex machina either. Instead, he discusses the invertibility of 1x1, 2x2, and then 3x3 matrices, and develops the concept naturally and straightforwardly. Three cheers!

All in all, it's hard for me to imagine why anyone would go on using linear algebra texts that are not free information when there's a free book as good as this one.

 

[Sankaku]

I have not read all the way through this textbook, but I did use portions of it heavily as a supplement to other assigned linear algebra books. I am very impressed by the high-quality of the writing and presentation. Because of its open-source nature, Hefferon has continued to add material and corrections, making it even better than when I first used it.

Its only drawback is that it is probably a just little too advanced at the beginning for someone who has not had a proof-based course before.

 

[mathwonk]

a nice low level book. not in the league with "linear algebra done wrong" by treil, or the classics by hoffman and kunze or shilov, but a good intro. just don't stop here.