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Linear Algebra: A First Course with Applications

Larry E. Knop
Chapman & Hall/CRC
Publication Date: 
Number of Pages: 
Textbooks in Mathematics 2
[Reviewed by
P. N. Ruane
, on

How many books on introductory linear algebra exceed the 725 pages of this one? At a guess, none. Of such books, how many exclude coverage of standard topics such as the Gram-Schmidt process, quadratic forms, hermitian and unitary matrices and Jordan forms etc? In my estimation, only a strict minority. Is there any other book that concentrates so heavily on establishing the concepts the basic concepts of linear algebra? I very much doubt it.

By way of illustration, the first four hundred pages of Larry Knop’s book are devoted to eliciting the concept of vector space. The treatment begins with revision of elementary vector algebra in R2 and generalises such properties to Rn. In fact, up to page 146, everything is kept at an intuitively practical level prior to the emergence of vector space axioms. Subsequently, the notions of subspace, spanning sets, linear independence, and bases are carefully elicited and not until page 391 is it felt safe to introduce the ideas and methodology of linear transformations.

So, after 520 pages of lively narrative, and hundreds of exercises and examples, one encounters the two remaining chapters on Determinants and Eigenspaces respectively. The remaining sixty pages are allocated to the solutions of selected exercises.

I have to say that I’m very much in favour of this approach to the teaching of linear algebra because, recalling my own introduction to the subject (about 200 years ago, it seems), and recalling courses that I have designed and taught to undergraduates, the mistake was to proceed too rapidly through the early stages, and move on to more advanced techniques too quickly.

Consequently, the thickness of this book is partly due to the nature of its expositional narrative, which is almost conversational, and it aims to get students to read more independently and of their own volition (this always begins with an element of directed reading of course).

Another appealing feature of Larry Knop’s book is the range and quality of the illustrations and the myriad of applications that appear from the earliest chapters onwards.

To summarize, by limiting the mathematical scope of this book, and by carefully smoothing the path from the particular to the general, the author has provided an alternative approach to linear algebra that will be of interest to many of  those who currently teach it.

The very first lecture on linear algebra that Peter Ruane received as a student began with a delineation of the axioms and a range of theorems with no motivational material whatever (and certainly no applications).


Preface for the Instructor 
A Little Logic 
Logical Foundations
Logical Equivalences
Sets and Set Notation
An Introduction to Vector Spaces 
The Vector Space R2—The Basics
The Vector Space R2—Beyond the Basics
The Vector Spaces Rn—The Basics
The Vector Spaces Rn—Beyond the Basics
The Vector Spaces Rn—Lines and Planes
Vector Spaces in General 
Vector Spaces: Setting the Rules
Vector Spaces: On the Wild Side
Subspaces and Linear Equations
Subspaces from Subsets
A Numerical Interlude—Systems of Linear Equations 
Solving Linear Systems
Systematic Solutions of Systems
Technology and Linear Algebra
The Structure of Vector Spaces 
Spanning Sets
Linear Independence
More on Linear Independence
Linear Independence and Span
Vector Space Bases
The Dimension of a Vector Space
Linear Transformations 
Transformation Fundamentals
Vector Space Isomorphisms
Linear Transformations and Matrices 
Matrix Representations of Transformations
Matrices and Associated Vector Spaces
Inverses in Matrix Multiplication
Elementary Matrices
An Introduction to Determinants
Properties of Determinants
Eigenvalues and Eigenvectors 
Eigenvalues, Eigenvectors, and Eigenspaces
More on Eigenvalues, Eigenvectors, and Eigenspaces
Forests, Digraphs, and PageRank
Answers to Selected Problems