Annotated Bibliography of Linear Algebra Books

Anton, Howard and Busby, Robert C., Contemporary Linear Algebra, Wiley, 2003, xviii + 633 pp.

An introductory text with lots of short historical notes and introductions to applications of linear
algebra, as well as information on numerical issues. The book focuses primarily on Rⁿ. The order of
topics is somewhat nonstandard—the notion of dimension first appears in Chapter 7 and the general
definition of a vector space does not appear until Chapter 9, the last chapter.

Axler, Sheldon, Linear Algebra Done Right (Second Edition), Springer-Verlag, 1997, paperback, xv + 251 pp.
Corrected printing, 2004.

A second course in linear algebra concentrating on real and complex vector spaces, linear maps, and
inner product spaces. Its central concern is the structure of a linear operator (a linear map from a vector
space to itself). The special feature of this book is that it proves the existence of eigenvalues for linear
maps on complex finite-dimensional vector spaces without using determinants. Much thought has gone
into this book’s clean, clear proofs. It is a good book for students to read and refer to on their own.

Bretscher, Otto, Linear Algebra with Applications (Second Edition), Prentice Hall, 2001, xv + 478 pp.

An introductory text with an unusual and interesting approach to determinants based on a pictorial
determination of the sign of a permutation. (The author would call it “counting the inversions of a
pattern”.) Cramer’s rule is also interpreted geometrically. Eigenvectors are introduced via linear dynamical
systems.

Broida, Joel G. and Williamson, S. Gill, A Comprehensive Introduction to Linear Algebra, Addison-Wesley,
1989, xiv + 734 pp.

Sophisticated linear algebra text emphasizing canonical forms, multilinear mappings and tensors, and
infinite-dimensional vector spaces. No coverage of numerical methods.

Brown, William C., A Second Course in Linear Algebra, Wiley, 1988, x + 264 pp..

This text is intended to provide material for a second one-term linear algebra course, pitched at the
senior or first-year graduate level. Written in theorem-proof style, it covers multilinear algebra, canonical
forms, normed linear vector spaces, and inner product spaces.

Cohn, Paul M., Elements of Linear Algebra, Chapman & Hall, 1994, paperback, xiii + 226 pp.

Concise, elegant introduction to linear algebra. A chapter on vectors precedes chapters on systems
of equations, matrices, and determinants. These are followed by chapters on coordinate geometry and
normal forms of matrices, then applications to algebra, geometry, calculus, mechanics, and economics.
Applications include the classification of central quadrics, positivity criteria, simultaneous reduction of
two quadratic forms, polar form, linear programming, the Morse lemma, normal modes of vibration, linear
differential equations with applications to economics, inversion by iteration, and difference equations.

Curtis, Charles W., Linear Algebra: An Introductory Approach (Fourth Edition), Springer Verlag, 1984, xvii
+ 347 pp.

This is a book on the theory of linear algebra. The only applications are at the end: finite symmetry
groups in three dimensions, differential equations, analytic methods in matrix theory, and sums of squares
and Hurwitz’s theorem. Curtis includes material on canonical forms, dual vector spaces, multilinear
algebra, and the principal axis theorem. The singular value theorem and pseudoinverse are not covered.

Fekete, Antal E., Real Linear Algebra, Marcel Dekker, 1985, xxi + 426 pp.

The author has used sketches by Norman Steenrod to create a book on real vector spaces (especially
R³) that emphasizes an intuitive geometric approach rather than the usual axiomatic algebraic approach.
This text stresses the geometry of linear transformations and regards matrices and determinants as tools
for computation rather than as primary objects of study. The role of Lie theory is explained. Complex
vector spaces are not covered at all. Unusual approach not found in any other book.

Friedberg, Stephen H., Insel, Arnold J., and Spence, Lawrence E., Linear Algebra (Fourth Edition), Prentice
Hall, 2003, xii + 601 pp.

This excellent text is a careful and thorough treatment of linear algebra that briefly covers a number of
applications, such as Lagrange interpolation, incidence matrices, Leontief’s model (economics), systems of
differential equations, Markov chains and genetics, rigid motions in R² and R³, conic sections, the second
derivative test, and Sylvester’s law of inertia. The main emphasis is on theory, including duality and canonical
forms, with two sections on Jordan canonical form. A distinguishing feature of this text is that vector
spaces and linear transformations are covered before systems of linear equations. The chapter on inner
product spaces is especially rich, with sections on the singular value theorem and pseudoinverse (including
the complex case, which I have not found elsewhere), bilinear and quadratic forms, Einstein’s special
theory of relativity, conditioning and the Rayleigh quotient, and the geometry of orthogonal operators.

Gantmacher, F. R., The Theory of Matrices (2 volumes), Chelsea. Volume One: 1959, 1960, and 1977, x
+ 374 pp.; Volume Two: 1959 and 1960, ix + 276 pp. English translation of the original Russian book.
 

This classic text goes much deeper than most books. Volume One includes chapters on functions
of matrices (including representation by series, systems of linear differential equations, and stability),
canonical forms, matrix equations (such as AX = XB, AX − XB = C, matrix polynomial equations,
m^th roots of matrices, and the logarithm of a matrix), and quadratic and Hermitian forms. Volume Two
covers complex symmetric, skew-symmetric and orthogonal matrices; singular pencils of matrices; matrices
with non-negative elements; applications to systems of linear differential equations; and the problem of
Routh-Hurwitz and related questions.

Gilbert, Jimmie and Gilbert, Linda, Linear Algebra and Matrix Theory (Second Edition), Thomson Brooks/
Cole, 2004, ix + 518 pp.

This text goes beyond eigenvalues and eigenvectors to the classfication of bilinear forms, normal matrices,
spectral decompostions, the Jordan canonical form, and sequences and series of matrices.

Greub, Werner, Linear Algebra (Fourth Edition, 2nd revised printing), Springer-Verlag, 1981, Graduate
Texts in Mathematics 23, xvii + 451 pp.

An elegant and detailed axiomatic treatment of linear algebra, written by a differential geometer. Topics
include duality, oriented vector spaces, algebras, gradations and homology, inner product spaces, quaternions,
rotations of Euclidean spaces of dimensions 2 through 4, differentiable families of linear automorphisms,
symmetric bilinear forms, pseudo-Euclidean spaces and Lorentz transformations, quadrics in affine
and Euclidean space, unitary spaces, polynomial algebras, and structure of linear transformations.

Greub, Werner, Multilinear Algebra (Second Edition), Springer-Verlag, 1978, paperback, vii + 294 pp.

Sequel and companion volume to the author’s Linear Algebra. Topics include tensor products, tensor
algebra, exterior algebra, applications to linear transformations, Clifford algebras and their representations.

Halmos, Paul R., Finite-Dimensional Vector Spaces (Second Edition), van Nostrand, 1958, viii + 200 pp.
Now available from Springer.

This is a classic text by a famous analyst and expositor. Its purpose is to treat the theory of linear
transformations on finite-dimensional vector spaces by the methods of more general theories. The book
emphasizes coordinate-free methods. The treatments of matrices and determinants are unusually brief.
The last chapter on analysis discusses convergence of vectors , norms of transformations, a minimax
principle for self-adjoint transformations, convergence of linear transformations, an ergodic theorem by
Riesz, and power series. There is an appendix on Hilbert space.

Herman, Eugene A. and Pepe, Michael D., Visual Linear Algebra, Wiley, 2005, xix + 550 pp.

This introductory text is a blend of interactive computer tutorials and traditional text. It comes with
a CD-ROM containing 30 Maple worksheets and 30 Mathematica notebooks. There are chapters on
systems of linear equations, vectors, matrix algebra, linear transformations, vector spaces, determinants,
eigenvalues and eigenvectors, and orthogonality. Some standard topics are treated briefly in tutorials.
Complex vector spaces and canonical forms are not covered. Applications include curve fitting, estimation
of temperature distribution in a thin plate, Markov chains, cryptology, computer graphics, networks, and
systems of linear differential equations.

Hoffman, Kenneth & Kunze, Ray, Linear Algebra (Second Edition), Prentice-Hall, 1971, viii + 407 pp.

Excellent junior/senior-level text. The chapter headings are: linear equations, vector spaces, linear
transformations, ploynomials, determinants, elementary canonical forms, the rational and Jordan forms,
inner product spaces, operators on inner product spaces, and bilinear forms. Emphasizes concepts, not
applications or numerical methods. Good exercises.

Jacobson, Nathan, Lectures in Abstract Algebra (Volume II—Linear Algebra), Van Nostrand, 1953, xii +
280 pp.

Middle volume of excellent high-level text on abstract algebra. Can be read independently of first
volume. The chapter headings are: finite dimensional vector spaces, linear transformations, the theory
of a single linear transformation, sets of linear transformations, bilinear forms, Euclidean and unitary
spaces, (tensor) products of vector spaces, the ring of linear transformations, and infinite dimensional
vector spaces. Jacobson drops the assumption that multiplication of scalars is commutative, defining his
vector spaces over a division ring.

Lang, Serge, Linear Algebra (Second Edition), Addison Wesley, 1971, xi + 400 pp. Third Edition, Springer
Verlag, 1987, ix + 296 pp.

The second edition is a rather abstract text on linear algebra, but the author does explain some geometric
concepts. The book is organized into three parts: basic theory, structure theorems, and relations with
other structures. The second part includes triangulation (i.e., triangularization) and diagonalization of
matrices, primary decomposition, and Jordan normal form. The third part discusses multilinear products,
groups, rings, and modules. There are appendices on convex sets, odds and ends (induction, algebraic
closure of the complex numbers, and equivalence relations), and angles. The third edition is considerably
shorter and is not divided into parts. It omits the first chapter on the geometry of vectors and Appendix
3 on angles, deletes the section on determinants as area and volume, rearranges the chapters and sections
of Part Two, and omits part three entirely, but includes a chapter on convex sets (the old Appendix 1)
and adds an appendix on the Iwasawa decomposition and others.

Lax, Peter D., Linear Algebra, Wiley, 1997, xiv + 250 pp.

Advanced treatment that covers all the standard elementary topics in the first 75 pages or so. Uses
quotient spaces. Topics include duality, interpolation, difference equations, law of inertia, Rayleigh quotients,
Rellich’s theorem, and avoidance of crossing. Whole chapters on matrix inequalities, kinematics
and dynamics, convexity, the duality theorem, normed linear spaces, positive matrices, and numerical
solution of linear systems of equations. Appendices on special determinants, Pfaff’s theorem, symplectic
matrices, tensor products, lattices, fast matrix multiplication, Gershgorin’s theorem, and the multiplicity
of eigenvalues.

Leon, Stephen J., Linear Algebra with Applications (Sixth Edition), Prentice Hall, 2002, xv + 544 pp.

This introductory text features brief accounts (with references) of a great variety of applications. There
are many MATLAB exercises, supported by an appendix on MATLAB. There is a chapter on numerical
linear algebra, and two extra chapters (on interative methods and Jordan canonical form) are available
for downloading from the book’s web page

Lipschutz, Seymour, Theory and Problems of Linear Algebra (Second Edition), Schaum’s Outline Series,
McGraw-Hill, 1991, paperback, vii + 453 pp.

This inexpensive text is a good source of numerous solved problems.

Meyer, Carl D., Matrix Analysis and Applied Linear Algebra, siam (Society for Industrial and Applied
Mathematics), 2000, xii + 718 pp., includes a multiplatform CD-ROM.

This big applied text has broad coverage and emphasizes matrices and numerical aspects of linear
algebra algorithms. The LU factorization is covered for square matrices only. Includes a chapter on
Perron-Frobenius theory. The CD-ROM contains the entire text and solutions manual in pdf format, plus
many extras, such as biographies of mathematicians, the history of mathematical notations, the history
of mathematics in China, and articles on numerical linear algebra.

Nakos, George and Joyner, David, Linear Algebra with Applications, brooks/Cole, 1998, xviii + 666 pp.

Introductory text with brief treatments of many interesting applications, some presented in the form of
miniprojects. Contains computer exercises with selected solutions in Maple, MATLAB, and Mathematica.

Noble, Ben and Daniel, James W., Applied Linear Algebra (Third Edition), Prentice Hall, 1988, xvi + 521 pp.

Fine applied text with many interesting applications and helpful discussion of practical numerical issues.
Includes coverage of canonical forms, the singular value decomposition, the pseudoinverse, Rayleigh’s
principle and the min-max principle for extremizing quadratic forms, and linear programming, as well as
inverses of perturbed matrices.

Olver, Peter J. and Shakiban, Chehrzad, Applied Linear Algebra, Pearson Prentice Hall, 2006, xxii + 714 pp.

This applied text has chapters on linear algebraic systems, vector spaces and bases, inner products
and norms, minimization and least squares approximation, orthogonality, equilibrium, linearity, eigenvalues,
linear dynamical systems, iteration of linear systems, and boundary value problems in one dimension.
The depth and variety of its applications exceed those of most texts. Its philosophy is that
of Strang’s text Linear Algebra and its Applications, but it covers more topics. The last chapter introduces
generalized functions and infinite-dimensional function space methods.

Poole, David, Linear Algebra: A Modern Introduction (Second Edition), Thomson Brooks/Cole, 2006, xxiv
+ 712 pp.

Large introductory text emphasizing geometry, applications, and technology. Explorations and applications
include error-detecting codes, LU factorization for square matrices, Markov chains, Leslie’s model
of population growth, graphs and digraphs, error-correcting codes, iterative methods for computing eigenvalues,
the Perron-Frobenius theorem, linear recurrence relations, systems of linear differential equations,
the modified QR factorization, dual codes, quadratic forms and graphs of quadratic equations in two
and three variables, tilings of the plane, linear codes, taxicab geometry, and approximation of functions.
Comes with CD-ROM containing data sets and manuals for using Maple, MATLAB, and Mathematica.

Prasolov, Viktor Vasilevich, Problems and Theorems in Linear algebra, Translations of Mathematical Monographs,
Volume 134, American Mathematical Society, 1994, paperback, xviii + 229 pp.

This intriguing book is filled with interesting results on finite-dimensional vector spaces, mostly real or
complex, that are hard to find elsewhere. It has chapters on determinants, linear spaces, canonical forms
of matrices and linear operators, matrices of special form, multilinear algebra, matrix inequalities, and
matrices in algebra and calculus. Computational linear algebra is not treated. Most essential results of
linear algebra appear here, often with nonstandard neat proofs. Solutions to all the problems are included.

Sadun, Lorenzo, Applied Linear Algebra: The Decoupling Principle, Prentice Hall, 2001, xvii + 349 pp.

Written from the point of view of physics and engineering, this book emphasizes one important aspect
of linear algebra, the diagonalization (or decoupling) of matrices and linear operators. It is designed
as a text for a second course in linear algebra for juniors and seniors. Chapters on crucial applications
(discrete-time evolution, first- and second-order continuous-time evolution, Markov chains and probability
matrices, linear analysis near fixed points of nonlinear problems), the wave equation, continuous spectra
and the Dirac delta function, Fourier transforms, and Green’s functions.

Shifrin, Theodore and Adams, Malcolm R., Linear Algebra: A Geometric Approach, Freeman, 2002, xviii +
439 pp.

This is a well-written introductory text with some unusual features. It contains historical notes at the
end of each chapter and covers some nonstandard topics such as Lagrange interpolation, Jordan canonical
form, isometries of Rⁿ (for n = 1, 2, and 3), and perspective projections. Roughly comparable to Strang’s
Introduction to Linear Algebra, but with more emphasis on definitions and proofs and less on numerical
linear algebra. Contains a short annotated bibliography.

Strang, Gilbert, An Introduction to Linear Algebra (Third Edition), Wellesley-Cambridge Press, 2003, viii
+ 568 pp.

Somewhat lower in level than Strang’s Linear Algebra and its Applications, this introductory text covers
the same topics in less detail and its exercises are more elementary. The web sites offer MATLAB “teaching codes”, interactive Java demos, and videos of Strang’s lectures. Useful items at the back of the text include a sample final exam, a two-page summary of matrix factorizations, conceptual questions for review, a glossary, a list of the MATLAB teaching codes, and a table called “Linear Algebra in a Nutshell” that lists many ways of distinguishing nonsingular square matrices from singular ones.

Strang, Gilbert, Linear Algebra and its Applications (Fourth Edition), Thomson Brooks/Cole, 2006, viii +
488 pp.

Excellent text on real and complex matrices and their applications, with chapters on matrices and
Gaussian elimination, vector spaces, orthogonality, determinants, eigenvalues and eigenvectors, positive
definite matrices, computations with matrices, and linear programming and game theory. Discusses the
singular value decomposition, the pseudoinverse, the fast Fourier transform, Rayleigh’s quotient and the
minimax principle, the finite element method, and numerical methods. There are appendices on the
intersection, sum, and product of spaces and on Jordan form. This book is the standard against which
modern texts on applied linear algebra are judged.

Szabo, Fred, Linear Algebra: An Introduction Using , Harcourt/Academic Press, 2002, xxiii + 788 pp.

This book is intended to serve as the main text for a traditional course in linear algebra, enriched and
facilitated using Maple V or Maple 6. (Note: As of this writing, the latest version of Maple is Maple 9 ,
but most of the material in this book is still current.) Often examples are solved using three methods:
the linalg package of Maple V, the LinearAlgebra package of Maple 6, and ordinary pencil and paper
calculation. It uses standard mathematical notation, but also incorporates Maple code throughout. There
is an appendix on Maple packages, as well as a long and helpful answer section that includes many
Maple-based solutions.

Uhlig, Frank, Transform Linear Algebra, Prentice Hall, 2002, xviii + 503 pp.

This text is organized around the idea of a linear transformation. An account of the philosphy underlying
Each of the fourteen chapters starts with a fundamental lecture, usually followed by sections on theory and applications.
There is enough material for a year-long course. The text is organized so that eigenvalues and eigenvectors can be covered without determinants (as in Axler’s book), with determinants, or (for purposes of comparison) in both ways.

Williams, Gareth, Linear Algebra with Applications (Fourth Edition), Jones and Bartlett, 2001, xvii +
647 pp.

This text integrates mathematics and computation with a wide variety of applications. Browsing through
the applications gives one a real appreciation for the usefulness of linear algebra. Manuals for the use of
calculators (TI-82/83) and MATLAB are included as appendices.