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
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
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
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
mth 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
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
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
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 +
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
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
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
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
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
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
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 +
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 +
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
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
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 +
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.