id: 06303763
dt: b
an: 2015a.00680
au: Baker, Roger; Kuttler, Kenneth
ti: Linear algebra with applications.
so: Hackensack, NJ: World Scientific (ISBN 978-981-4590-53-2/hbk). ix, 320~p.
(2014).
py: 2014
pu: Hackensack, NJ: World Scientific
la: EN
cc: H65
ut: textbook; numbers; vectors; fields; matrices; row operations; vector
spaces; linear mappings; inner product spaces; similarity;
determinants; characteristic polynomial; eigenvalue; unitary matrix;
orthogonal matrix; Hermitian matrix; symmetric matrix; singular value
decomposition; cross-product; Cayley-Hamilton theorem; linear
differential equations; difference equations; Markov process; least
squares; pseudo-inverse
ci:
li: doi:10.1142/9111
ab: Linear algebra is the most widely studied branch of mathematics after
calculus. There are good reasons for this as it plays an important role
in many fields. The readership the authors had in mind for this book
includes majors in disciplines such as mathematics, engineering and
social sciences. It will be useful for beginners and also as a
reference for graduates. It introduces linear algebra, but at the same
time it is tersely written and includes a great deal of material, a
good lot contained in the exercises which, however, also contain many
numerical ones. The chapter headings are: (1) Numbers, vectors and
fields, (2) Matrices, (3) Row operations, (4) Vector spaces, (5) Linear
mappings, (6) Inner product spaces, (7) Similarity and determinants,
(8) Characteristic polynomial and eigenvalues of a matrix, (9) Some
applications, (10) Unitary, orthogonal, Hermitian and symmetric
matrices, (11) The singular value decomposition, and an appendix on
using Maple. Several comments are necessary. Chapter (1) introduces
3-dimensional vectors, including their inner product and also their
cross-product which is not mentioned again. The angle between two
3-dimensional vectors enters here without formal introduction and is
not introduced later via the Cauchy-Schwarz inequality where it arises
naturally. Moreover, the cosine rule is fished out of trigonometry. In
the other chapters scalars are in an arbitrary field, with the special
cases of the real and complex numbers in mind. This poses no pitfalls
until one comes to roots of polynomials. In the section on the
Cayley-Hamilton theorem, eigenvalues are briefly defined for the first
time as roots of the characteristic polynomial of a linear mapping.
Then, in Chapter (8), an eigenvalue of a linear mapping is defined as a
scalar with the property that there are vectors which are just
multiplied by this scalar. Eigenvectors are not defined before
eigenvalues, namely as vectors which are just scaled. It is then shown
that eigenvalues are roots of the characteristic polynomial, all this
without mention of a particular field, without discussing what can
happen in the case of real numbers. Thus the student is misled into
thinking that in the real case $n\times n$ matrices have $n$ real
eigenvalues. These are, however, small quibbles with a well-written
book packed with rigorously developed information, plus a great amount
of extra theory in the exercises. It also deals with more specialized
topics like linear differential equations, difference equations, Markov
processes, least squares and the pseudo-inverse. It will thus provide
the reader with a solid grounding in linear algebra.
rv: Rabe von Randow (Bonn)