id: 06206220
dt: b
an: 2016c.00796
au: Singh, Kuldeep
ti: Linear algebra. Step by step.
so: Oxford: Oxford University Press (ISBN 978-0-19-965444-4/pbk). viii, 608~p.
(2014).
py: 2014
pu: Oxford: Oxford University Press
la: EN
cc: H65
ut: textbook; linear equations; Euclidean space; vector space; inner product
space; linear transformation; determinant; inverse matrix; eigenvalue;
eigenvector
ci:
li:
ab: Linear algebra plays an important role in many fields where students are
not normally strong in mathematics. The author has written this text
with these students in mind. To this end he has incorporated a large
number of examples and step-by-step explanations of every new concept
and result, as well as many numerical and also theoretical exercises
with solutions at the back of the book or available online. The
students are thus led by the hand and spoon-fed all the way which in
the reviewer’s opinion makes the book not very suitable for
mathematics students. The chapter headings are: 1. Linear equations and
matrices, 2. Euclidean space, 3. General vector spaces, 4. Inner
product spaces, 5. Linear transformations, 6. Determinants and the
inverse matrix, 7. Eigenvalues and eigenvectors. The book is rigorous
in the sense that definitions, results and proofs are stated carefully.
There are, however, some inconsistencies and shortcomings. Chapter 2
bridges the gap between the vector geometry learnt at school and the
concept of an $n$-dimensional vector space, while Chapter 3 presents
the formal definition of vector spaces. In Chapter 2, the concept of
the angle (measured in degrees) between two vectors (two-dimensional)
suddenly appears from nowhere and the cosine rule is fished out of
trigonometry. This is then generalized to $n$ dimensions by means of
the formula normally derived from the Cauchy-Schwarz inequality, now
using radians which remain unexplained, while the Cauchy-Schwarz
inequality is then proved using the above-mentioned formula! In Chapter
3, the Cauchy-Schwarz inequality reappears and is now proved in the
usual way, while angles make no appearance. The student will no doubt
ask why the earlier simple proof does not work here. An important
shortcoming is that the scalars are assumed to be real numbers. Complex
numbers are nowhere mentioned. In the last chapter, eigenvectors are
introduced as those non-zero vectors that are just scaled by the
matrix, the scaling factor being the associated eigenvalue. In the
reviewer’s opinion a statement like ‘It is a relationship like
mother and child because eigenvalues give birth to eigenvectors’ is
not very helpful. On the one hand, the direction wrong, on the other,
mothers can have several children and in this way the statement makes
sense, but this fact is not mentioned. Then eigenvalues are determined
via the characteristic equation of a matrix. Here, the above-mentioned
shortcoming comes to bear: nowhere is there a caveat about complex
roots. The student is thus led to expect all eigenvalues to turn out
real! This leads to other pitfalls. The author defines a matrix to be
diagonal if all its off-diagonal entries vanish, thus implicitly
assuming that all the diagonal elements are real numbers. Next, the
author defines a matrix $A$ to be orthogonally diagonalizable if there
is an orthogonal matrix $Q$ such that $Q-1AQ$ is diagonal. This leads
the author to the spectral theorem: A matrix is orthogonally
diagonalizable if and only if it is symmetric, which is, of course,
correct with the given definitions. But it is misleading because the
usual spectral theorem is: A square matrix over $\Bbb C$ is unitarily
similar to a diagonal matrix if and only if it is normal, with the
corollary: A square matrix over $\Bbb R$ is orthogonally similar to a
diagonal matrix if and only if it is symmetric (cf. the excellent
textbooks by Blyth and Robertson). Here the word ‘diagonal’ is tied
to the field considered. It would help the student a great deal if this
more general context had been elucidated. The book also contains brief
historical biographies of those mathematicians involved in the
development of linear algebra, as well as a number of interviews with
people actively using linear algebra. It will certainly be welcomed by
those who find more formal introductions too daunting.
rv: Rabe von Randow (Bonn)