Linear Algebra for Physics – Mathematical Association of America

Author: Nikolaos A. Papadopoulos and Florian Scheck
Publisher: Springer
Publication Date: 10/14/2024
Number of Pages: 401
Format: Hardcover
Price: $84.99
ISBN: 978-3-031-64907-3
Category: textbook

Linear algebra is indispensable for physicists. Besides being of great utility, it is the mathematical prelude to functional analysis, tensors, differential geometry, and representation theory that undergird modern fundamental physical theories. It is also needed for Markov chains, linear regression, and various statistical learning techniques that are increasingly used in physics. Texts on applied mathematical methods often devote a portion to linear algebra, and physics students are likely to encounter matrix algebra and eigentheory in courses on special relativity, mechanics, or elementary quantum mechanics. There are a multitude of linear algebra textbooks, but only a small fraction are specifically addressed to physics students. In some of these cases, linear algebra shares space with group theory, an unsurprising arrangement given the importance of group representation theory in modern physics.

The book under review is addressed to undergraduate and graduate physics students, and the authors aim for a sophistication comparable to mathematical texts but from a physicist's perspective. For example, there is a stated bias towards the use of bases ("a basis is our best friend in linear algebra!" p95), inner product spaces are prominent, and many results are stated both invariantly and in coordinate form. The authors carefully distinguish linear transformations from their representations under choice of bases and provide multiple perspectives on the determinant.

The layout of the book is helpful: definitions, propositions, theorems, and examples are in separate shaded boxes, and lesser comments are in unshaded boxes. All boxes and every exercise are tagged with a short description. This aids locating a particular concept or result since the provided index is minimal.

As with most other linear algebra texts at this level, attention is essentially restricted to finite-dimensional vector spaces over either the field of real or complex numbers. All standard elementary topics are presented; inner products and dual spaces appear early, though their systematic development follows much later. The full Jordan canonical form is not proved, but only a block-diagonal "pre-Jordan" form with blocks $\lambda I+N$ for eigenvalues $\lambda$ and $N$ nilpotent. Normal operators over both complex and real inner product spaces are discussed, the latter via complexification. Other advanced topics include sums, direct sums, dual spaces, annihilators, Householder reflections, the Cartan-Dieudonne Theorem, positive operators, and isometries.

There is also a chapter devoted to a detailed discussion of two forms of duality (that from a choice of basis and that from a choice of inner product) and their interrelations. The text concludes with a chapter on tensors (via vector-valued multilinear maps), mixed tensors, the universal property of tensors, and tensor contractions. The effect of basis change on tensor coordinates is carefully explained. These two chapters again reflect attention to the needs of the physics audience.

Somewhat unusual is the inclusion of group actions, appearing already on the sixth page of the book. Their inclusion is commendable, though most readers may not appreciate their role until much later. Mention of rings and k-algebras follow a similar inclination to highlight various mathematical structures. In a more geometric vein, there are sections on affine spaces and tangent/cotangent vectors in $\mathbb{R}^n$.

Most unusual among the included topics is a section on what the authors call "basis dependent coordinate free'' representations of a vector space. They highlight two such: for a $n$-dimensional vector space $V$ over scalar field $k$, let $B(V)$ denote the set of ordered bases (i.e. frames) and $G=GL(n,k)$ the general linear group over $k$. Taking coordinates yields a pairing $\varphi\colon B(V)\times V \to k^n$. From change of basis, one may define a right $G$-action on $B(V)$ and a left $G$-action on $k^n$ such that $\varphi(\cdot,v)\colon B(V)\to k^n$ is an equivariant map. One shows that $V$ is isomorphic to the vector space of $G$-equivariant maps from $B(V)$ to $k^n$ via $v\leftrightarrow \varphi(\cdot,v)$. Similarly, $B(V)\times k^n$ is a $G$-space via $(Bg,g^{-1}x)$ for $g\in G$, and the orbit space is isomorphic to $V$. This is the algebraic crux of the construction by which one associates a vector bundle to a principle $G$-bundle, where $G$ is a matrix Lie group. The present reviewer is not aware of another linear algebra text for undergraduates which discusses these two isomorphisms.

More broadly, the text's overall perspective is largely algebraic or geometric, and the choice of topics is traditional. The analytic perspective is not highlighted; only norms induced by an inner product appear. The variational characterization of orthogonal projections is discussed, but not the variational approach to eigenvalues and singular values via the Rayleigh quotient. Neither is there mention of the matrix exponential and its connection with matrix Lie groups and Lie algebras. Only a basic description of the SVD is included but not the useful Eckart-Young Theorem.

Similarly, there is less discussion of algorithmic aspects of linear algebra than is found in comparable mathematical texts. Some topics such as Gram-Schmidt and cofactor expansion are present. Others such as Gaussian elimination, RREF, matrix inverse via row operations, or the PLU and QR factorizations are not. Applications to linear regression or Markov chains are also absent. (For a more applied approach to linear algebra, see Boyd and Vandenberghe's Introduction to Applied Linear Algebra or Lukas's The Oxford Linear Algebra for Scientists.) Perhaps these are missed opportunities given the importance of stochastic and statistical learning methods in physics. These in turn may also prompt reconsideration for the analytic and algorithmic aspects of linear algebra.

This outlook extends to the problem sets which conclude most chapters. They are generally straightforward but almost all of them ask either for a proof or a counterexample. Though there are many worked examples in the text, the inclusion of exercises which ask a student to compute a specific similar case or which introduce an application would likely aid students' grasp and appreciation for the material.

The choice of preposition in the book's title is telling; the book's focus is linear algebra for physics rather than its use in physics. Though there are sections on special relativity, Euler-Lagrange equations, and the Dirac bra-ket formalism, these are rather cursory. In particular, connections with quantum mechanics are muted. For example, the applicability of the non-commutativity of operators and the Cauchy-Schwarz Inequality to analyzing the Stern-Gerlach experiment or to deriving the Heisenberg Uncertainty Principle are not mentioned. (For an elementary mathematical treatment, see Isham's Lectures on Quantum Theory, for example.) Perhaps the authors assume physics students will (eventually) encounter these in their physics coursework.

The text also assumes prior acquaintance with some linear algebra ideas (perhaps from physics settings). In the early chapters, many terms are used without a formal definition. These are rigorously defined in subsequent chapters, but the presence of such forward references renders the text less friendly for those new to linear algebra.

In summary, the book is essentially an advanced linear algebra text informed by the needs of physics students. It would be a good supplement for physics students enrolled in linear algebra or as a second text for those with prior background in linear algebra. Its greater scope and mathematical sophistication will be particularly appealing and profitable to prospective mathematical physicists and even math students. In a crowded field of linear algebra titles, I warmly commend the authors for an inviting, well-written addition with a distinctive vision.

Jer-Chin Chuang is a Lecturer at the University of Illinois.