*Linear Algebra for the Young Mathematician* is a rigorous introduction to linear algebra which, according to the author, presupposes no prior knowledge of the subject. The author wrote the book for "young mathematicians," by which he means people that are just beginning a serious study of mathematics. These are not necessarily undergraduate or graduate students in mathematics, but could be physicists, engineers, statisticians, or other professionals wishing to gain a more in depth knowledge of linear algebra.

The author believes that linear algebra is, at its core, about linear transformations between vector spaces, not matrices. This viewpoint permeates the book. Although the first two chapters define matrices and discuss row reduction and solving systems of linear equations, beginning in chapters three and four vector spaces and linear transformations take center stage. (These are defined over an arbitrary field; readers are told that \(\mathbf Q, \mathbf R, \mathbf C\) are all fields and are referred to an appendix for the field axioms.) Although matrices don't disappear entirely, their role largely becomes to provide concrete illustrations of more general results stated for linear transformations between (not necessarily finite dimensional) vector spaces. Matrices still appear from time to time in examples, but the routine matrix based examples that one would expect to see in an introductory linear algebra text are largely absent. An illustrative series of examples appears just after the definition of eigenvalues and eigenvectors. The first example concerns the eigenvalues and eigenvectors of an explicitly given \(2\times 2\) matrix. The second example finds the eigenvalues of a shift operator on the infinite dimensional vector space \(\mathbf F^{\infty\infty}\) of all "infinite" column vectors over a field \(\mathbf F\). In the third example the author finds the eigenvalues of a certain linear transformation defined over the vector space of all real polynomials. The final example concerns the eigenvalues of the derivative map on the vector space of smooth functions on an open interval.

After an initial discussion of vector spaces and linear transformations (covering things like subspaces, dimension, bases, and dual spaces), the author defines determinants in terms of signed volumes of \(n\)-dimensional parallelepipeds (cofactor expansion and the definition involving permutations are also discussed). The chapter on determinants contains exactly two examples, both of which compute the determinants of \(4\times 4\) matrices. The formula for the determinant of a \(2\times 2\) matrix does not explicitly appear in the chapter.

The seventh chapter is likely to be the toughest in the book for beginners. In this chapter the theory of eigenvectors and of generalized eigenvectors are developed in tandem, along with the minimal and characteristic polynomials of a linear transformation, triangularizability, and a section relating these concepts to the study of linear differential equations. As is the case for most concepts in the book, eigenvalues are first defined for linear transformations. We are later told that if \(A\) is an \(n\times n\) matrix, then the eigenvalues of \(A\) are defined to be the eigenvalues of the associated linear transformation. The equation \(Av=\lambda v\) does not appear at all, although to be fair the equation \(\mathcal T(v)=\lambda v\), for \(\mathcal T\) a linear transformation, does.

The high point of the book is chapter eight, on the Jordan Canonical Form (JCF). The author's treatment of the JCF is rigorous and clear, making extensive use of certain graphs that the author calls *labeled eigenstructure pictures* (which very helpfully encode the structure of the generalized eigenvalues of a linear transformation). The chapter includes two extended examples which I found incredibly helpful, and finishes with an application to first-order differential equations.

Chapters 10 and 11 constitute Part II of the book and concern forms on vector spaces (symmetric, skew-symmetric, Hermitian, etc) and real and complex inner product spaces. These chapters build to the classification of forms and the spectral theorem (respectively), and while interesting and well-written, seemed a bit tacked on to me.

I enjoyed this book. It contains as clear an exposition of the JCF as I've seen anywhere, many applications which illustrate how transparent certain facts from calculus and differential equations are when viewed in the context of linear algebra, and is very well-written. I do not, however, have a clear sense of who its true audience is. As I believe my review makes clear, I do not believe that the book is appropriate for readers lacking a previous exposure to linear algebra. On the other hand, it is not quite appropriate for a second linear algebra course or a graduate course either, as it omits many of the topics that the instructor of such a course would want to mention. Because the book largely ignores the computational side of linear algebra and focuses almost exclusively on linear transformations between vector spaces rather than on matrices, I find it hard to believe that readers whose interests are not firmly in pure mathematicians will find much value here. Perhaps the best I can say is that this book is likely to be a useful reference for readers desiring a rigorous, well-written, vector space oriented treatment of the standard topics covered in a strong undergraduate linear algebra course.