This book is intended for sophisticated and successful juniors intending to deepen their study of mathematics. It is an advanced course which embeds the usual topics of a serious course of linear algebra in a very general mathematical context. Occasionally, however, this results in obscure definitions like the order ideal of a vector \({\mathbf v}\), which is the set of polynomials \(f(x)\) such that \({\mathbf v}\in\ker(f(T))\), where \(T\) is a linear operator. Here it should say ‘the order ideal of a vector \({\mathbf v}\) w.r.t. a linear operator \(T\)’. This object is denoted by \(\text{Ann}(T,{\mathbf v})\), but there is no word of explanation for this nomenclature or that Ann is short for annihilator. Then the minimal polynomial of \(T\) w.r.t. \({\mathbf v}\) is defined. Four pages on, the author defines \(\text{Ann}(T)\) as the set of polynomials \(f(x)\) such that \(f(T)\) is the zero operator, and then defines the minimal polynomial of \(T\). The advantage of introducing the earlier objects remains unclear. Also, the characteristic polynomial of a linear operator is defined much later as the product of the invariant factors of \(T\). The student thus has to learn all about invariant factors and elementary divisors before meeting the characteristic polynomial, and the famous Theorem of Cayley and Hamilton is not named, although it is, of course, there. On the other hand, if one makes the effort of studying this book in detail, one will be rewarded with a sound and thorough basis of the more advanced topics of linear algebra.
The chapter headings are: 1. Vector spaces, 2. Linear transformations, 3. Polynomials, 4. Theory of a single linear operator, 5. Inner product spaces, 6. Linear operators on inner product spaces, 7. Trace and determinant of a linear operator, 8. Bilinear maps and forms, 9. Tensor products. The text contains many worked examples and each section ends with a set of exercises, both numerical and theoretical.