This is the substantially extended second edition of a book comprising an advanced course in linear algebra (cf. the review of the first edition in [(2010; Zbl 1204.15002)]). Chapters 1 to 9 of the first edition are essentially the same as Chapters 1 to 8 and Chapter 10 in the second edition, except that Chapter 5 has a new section on normed vector spaces (and is now titled “Normed and inner product spaces”), Chapter 8 has a new section on orthogonal spaces over perfect fields of characteristic two and a proof of Witt’s extension theorem for such spaces, and Chapter 10 has a new section on Clifford algebras.
There are four new chapters. Chapter 9 (Sesquilinear forms and unitary geometry) studies sesquilinear forms, introduces unitary spaces and their isometries, and proves Witt’s extension theorem for nondegenerate unitary spaces. The first section of Chapter 11 (Linear groups and groups of isometries) introduces the special linear group and “transvections”. This concept is not encountered that often and is, moreover, not very easy to picture geometrically. A fuller description would have been very welcome, as for example in the paper of B. B. Phadke [Can. J. Math. 26, 1412–1417 (1974; Zbl 0254.20028)]. The author proves that the special linear group is generated by transvections and that, with three small exceptions, it is perfect, in which case the quotient group by its center is simple. The second section is concerned with the symplectic group, the third investigates the isometry group of a nondegenerate singular orthogonal space over a field of characteristic not two, and the final section deals with the isometry group of a nondegenerate isotropic unitary space.
Chapter 12 (Additional topics in linear algebra) is devoted to selected matrix topics: the concept of a matrix norm and its properties, the Moore-Penrose pseudoinverse, real nonnegative square matrices, the location of eigenvalues of a complex matrix including Gershgorin’s circle theorem, and matrix polynomials and power series. The last chapter (Applications of linear algebra) treats three important areas of application: the method of least squares, an introduction to coding theory, and the design of a page rank algorithm for a web search engine, for example.