Igor R. Shafarevich

Linear Algebra and Geometry

Preface.- Preliminaries.- 1. Linear Equations.- 2. Matrices and Determinants.- 3. Vector Spaces.- 4. Linear Transformations of a Vector Space to Itself.- 5. Jordan Normal Form.- 6. Quadratic and Bilinear Forms.- 7. Euclidean Spaces.- 8. Affine Spaces.- 9. Projective Spaces.- 10. The Exterior Product and Exterior Algebras.- 11. Quadrics.- 12. Hyperbolic Geometry.- 13. Groups, Rings, and Modules.- 14. Elements of Representation Theory.- Historical Note.- References.- Index

What is Algebra

What is algebra? Is it a branch of mathematics, a method or a frame of mind? Such questions do not of course admit either short or unambiguous answers. One can attempt a description of the place occupied by algebra in mathematics by drawing attention to the process for which Hermann Weyl coined the unpronounceable word ‘coordinatisation’ (see [H. Weyl 109 (1939), Chap. I, § 4]). An individual might find his way about the world relying exclusively on his sense organs, sight, feeling, on his experience of manipulating objects in the world outside and on the intuition resulting from this. However, there is another possible approach: by means of measurements, subjective impressions can be transformed into objective marks, into numbers, which are then capable of being preserved indefinitely, of being communicated to other individuals who have not experienced the same impressions, and most importantly, which can be operated on to provide new information concerning the objects of the measurement.

Algebraic Geometry I

These notes follow a first course in algebraic geometry designed for second year graduate students at the University of Michigan. The recommended texts accompanying this course include Basic Algebriac Geometry I by Igor R. Shafarevich, Algebraic Geometry, A First Course by Joe Harris, An Invitation to Algebraic Geometry by Karen Smith, and Algebraic Geometry by Robin Hartshorne. These notes were typed during class and then edited somewhat, and so they may not be error free. Please email me any comments, corrections, or suggestions you have at djbruce@umich.edu.

Elements of Representation Theory

The final chapter of the book presents the basic elements of representation theory, especially finite-dimensional representations of finite groups. For instance, it is proved that every representation of a finite group is a direct sum of irreducible representations, and that a finite group has only a finite number of distinct (up to equivalence) irreducible representations. At the end of the chapter, irreducible representations (characters) of abelian finite groups are considered.

Divisors and Differential Forms

In algebraic geometry, divisors are codimension 1 subvarieties, whose significance is to serve as the locus of zeros and poles of rational functions. The chapter discusses Weil divisors and locally principal divisors (or Cartier divisors), and linear equivalence between them. A divisor gives rise to a linear system, that one visualises as a family of codimension 1 subvarieties parametrised by a vector space. Algebraic groups are varieties with a group law given by regular maps. This includes linear algebraic groups (or matrix groups), but also elliptic curves and their higher dimensional generalisations, the Abelian varieties. Differential forms are dual to tangent fields, and have many applications. A principal aim is to discuss the canonical class of a variety. The chapter includes a complete discussion and proof of the Riemann–Roch theorem for curves.

The Topology of Algebraic Varieties

A variety over \(\mathbb{C}\) has a Euclidean topology defining an underlying topological space which, for a nonsingular variety, is a real differentiable manifold, that is orientable and of twice the complex dimension. The variety inherits the usual topological invariants such as fundamental group and cohomology. In this context, complete implies compact. A set of less elementary questions centres around the connectedness of the topological space underlying an irreducible variety. The chapter includes a sketch of the local and relative versions of this problem, that includes the Zariski connectedness theorem.

Groups, Rings, and Modules

This chapter presents an excursion in abstract algebra. It begins with the notions of group, subgroup, direct sum, homomorphism, isomorphism, etc., their basic properties, and numerous examples. This is justified by the main aim of the chapter: to establish the decomposition of a finite abelian group as a direct sum of cyclic subgroups, which is very similar to the decomposition of a vector space as a direct sum of cyclic subspaces (the Jordan normal form of a linear transformation). Moreover, the final part of the chapter presents a more general fact—the decomposition of a finitely generated torsion module over a Euclidean ring as a direct sum of cyclic submodules, which contains the decomposition of a vector space and the decomposition of a finite abelian group as partial cases.

Jordan Normal Form

The goal of this chapter is a more complete study of linear transformations of a complex or real vector space to itself, including the investigation of nondiagonalizable transformations. The Jordan normal forms for complex and real vector spaces are established. The final part of the chapter contains applications of the Jordan normal form: raising a matrix to a power, analytic functions of matrices, solution of systems of linear differential equations with constant coefficients. Linear differential equations in the plane and their singular points are investigated in greater detail.

The Exterior Product and Exterior Algebras

The chapter begins with notion of the Plucker coordinates of a subspace in a vector space. Then the Plucker relations are derived, and the Grassmann varieties are described. Then an exterior product of vectors is defined, and the connection between the exterior product and Plucker coordinates is explained: the Plucker relations give necessary and sufficient conditions for an m−vector to be represented as an exterior product of m vectors of the initial vector space. The properties of an exterior product are investigated in greater detail; the notion of exterior algebra is introduced and discussed. Finally, several applications of the obtained theoretical results are considered, for instance, Laplace’s formula for determinant of a square matrix and the Cauchy–Binet formula for the determinant of the matrix product are proved using the exterior product.

Quadratic and Bilinear Forms

Quadratic and bilinear forms on vector spaces are considered. A connection between the notion of bilinear form and that of linear transformation is established, based on the isomorphism between the space of bilinear forms and the space of linear transformations of the vector space to the dual space. A theorem on reducing a quadratic form to canonical form is proved, and the corresponding normal forms for symmetric and antisymmetric bilinear forms are established. Complex, real, and Hermitian forms are investigated in greater detail. For illustration of the obtained results, we consider an application of Sylvester’s criterion to algebraic equations (necessary and sufficient conditions for a real polynomial to have only real roots).