Synopsis for Algebra

Overview

Linear Algebra
The core of linear algebra comprises the theory of linear equations in many variables, the theory of matrices and determinants, and the theory of vector spaces and linear maps. All these topics were introduced in the Moderations course. Here they are developed further to provide the tools for applications in geometry, modern mechanics and theoretical physics, probability and statistics, functional analysis and, of course, algebra and number theory. Our aim is to provide a thorough treatment of some classical theory that describes the behaviour of linear maps on a finite-dimensional vector space to itself, both in the purely algebraic setting and in the situation where the vector space carries a metric derived from an inner product.

Rings
The rings part of the course introduces the student to some classic ring theory which is basic for other parts of abstract algebra, for linear algebra and for those parts of number theory that lead ultimately to applications in cryptography. The first-year algebra course contains a treatment of the Euclidean Algorithm in its classical forms for integers and for polynomial rings over a field; here the idea is developed in abstracto.

Learning Outcomes

Linear Algebra
Students will deepen their understanding of Linear Algebra. They will be able to define and obtain the minimal and characteristic polynomials of a linear map on a finite-dimensional vector space, and will understand and be able to prove the relationship between them; they will be able to prove and apply the Primary Decomposition Theorem, and the criterion for diagonalisability. They will have a good knowledge of inner product spaces, and be able to apply the Bessel and Cauchy–Schwarz inequalities; will be able to define and use the adjoint of a linear map on a finite-dimensional inner product space, and be able to prove and exploit the diagonalisability of a self-adjoint map.

Rings
By the end of the course students will have extended their knowledge of abstract algebra to include the key elements of classical ring theory. They will understand and be able to prove and use the Isomorphism Theorem. They will have a good knowledge of Euclidean rings, and be able to apply it.

Synopsis

1. Linear Algebra MT (17 lectures)
   Vector spaces over an arbitrary field, subspaces, direct sums; quotient vector spaces; induced linear map; projection maps and their characterisation as idempotent operators.
   
   [2 Lectures]
   
   Dual spaces of finite-dimensional spaces; annihilators; the natural isomorphism between a finite-dimensional space and its second dual; dual transformations and their matrix representation with respect to dual bases. [2-3 Lectures] Some theory of a linear map on a finite-dimensional space to itself: characteristic polynomial, minimal polynomial, Primary Decomposition Theorem, the Cayley-Hamilton Theorem (economically); diagonalisability; triangular form. Statement of the Jordan normal form.
   
   [4-5 Lectures]
   
   Real and complex inner product spaces: examples, including function spaces [but excluding completeness and ]. Orthogonal complements, orthonormal sets; the Gram–Schmidt process. Bessel's inequality; the Cauchy–Schwarz inequality. [4 Lectures] Some theory of a linear map on a finite-dimensional inner product space to itself: the adjoint; eigenvalues and diagonalisability of a self-adjoint linear map.
   
   [4 Lectures]
   
   
2. Rings MT (7 Lectures)
   
   Review of commutative rings with unity, integral domains, ideals, fields, polynomial rings and subrings of and , Isomorphism Theorems. The Chinese Remainder Theorem; the quotient ring by a maximal ideal is a field.
   
   [2 lectures]
   
   Euclidean rings and their properties : units, associates, irreducible elements, primes. The Euclidean Algorithm for a Euclidean ring; and as proto­types; their ideals are principal; their irreducible elements are prime; factorisation is unique (proof not examinable).
   
   [3 Lectures]
   
   Examples for applications: Gauss's Lemma and factorisation in ; Eisenstein's criterion.
   
   [2 lectures]
   
   

1. Richard Kaye and Robert Wilson, Linear Algebra (OUP, 1998) ISBN 0-19-850237-0. Chapters 2–13. [Chapters 6, 7 are not entirely relevant to our syllabus, but are interesting.]
   
2. Peter J. Cameron, Introduction to Algebra (OUP, 1998) ISBN 0-19-850194-3. Chapter 2.

Alternative and further reading:

1. Joseph J. Rotman, A First Course in Abstract Algebra (Second edition, Prentice Hall, 2000), ISBN 0-13-011584-3. Chapters 1, 3. I. N. Herstein, Topics in Algebra (Second edition, Wiley, 1975), ISBN 0-471-02371-X. Chapter 3. [Harder than some, but an excellent classic. Widely available in Oxford libraries; still in print.]
2. P. M. Cohn, Classic Algebra (Wiley, 2000), ISBN 0-471-87732-8. Various sections. [This is the third edition of his book previously called Algebra I.]
3. David Sharpe, Rings and Factorization (CUP, 1987), ISBN 0-521-33718-6. [An excellent little book, now sadly out of print; available in some libraries, though.]
4. Paul R. Halmos, Finite-dimensional Vector Spaces, (Springer Verlag, Reprint 1993 of the 1956 second edition), ISBN 3-540-90093-4. 1–15, 18, 32–51, 54–56, 59–67, 73, 74, 79. [Now over 50 years old, this idiosyncratic book is somewhat dated but it is a great classic, and well worth reading.]
5. Seymour Lipschutz and Marc Lipson, Schaum's Outline of Linear Algebra (3rd edition, McGraw Hill, 2000), ISBN 0-07-136200-2. [Many worked examples.]
6. C. W. Curtis, Linear Algebra—an Introductory Approach (4th edition, Springer, reprinted 1994).
7. D. T. Finkbeiner, Elements of Linear Algebra (Freeman, 1972). [Out of print, but available in many libraries.]
8. J. A. Gallian, Contemporary Abstract Algebra (Houghton Mifflin Company, 2006).

There are very many other such books on abstract and linear algebra in Oxford libraries.