## MAS220 Algebra

Both semesters, 2022/23 | 20 Credits | ||||

Lecturer: | Dr Sam Marsh | Timetable | Reading List | ||

Outcomes | Assessment | Full Syllabus |

The definition of abstract algebraic structures such as groups, rings, and vector spaces, is dependent on the concepts of set theory, and did not occur until the late nineteenth and early twentieth centuries. That many of the things mathematicians had already been studying turned out to be examples shows that these are the right definitions, not just exercises in playing with axioms. Through them we achieve a stunning unification of diverse areas of mathematics. The aim of this module is not only to build abstract theories, but to use them to obtain a deeper understanding of familiar mathematics, including arithmetic, coordinate geometry, vectors, calculus, linear and differential equations, with one eye on applications, to underline the significance of this mathematics.

**Prerequisites:** MAS110 (Mathematics Core I); MAS111 (Mathematics Core II); MAS114 (Numbers and Groups)

**Corequisites:** MAS211 (Advanced Calculus and Linear Algebra)

The following modules have this module as a prerequisite:

MAS333 | Fields |

MAS345 | Codes and Cryptography |

MAS346 | Groups and Symmetry |

MAS435 | Algebraic Topology |

MAS436 | Functional Analysis |

MAS438 | Fields |

MAS439 | Commutative Algebra and Algebraic Geometry |

## Outline syllabus

- Quotient Groups
- Conjugation in Groups
- Group Homomorphisms
- Introduction to Rings
- Ring Homomorphisms
- Divisibility and Factorisation
- Vector Spaces
- Linear Maps
- Conjugation of Matrices
- Inner Products
- Self-adjoint Operators

## Office hours

## Learning outcomes

With the successful completion of MAS220, a student will be able to:- Understand the quotient construction in groups, rings and vector spaces,
- Understand the notions of conjugacy class and centre in groups and be able to make simple deductions about conjugacy classes, centralisers and structure of a group using the orbit-stabiliser formula,
- Understand the notion of homomorphisms, kernels (in group, ring and vector space settings), be able to use these notions and the First Isomorphism Theorem effectively to establish relationships between different groups, rings and vector spaces,
- Understand the notion of divisibility, irreducibility and congruence in Euclidean domains,
- Understand the notions of spanning and linear independence in vector spaces, be able to determine linear independence in standard examples,
- Understand the connection between matrices and linear maps and change of basis,
- Understand the notion of linear operator on a vector space, eigenvectors and eigenvalues, be able to determine eigenvectors/eigenvalues of linear operators in standard examples,
- Understand the notion of orthogonality in inner product space, be familiar with its applications to linear algebra, such as the Gram-Schmidt process,
- Understand the notion of self-adjointness for a linear operator and its implications on the eigenvalues of the linear operator

43 lectures, 11 tutorials

## Assessment

10 online tests, 1% each (10%). One formal examination (90%).

## Full syllabus

**1. Quotient Groups**

**Z**, O

_{2}, GL

_{2}(

**R**) and S

_{n}as examples. Normal subgroups, quotient groups.

**2. Conjugation in Groups**

Conjugation as a group action, conjugacy classes as orbits. Conjugation in GL

_{2}(

**R**), O

_{2}and S

_{n}. Normal subgroups as unions of conjugacy classes. The centre, the class equation, application to p-groups.

**3. Group Homomorphisms**

Homomorphisms, image subgroups and kernel normal subgroups. First Isomorphism Theorem for groups.

**4. Introduction to Rings**

Ring axioms. Commutative and non-commutative rings, units, division rings, fields. Examples: polynomial rings, Gaussian integers, Hamilton's quaternions, matrix rings.

**5. Ring Homomorphisms**

Ring homomorphisms, inclusion and evaluation examples. First Isomorphism Theorem for rings.

**6. Divisibility and Factorisation**

Divisibility, integral domains, Euclidean domains, Euclid's algorithm. Irreducibles, associates, modular arithmetic in Euclidean domains, rings of congruence classes. Unique factorisation in Euclidean domains.

**7. Vector Spaces**

Vectors in the plane and space. Cartesian coordinates. Spaces of linear functions, lines and planes and their bases. n-dimensional space,

**R**

^{n}and linear equations. Vector spaces. Subspaces, including null spaces and spans. Basis and dimension.

**C**,

**F**

_{4}and

**H**as vector spaces. Infinite-dimensional spaces of continuous functions.

**8. Linear Maps**

Evident isomorphisms. Homomorphisms of vector spaces, example F

^{n}→ F, re-name linear maps. Example F

^{n}→ F

^{m}, matrices. Linear coordinate changes, geometrical transformations. Evaluation, differentiation and integration of functions as linear maps, linear differential equations. Ring of linear operators. Image and kernel subspaces for linear maps. Null spaces, column spaces, rank. Quotient spaces, restriction of functions. First Isomorphism Theorem for vector spaces. Rank-Nullity Theorem.

**9. Conjugation of Matrices**

Matrix of a linear operator with respect to a basis. Change of basis. Trace, determinant, eigenvalues and eigenvectors of a linear operator.

**10. Inner Products**

Dot product of vectors in the plane or space, geometrical demonstration of symmetry and bilinearity. Deduction of algebraic formula, Pythagoras' Theorem. Dot product, lengths and angles (well-defined?) in

**R**

^{n}. Real inner product spaces, Cauchy-Schwarz Inequality, Triangle Inequality. Substitution of integration for addition for inner products of functions. Orthogonality of trigonometric functions. Orthogonal complements. Linear functions, lines and planes, and Rank-Nullity, all revisited. Orthogonal projection, Gram-Schmidt process, Fourier coefficients as inner products.

**11. Self-adjoint Operators**

The adjoint property of the transpose of a matrix, general definition of the adjoint of an operator. Self-adjoint operators, real eigenvalues and orthogonal eigenvectors for distinct eigenvalues. Integration by parts, self-adjoint differential operators, orthogonality of trigonometric functions.

## Reading list

Type | Author(s) | Title | Library | Blackwells | Amazon |
---|---|---|---|---|---|

C |
Allenby | Rings, Fields and Groups | |||

C |
Cameron | Introduction to Algebra | |||

C |
Carter | Visual group Theory | |||

C |
Chatters and Hajarnavis | An Introductory Course in Commutative Algebra | |||

C |
Halmos | Finite-Dimensional Vector Spaces | |||

C |
Herstein | Abstract Algebra | |||

C |
Jordan and Jordan | Groups | |||

C |
Kaye and Wilson | Linear Algebra | |||

C |
Lay | Linear Algebra and its Applications | |||

C |
Nicholson | Linear Algebra with Applications |

(

**A**= essential,

**B**= recommended,

**C**= background.)

Most books on reading lists should also be available from the Blackwells shop at Jessop West.