Courses / Module

ABSTRACT ALGEBRA

Module code: MT242P
Credits: 5
Semester: 2
Department: MATHEMATICS AND STATISTICS
International:

	Overview
	Module Objective: To introduce the student to abstract algebra, with a firm emphasis on linear algebra over R and C. The rationals, reals and complex numbers as examples of fields, the field with 2 elements and field of rational functions; Rn and Cn as vector spaces, vector spaces of functions; the axioms of a vector space; subspace, direct sums, linear maps; kernel, image and quotient spaces; span, linear independence, basis and dimension; linear maps and matrices, change of basis, rank and nullity; normed vector spaces, orthonormal bases, orthogonal projection, Gram-Schmidt process; eigenvalues and eigenvectors, diagonalisation; dual spaces, tensor products, endomorphism algebras; other topics may include exterior algebra and determinants.

Overview

Module Objective: To introduce the student to abstract algebra, with a firm emphasis on linear algebra over R and C.

The rationals, reals and complex numbers as examples of fields, the field with 2 elements and field of rational functions; Rn and Cn as vector spaces, vector spaces of functions; the axioms of a vector space; subspace, direct sums, linear maps; kernel, image and quotient spaces; span, linear independence, basis and dimension; linear maps and matrices, change of basis, rank and nullity; normed vector spaces, orthonormal bases, orthogonal projection, Gram-Schmidt process; eigenvalues and eigenvectors, diagonalisation; dual spaces, tensor products, endomorphism algebras; other topics may include exterior algebra and determinants.

	Learning Outcomes
	On successful completion of the module, students should be able to: Define a field and a vector space. Prove simple consequences of the axioms. Understand subspaces, span and linear independence. Compute a basis and the dimension of a vector space. Represent and manipulate linear maps using matrices. Compute the rank and nullity of a linear map and a matrix. Understand quotient spaces and the first isomorphism theorem. Find eigenvalues and eigenvectors of linear maps. Define inner products spaces. Perform the Gram-Schmidt process and diagonalization. Use methods from MT251P and the course to solve problems. State definitions and theorems covered in the course. Understand the role of axioms in algebra.

Learning Outcomes

Teaching & Learning methods

Assessment

	Autumn Supplementals/Resits
	Are supplemental/resit registrations permitted? Yes University scheduled written examination (Autumn): 120 minutes, On-campus

	Pre-Requisites
	MT251P or equivalent. Exposure to Modular Arithmetic, as in MT241P, helpful but not essential.

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Courses / Module

ABSTRACT ALGEBRA

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