	Department of Mathematics & Computer Science

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MA2102 Linear Algebra

MC2102(=MC241) Linear Algebra

Credits: 10

Convenor: Dr. N. J. Snashall

Semester: 1 (weeks 1 to 6)

Prerequisites:	essential: MA1152(=MC147)
Assessment:	Individual and group coursework: 20%	One and a half hour exam in January: 80%

Lectures:	18	Problem Classes:	5
Tutorials:	none	Private Study:	52
Labs:	none	Seminars:	none
Project:	none	Other:	none
Surgeries:	none	Total:	75

Explanation of Pre-requisites

The student will be assumed to be familiar with the general notion of a vector space, and to understand the concepts of spanning set, basis and linear independence.

Course Description

The course continues the study of linear algebra, taking a more conceptual point of view than MA1152. There is a review of some of the basic notions of vector spaces, linear independence, spanning sets and bases. A detailed study of linear transformations, their matrices and some of their major properties then commences. A central theme is whether a vector space has a basis consisting of eigenvectors for a given linear transformation, or equivalently, whether it has a diagonal matrix with respect to a suitable basis. The geometrically motivated notion of inner product space is then studied, continuing with the study of linear transformations which respect the inner product structure of such a space.

Aims

The aim of this module is to introduce the student to linear algebra from a conceptual point of view, developing the introductory module MA1152. It should enable the student to gain an appreciation of the importance of mathematical structure and the theory of mappings which preserve that structure. The group work aims to encourage mathematical thinking and investigation of the material covered in this course.

Objectives

To know the definitions of and understand the key concepts introduced in this module.

To understand, reconstruct and apply the main results and proofs covered in the module.

To decide whether a vector space has a basis of eigenvectors for a given linear transformation.

To choose a basis with respect to which the matrix of a linear transformation has a particularly manageable form.

To construct an orthonormal basis for a given finite-dimensional inner product space.

To work in a group context.

Transferable Skills

The development of abstract mathematics and the axiomatic method.

The application of mathematical principles and concepts to new situations.

Written presentation of mathematical arguments in a coherent and logical form.

Use of techniques from the module to solve problems.

Experience of working as part of a team.

Syllabus

Review of definitions of field and vector space. Vector subspaces, linear independence, spanning sets, basis and dimension. Finite-dimensional spaces. Direct sum decompositions.

Linear transformations, and the kernel and image of a linear transformation. Rank and nullity and their relationship.

The matrix of a linear transformation. Change of basis matrix. Eigenvalues, eigenvectors and eigenspaces.

Characteristic polynomial (including some discussion of the fundamental theorem of algebra and working over the complex field). The Cayley-Hamilton theorem.

The minimum polynomial and its relationship to the characteristic polynomial. Relationship between the minimum polynomial and the existence of a basis of eigenvectors and diagonalizability of matrices.

Inner product spaces (real and complex), Gram-Schmidt Process.

Reading list

Essential:

Background:

S. Andrilli and D. Hecker, Elementary Linear Algebra, 2nd Ed., PWS-Kent.

H. Anton, Elementary Linear Algebra, Wiley.

W. K. Nicholson, Linear Algebra, with Applications, 3rd Ed., PWS-Kent.

Details of Assessment

There will be four pieces of work set for assessment which all carry equal weight and together count for 20% of the final mark. One of the pieces of work will be done within groups.

There are four questions on the examination paper; full marks are available for three perfect answers. All questions carry equal weight.