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MC241 Linear Algebra

MC241 Linear Algebra

Credits: 10 Convenor: Dr. N. J. Snashall Semester: 1 (weeks 7 to 12)

Prerequisites: essential: 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 MC147. 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 MC147. 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 recognise Hermitian, unitary and normal matrices.

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 dual of a vector space.

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. Hermitian, unitary and normal transformations (with discussion of the existence of an orthonormal basis of eigenvectors for such transformations, and application to quadratic forms).

Reading list

Essential:

Recommended:

R .B. J. T. Allenby, Linear Algebra, Edward Arnold.

C.W. Curtis, Linear Algebra, an Introductory Approach, Springer.

J .B. Fraleigh and R. B. Beauregard, Linear Algebra, 3rd Ed., Addison-Wesley.

Background:

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

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; all marks gained will be counted. All questions carry equal weight.

Next: MC243 Aspects of Linear Analysis Up: Year 2 Previous: MC240 Abstract Analysis

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Last updated: 10/4/2000
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