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MA2102 Linear Algebra

MA2102 Linear Algebra

Credits: 10 Convenor: Prof. S. C. Koenig Semester: 1 (weeks 1 to 6)
Prerequisites: essential: MA1152(=MC147)
Assessment: Coursework:Exam: 20:80% Examination: 0%
Lectures: 18 Problem Classes: 5
Tutorials: none Private Study: 47
Labs: none Seminars: none
Project: none Other: none
Surgeries: 5 Total: 75

Subject Knowledge

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.

Learning Outcomes

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 work in a group context.

Methods

Lectures, problem classes and surgeries together with some handouts.

Assessment

Exam, marked problem sheets, group work.

Subject Skills

Aims

Being able to handle abstract concepts and apply them in concrete examples.

Experience of working as part of a team.

Learning Outcomes

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.

Methods

Lectures, problem classes, surgeries, group work.

Assessment

Exam, marked problem sheets, group work.

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.

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.

Reading list

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.

H. Anton, Elementary Linear Algebra, Wiley.

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

Resources

Lectures, problem sheets, surgeries, problem classes, additional handouts.

Module Evaluation

Module questionnaires, module review, year review.

Next: MA2111 Algebra I Up: ModuleGuide03-04 Previous: MA2101 Real Analysis

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Last updated: 2004-02-21
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