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MC147 Introductory Linear Algebra

MC147 Introductory Linear Algebra

Credits: 10 Convenor: Dr Dietrich Notbohm Semester: 2
Prerequisites: essential: MC144, MC145
Assessment: Coursework, Maple project: 20% One and a half hour exam: 80%
Lectures: 18 Problem Classes: none
Tutorials: 5 Private Study: 46
Labs: 6 Seminars: none
Project: none Other: none
Surgeries: none Total: 75

Explanation of Pre-requisites

The modules MC144 and MC145 provide the axiomatic experience on which this module builds. The concept of a field, introduced in MC145, is used in defining an abstract vector space. MC145 also provides the introduction to Maple which is used in this module.

Course Description

Vector spaces arise in many areas of mathematics. This course begins by defining vector spaces over a field and introduces the ideas which lead to the concept of dimension. Dimension depends on the notion of a basis, and to identify a basis we need to be able to solve systems of linear equations. This course explores how to solve such systems by a methodical use of matrices and elementary row operations. The course concludes with an introduction to eigenvalues and eigenvectors of matrices for which we need to provide some basic material on determinants. Maple laboratories run for about six weeks to support the lectures.

Aims

This module introduces abstract vector spaces and the concepts of linear independence, spanning, bases and dimension. Techniques for solving systems of linear equations, the theory underpinning these techniques, and the role of matrices in such problems are presented. The module also aims to provide some facility in finding eigenvalues and eigenvectors of matrices. Maple is used alongside the lectures to provide both pedagogical support and to show how a number of the tasks of linear algebra can be carried out by Maple. Material taught in this module provides the necessary foundation for the continued treatment of abstract vector spaces in MC241.

Objectives

To understand the definition and fundamental examples of vector spaces and subspaces; to be able to verify that certain subsets of vector spaces are subspaces; to understand the ideas of a linear combination of vectors, the space spanned by a set of vectors, linear independence, basis and dimension; to be able to find a basis for the space spanned by a given set of vectors.

To understand how to use elementary row operations on matrices to solve systems of linear equations; to know how to use Maple to carry out these computations; to understand the role of the rank and the row-reduced echelon form; to understand the varying nature of solution sets.

To be able to perform the operations of matrix algebra, including finding determinants and inverses; to know how to use Maple to carry out such tasks.

To be able to compute the eigenvalues and eigenvectors of a matrix; to know how to diagonalise matrices in simple cases.

Transferable Skills

An understanding of abstraction and the axiomatic method.

The ability to solve systems of linear equations methodically.

The ability to perform matrix computations including diagonalisation.

Knowledge of Maple as a tool for doing linear algebra.

The ability to present logical arguments in written form.

Syllabus

Definition of a vector space over a field $F$, fundamental examples of vector spaces, elementary consequences of the definition, subspaces, intersections of subspaces, linear combinations of vectors, the space spanned by a set of vectors, linear independence, basis, dimension, the exchange process

Homogeneous and inhomogeneous systems of linear equations, the matrix form of a system of linear equations, elementary row operations, Gaussian elimination, row equivalence, row-reduced echelon form of a matrix, rank of a matrix, parametric and non-parametric descriptions of the space spanned by a set of vectors in ${\bf R}^n$, row space and row rank, column space and column rank, null space and nullity

Addition and multiplication of matrices, invertibility, using row operations to find the inverse of an invertible matrix, elementary matrices, rank and invertibility, cofactors and minors of a matrix, determinants, the determinant of a product, the adjoint of a matrix

Eigenvalues, eigenvectors, the characteristic polynomial and characteristic equation of a matrix, similarity of matrices, diagonalisation, the Cayley-Hamilton Theorem, powers of a matrix

Reading list

Recommended:

R. B. J. T. Allenby, Linear Algebra, Edward Arnold, 1995.

J. B. Fraleigh and R. A. Beauregard, Linear Algebra, 3rd edition, Addision-Wesley, 1995.

L. W. Johnson, R. P. Riess, and J. T. Arnold, Introduction to Linear Algebra, 3rd edition, Addison-Wesley, 1993.

S. Lang, Introduction to Linear Algebra, Springer Verlag, Undergraduate Texts in Mathematics.

L. Smith, Linear Algerba, Springer Verlag, Undergraduate Texts in Mathematics.

G. Strang, Linear Algebra and its Applications, 3rd edition, Harcourt Brace Jovanovich, 1988.

Background:

S. Lipschutz, Schaum's Outline of Theory and Problems of Linear Algebra, 2nd edition, McGraw-Hill, 1991.

Details of Assessment

Details of Assessment

The final assessment of this module will consist of 10% coursework, 10% from a MAPLE project and 80% from a one and a half hour examination during the Summer exam period. The 10% coursework contribution will be determined by students' solutions to coursework problems. The examination paper will contain 4 questions with full marks on the paper obtainable from 3 complete answers.

Next: MC148 Pure Mathematics at Work Up: Year 1 Previous: MC146 Introductory Real Analysis

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