MC147 Introductory Linear Algebra

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MC147 Introductory Linear Algebra

Credits: 10

Convenor: Dr. W. Wheeler

Semester: 2

Prerequisites:	essential: MC144, MC145	desirable: MC120
Assessment:	Supervision work, Maple project: 20%	One and a half hour exam: 80%

Lectures:	18	Classes:	6
Tutorials:	6	Private Study:	45
Labs:	none	Seminars:	none
Project:	none	Other:	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. The module MC120 allows for a geometrical interpretation of problems in linear equations and, as such, is a desirable prerequisite.

Course Description

Vector spaces arise in many areas of mathematics. This course begins by defining vector spaces over a field (e.g. the set of vectors with n entries in the field of real numbers) and introduces the ideas which lead to the concept of dimension (so that the above example has dimension n). Dimension depends on the notion of a basis and to identify a basis we need to be able to solve systems of linear equations, so this course explores how to do this systematically using 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 six weeks to support the lectures.

Aims

This module introduces abstract vector spaces and the concepts of independence, spanning, basis 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 at finding eigenvalues and eigenvectors of matrices. MAPLE is used alongside the lectures to provide both pedagogical support and to show how a number of linear algebra tasks 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 recognise vector spaces and subspaces.

To decide independence and spanning properties of sets of vectors and find bases of spaces.

To solve systems of linear equations using Gaussian elimination and MAPLE, recognising the role of rank and the rre form in the underlying theory, and the varying nature of solution sets.

To perform operations of matrix algebra including finding determinants and inverses; to know how to use MAPLE to carry out such tasks.

To use eigenvalues and eigenvectors to diagonalise matrices in simple cases and to know how to use MAPLE to do this.

Transferable Skills

Developing understanding of abstraction and the axiomatic method.

The ability to solve systems of linear equations systematically.

The ability to carry out matrix algebra including diagonalisation.

Knowledge of MAPLE as a tool for doing linear algebra.

The ability to present logical argument in written form.

Syllabus

Define a vector space over a field F, be familiar with fundamental examples, and prove elementary consequences of the definition. Define a subspace and recognise when subsets of spaces are subspaces. Prove that the intersection of subspaces is a subspace. Write a vector as a linear combination of others when possible and decide if a set of vectors is linearly independent. Understand what the space spanned by a set of vectors is. Define the concepts of a basis and the dimension of a space. Understand the principle of the Exchange Process and its relevance to dimension. Use matrices and vectors in formulating systems of linear equations, distinguishing homogeneous and non-homogeneous systems. Elementary row operations. Use the process of Gaussian elimination to solve systems of m linear equations in n unknowns. Find the reduced row echelon form of a given matrix. Be able to describe in parametric and non-parametric form the space spanned by a set of vectors in a finite real vector space. Use MAPLE to solve systems of linear equations, to carry out row operations, to find rre form and inverse. Describe the row space of a matrix in either non-parametric or parametric forms using row operations. Find a basis for a space given either in non-parametric form or by a spanning set and find its dimension. Be aware of row space, row rank, column space, column rank, null space as solutions, and nullity. State the relationship between rank, nullity, and number of unknowns. Know the rules of matrix algebra and how it differs from the algebra of numbers. Know how to use row operations to find the inverse of an invertible matrix. Describe and compute elementary matrices. Express an inverse as a product of elementary matrices. Know the relationship between rank and invertibility. Find the cofactors and minors of a matrix. Define determinants in terms of cofactors. Evaluate determinants using cofactor expansion or row and column operations. Know that the determinant of a product is the product of the determinants and how to prove this result. Use the adjoint matrix to find an inverse when it exists. Use MAPLE to find the determinant of a matrix.

Define eigenvalues, eigenvectors, the characteristic polynomial, and the characteristic equation of a matrix. Find eigenvalues and eigenvectors in simple cases. Define the concept of matrix similarity and recognise the need to have n linearly independent eigenvectors to get similarity to a diagonal matrix. Use eigenvalues and eigenvectors to find a diagonal matrix similar to a given matrix. Use MAPLE to find eigenvalues and eigenvectors of a matrix. Know the statement of the Cayley-Hamilton Theorem. Use the C-H Theorem to find powers of a matrix.

Reading list

Background:

S. Lipschutz, Linear Algebra, Schaum.

Details of Assessment

Coursework - four pieces of set work provide, in total, 10% of final mark;
MAPLE Assignment - one piece of work handed in just before the Easter vacation provides 10% of final mark;
Examination - one and a half hours duration with four questions, all to be answered for full marks, all of equal weight. The Casio FX82 calculator is the only calculator allowed in this examination.

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

Roy L. Crole
10/22/1998