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MA1152 Introductory Linear Algebra

MA1152 Introductory Linear Algebra

Credits: 10 Convenor: Dr. D. Notbohm Semester: 2 (week 15 to 26
Prerequisites: essential: MA1101, MA1102
Assessment: Coursework and class test: 100% 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

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. Material taught in this module provides the necessary foundation for the continued treatment of abstract vector spaces in MA2102.

Learning Outcomes

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 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.

Methods

Class sessions, problem classes and handouts.

Assessment

Marked problem sheets, class test.

Subject Skills

Aims

An understanding of abstraction and axiomatic methods.

Learning Outcomes

The ability to solve systems of linear equations methodically and to perform matrix computations.

The ability to present logical arguments in written form.

Methods

Class sessions, problem classes, handouts.

Assessment

Marked problem sheets, class tests.

Explanation of Pre-requisites

The modules MA1101 and MA1102 provide the axiomatic experience on which this module builds. The concept of a field, introduced in MA1102, is used in defining an abstract vector space.

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.

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.

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 Algebra, 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.

Resources

Problem sheets, additional handouts, lecture rooms.

Module Evaluation

Module questionnaires, module review, year review.

Next: MA1201 Mathematical Modelling Up: ModuleGuide03-04 Previous: MA1151 Introductory Real Analysis

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Last updated: 2004-02-21
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