![[The University of Leicester]](http://www.le.ac.uk/corporateid/departmentresource/000066/unilogo.gif) | Department of Mathematics |
 |
Next: MA1201 Mathematical Modelling
Up: ModuleGuide03-04
Previous: MA1151 Introductory Real Analysis
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
, 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
, 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
Author: C. D. Coman, tel: +44 (0)116 252 3902
Last updated: 2004-02-21
MCS Web Maintainer
This document has been approved by the Head of Department.
© University of Leicester.