64624 ADVANCED MATHEMATICS I

Year	No.	Offer	Mode	Description			Cred. Pts
98	64624 	S2  	X 	ADVANCED MATHEMATICS I    	1.00

Contents

• Staffing
• Pre-requisite(s)
• Rationale
• Synopsis
• Objectives
• Topics
• Texts
• Recommended References
• Student Workload
• Assessment Details
• Other Requirements

STAFFING:

PRE-REQUISITE(S)

64622/75622

RATIONALE:

A thorough introduction to the unifying concepts of, and computational
techniques in, linear mathematics is an essential part of the training
of  the  applied  mathematician.  An  introduction  to  the  extensive
applications  of  these  techniques is also desirable.  Number  theory
provides  certain fundamental mathematical knowledge and is  important
in many areas of modern mathematics and its applications.

SYNOPSIS:

Topics  studied  include  linear differential operators  and  boundary
value  problems  for  ordinary  and  partial  differential  equations,
techniques and applications of linear algebra, and an introduction  to
number theory.

OBJECTIVES:

On completion of this unit the student will be able to:

1. describe linear operators and find matrix representations for
   linear operators on finite dimensional spaces;
   
2. find orthogonal bases for finite dimensional spaces;
   
3. use Householder transformations to find the QR decomposition
   of a matrix;
   
4. review and implement various techniques for the solution of
   linear algebraic equations;
   
5. reduce a normal matrix to diagonal form by a unitary
   transformation;
   
6. perform the Singular Value Decomposition of a matrix;
   
7. show an understanding of the special properties of defective
   matrices;
   
8. find the Jordan canonical form of a matrix;
   
9. classify quadratic forms in terms of their eigenvalues, and in
   terms of their geometrical properties;
   
10. identify self-adjoint linear differential operators;
    
11. find eigenfunction expansions for functions involved in the
    solution of linear partial differential equations;
    
12. demonstrate knowledge of elementary number theory;
    
13. solve problems with modulus systems;
    
14. solve elementary problems in number theory.
    

TOPICS:

 Description                                                    Weighting(%)

Vector Spaces                                                         25.00
- linear dependence and independence
- basis and dimension
- vector products and orthogonality
Linear Operators
- linear transformation
- representation in terms of a basis
- change of basis

Eigenvalues and Transformations                                       25.00
- eigensystems
- similarity transformations - simple cases
- unitary transformations
- singular value decomposition
Similarity Transformations
- defective matrices
- Jordan canonical form
Quadratic Forms
- conic sections
- quadrics
- canonical form; index
- positive definite forms

Linear Differential Operators                                         25.00
- self-adjoint problems
- eigenfunction expansions
- linear partial differential equations and separation of
  variables

Number Theory                                                         25.00
- moduli
- residue classes
- Chinese Remainder Theorem

TEXT and MATERIALS to be PURCHASED:

RECOMMENDED REFERENCE MATERIALS:

Amazigo, J.C. & Rubenfeld, L.A. 1980, Advanced Calculus, John Wiley,
New York.

Greenberg, M.D. 1978, Foundations of Applied Mathematics, Prentice-
Hall, NJ.

STUDENT WORKLOAD REQUIREMENTS:

	ACTIVITY				HOURS
Directed Study                                	42
Private Study                                 	52
Examinations                                  	4
Assessments                                   	16

ASSESSMENT DETAILS:

No  *F/S Marks     Due        Description                              Wtg(%)    LBL WWW
1   S              17/08/98  ASSIGNMENT 1                              5.00      Y   N
2   S              07/09/98  ASSIGNMENT 2                              5.00      Y   N
3   S              12/10/98  ASSIGNMENT 3                              5.00      Y   N
4   S              26/10/98  ASSIGNMENT 4                              5.00      Y   N
5   S              END S2    2 HR RESTRICTED EXAM - MODULES 1 & 2      40.00     N   N
6   S              END S2    2 HR RESTRICTED EXAM - MODULES 3 & 4      40.00     N   N

OTHER REQUIREMENTS:

1    To   obtain   a   pass  in  the  unit,  students   must   perform
     satisfactorily in all aspects of assessment.
2    The  due date for assessments is the date by which a student must
     despatch an assignment to the USQ. The onus is on the student  to
     provide proof of the despatch date, if requested by the Examiner.
3    Students  MUST  retain a copy of all assignments  which  must  be
     produced if and when required by the Examiner.
4    In accordance with University Policy and Guidelines,
4.1  an  Examiner  may  grant  an extension of  the  due  date  of  an
     assignment in extenuating circumstances;
4.2  no  assignments  will be accepted for assessment  purposes  after
     assignments  or  model  solutions have been  released  except  in
     extenuating circumstances;
4.3  assignments  submitted after the due date without any extenuating
     circumstances  will  attract a penalty of  at  most  20%  of  the
     assigned mark for each working day late;
4.4  students who submit an assignment after the due date and wish  to
     claim   extenuating   circumstances,  must  provide   documentary
     evidence with the assignment explaining the circumstances;
4.5  the   unit  examiner  shall  consider  a  claim  for  extenuating
     circumstances and decide on the outcome;
4.6  the  decision of the Dean shall be final in any dispute that  may
     arise in the implementation of these guidelines.
5    Restricted   Examination:   a  restricted   examination   is   an
     examination   where  only  those  materials  specified   in   the
     examination paper are permitted during the examination.