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Linear transformations


Multiplication by a 2x2 matrix can be used to transform points in the x-y plane — such transformations are called linear since they map straight lines to straight lines.
See what happens to the kiwi for each of these matrices. (Move the cursor over each matrix.)

Which of these do you think is reversible?


Sample problem


After an evening’s competition the six couples from the “Twirling Dervishes” Ballroom Dancing team headed off to the local pub for a drink. A fairly simple tavern, their local offers 6 different beers and 4 types of wine. Having crammed themselves around a table their captain, Fred, volunteered to get in the first round. Armed with the numbers of each type of drink required he set off for the bar. Being a bit of a wag, Fred decided to perform a pirouette on the way and became dizzy. When he reached the bar he had completely forgotten the order. Too embarrased to go back and ask for the order again, he just made up a selection of twelve drinks from those on offer. What is the probability that he got the order right?



Arthur Cayley, 1821-1895, published over 900 papers and notes covering nearly every aspect of modern mathematics. The most important of his work was in developing the algebra of matrices, work in non-euclidean geometry and n-dimensional geometry.



Sample problem

How can we tell the age of paintings? Some pigments contain a mineral which has two unstable isotopes occurring naturally. These isotopes decay at different rates, and one of them is a biproduct of the other’s decay process. By knowing their original relative proportions in the pigment and solving a differential equation, it is possible to give quite accurate estimates of age, and hence to detect forgeries.



Brook Taylor, 1685-1731, contributed greatly to 18th century mathematics — much more so than the single result that bears his name would suggest. He was a champion of Newton’s approach to calculus and produced many important works developing that area of mathematics. He invented the “calculus of finite differences” and “integration by parts”, as well as discovering “Taylor’s series”.



Sample problem


The interaction of two species, one a predator and the other its prey, can be studied using linked first order differential equtions. By varying the initial conditions (essentially the numbers of each species) different scenarios can be considered, the critical one being the circumstances under which the prey becomes extinct. We will solve the equations in class and also graph the changing populations using computer software.

MATH170 Mathematics 2

First Semester, Second Semester, 18 points
Algebra and Calculus form the basic tools used to produce most mathematical frameworks for modelling quantifiable phenomena. For example, to model the movement of an object through space we need first to create an algebraic structure in which to specify where our object is, and then we can study how that position changes with time (i.e. its movement) using calculus.

Many other problems arising in areas such as Economics or Chemistry, can be examined in a mathematical way using the same basic ideas. For example, we may need to minimize a manufacturing cost, or the time for a chemical reaction to take place, or the effects of river pollution; in each case the techniques used for the minimization are based on a mixture of algebra and calculus theories.

This course aims to develop skills with these tools both for use in other subjects and in preparation for further study of Mathematics.

Paper details

Like MATH 160, MATH 170 is divided between algebra and calculus, and focuses on both ideas and methods.

The algebra component first expands on the material on vectors and matrices begun in MATH 160. (Note, however, that sufficient background is provided on these topics to enable MATH 170 to be taken directly from school.) The course goes on to consider the determinant (a number associated with a matrix) and linear transformations (special mappings associated with computer graphics). The final section considers numbers and factorization, induction (used to prove sequential statements involving integers) and various counting techniques.

The calculus component extends some of the topics covered in MATH 160, and introduces others that are new. It starts with sequences (an ordered list of numbers, possibly infinite) and series (the sum of all the numbers in a sequences). The course then introduces special functions such as the natural logarithm, hyperbolic functions, and inverse trigonometric and hyperbolic functions. After further methods of integration and applications of integration to arclength and volumes, the course concludes with the study of differential equations (and examples of their many uses).

Potential students

This paper should appeal to a wide variety of students including Mathematics and Statistics majors, those studying Computer Science, Physics, Chemistry, Surveying or any discipline with a quantitative component requiring competent manipulation of mathematical formulae and interpretation of mathematical representations of systems.

Main topics

Algebra:
  • Algebra and geometry of 3 dimensional vectors
  • Manipulation of matrices and matrix equations
  • Introduction to linear transformations
  • Eigenvalues and eigenvectors
  • Discrete mathematics, including mathematical induction, Diophantine equations and basic counting techniques

Calculus:
  • Sequences, series and Taylor series
  • Natural log, exponential, hyperbolic, inverse trigonometric and hyperbolic functions
  • Methods of integration
  • Arc length; volumes and surfaces of revolution
  • Solving differential equations

Prerequisites

MATH 160 or high achievement (mostly Excellences and Merits) in NCEA Level 3 Mathematics with Calculus

Required texts

MATH 170 Algebra Outline Notes
MATH 170 Calculus Outline Notes
(Available from the Print Shop)
Caluclus by James Stewart, metric edition 7e (available from the Book Store)

Useful references

Several standard texts are suitable for reference. For example:

Elementary Vector Algebra by A.M. MacBeath
Algebra, Geometry and Trigonometry by M.V. Sweet
Elementary Linear Algebra (Applications version) by H. Anton and C. Rorres (7th edition)
Introductory Linear Algebra (with applications) by B. Kolman (6th edition)
Calculus with Analytic Geometry by Howard Anton (Wiley)
Calculus and Analytic Geometry by George Thomas and Ross Finney (Addison Wesley)

Lecturers

1st semester:
Algebra: Dr John Shanks (Room 221)
Calculus: Prof Astrid an Huef (Room 232A)
2nd semester:
Algebra: Prof Robert Aldred (Room 233)
Calculus: Prof Joerg Frauendiener (Room 223)

Lectures

1st semester:
Algebra: approximately 25 lectures, Mon and Wed at 12 noon
Calculus: approximately 25 lectures, Tues, Thurs and Fri at 12 noon
2nd semester:
Algebra: approximately 25 lectures, Mon and Wed at 12 noon
Calculus: approximately 25 lectures, Tues and Thurs at 12 noon, and Fri at 9

Tutorials

Attendance at tutorials is voluntary. An open system operates: tutorial classes run for up to 10 hours per week (depending on demand), and students may attend as many as they need to and are able to.

Office Hours

TBA

Internal Assessment

Five computer Skills Tests (in each of Algebra and Calculus) make up 20% (T) of your final mark. The other 80% comes from a mix of your exam mark (E) and the internal assessment mark (A) which is based solely on the ten marked weekly assignments.

You can check your marks by clicking on the Resources link at the top of this page.

Terms Requirement

You have to fulfil the terms requirement in order to be allowed to sit the final exam. In this paper, to pass “terms” you need to gain at least 6/10 in each of the first four Skills Tests.

Exam format

The 3-hour final exam is answered in spaces provided on the question booklet. All questions should be attempted and the number of marks available for each question is indicated on the paper. There are usually from 10 to 15 questions for each of Algebra and Calculus. You may allocate your time between the two sections as you wish.

Final mark

The final mark F is calculated from:
F = max { E, (4E + A)/5 } + T
where E (exam mark) is out of 80, A (internal assessment) is out of 80, T (test mark) is out of 20.

The “max” corresponds to plussage: if your internal assessment mark is greater than your exam mark then it is combined in the proportion shown. If it is less then it is ignored and the exam mark itself is used.

So your internal assessment counts at 1/5 weighting if that helps you.

Plagiarism

Students should make sure that all submitted work is their own. “Plagiarism is a form of dishonest practice. Plagiarism is defined as copying or paraphrasing another’s work and presenting it as one’s own” (University of Otago Calendar). In practice this means that plagiarism includes any attempt in any piece of submitted work (e.g. an assignment or test) to present as one’s own work the work of another (whether of another student or a published authority). Any student found to be responsible for plagiarism in any piece of work submitted for assessment shall be subject to the University’s dishonest practice regulations which may result in various penalties, including forfeiture of marks for the piece of work submitted, a zero grade for the paper, or in extreme cases exclusion from the University. The University of Otago reserves the right to use plagiarism detection tools.

While we strive to keep details as accurate and up-to-date as possible, information given here should be regarded as provisional. Individual lecturers will confirm teaching and assessment methods.