Module Content

The module covers
  • vectors and linear transformations,
  • matrices, their inverses and eigenvectors,
  • modular arithmetic and number theory,
  • and explains how these topics can be applied to
  • computer graphics,
  • linear models in economics,
  • data encryption.

Module Coordinates

  • Lecturer: John Burns
  • Lectures: Wednesday 10-11 AC213, Thursday 10-11 Dillon lecture theatre
  • Tutorials: Monday 12-1 , AC213
  • Recomended text: Algebra and Geometry: An introduction to university mathematics by Mark V. Lawson (available on the module Blackboard page).
  • Problem sheet: available here.
  • Module Website: Information and module documents will be posted to this site, which is linked from the Blackboard MA133A Analysis & Algebra1 (Algebra section) pages. Blackboard will also be used for announcements and for posting grades.

Module Assessment

The assessment consists of a 2 hour exam (covering both algebra and calculus). The continuous assessment for this course is done as part of the module MA131. The continuous assessment for MA133 comprises 50% of the mark for MA131 (the other 50% is the CA for MA135 whch takes place in semester 2.

The continuous assessment is submitted online. A link will be posted here and on Blackboard when that site is available.

Supplementary Material and News

Clicker opinion polling may be used in some lectures.

Lecture Notes

Lecture Notes
(click on number)

Lecture Summaries
1
We define and learn to visualise vectors in the (two dimensional Euclidean) plane. We define the operations of scalar multiplication of a vector by a real number (a scalar) and addition of two vectors. Using these two operations we obtain the parametric equation of any line in the plane and we relate it to the usual equation of a line, familiar to us from school.
2
We study transformations (or maps or functions) from the plane to itself that respect addition of vectors and scalar multiplication, i.e. Linear Transformations. We derive a formula for such transformations in terms of the components (or co-ordinates) of a vector. We store this information that describes a linear transformation in a 2 by 2 matrix.
3
We look at some examples of linear transformations and transformations that are not linear. We find the matrices for rotations and reflections. Encoding the geometry of linear transformations into the algebra of matrices allows Pixar to manipulate computer images effectively.
4
We learn how to multiply two 2x2 matrices and see that the answer corresponds to the matrix for the composition of the two linear transformations corresponding to the original two matrices. As an application we write systems of linear equations as one matrix equation and we use matrix multiplication to find the image of a vector under a linear transformation.
5
We learn how to multiply any mxn matrix A by any nxp matrix B to obtain an mxp matrix AB. We note that if the numbers in a row were the prices per unit of the components needed to manufacture a product and the numbers in a column were the numbers of units of each component needed then multiplying the row by the column gives the price of manufacturing the product. As an application we write a system of two linear equations in two unknowns as one matrix equation and use the inverse matrix to solve it.
6
We recall how to get the inverse matrix of a 2x2 matrix whose determinant is nonzero. We use the inverse matrix to solve a system of linear equations. We matrix multiplication to find the image of a given line under a linear transformation.
7

8
We define the dot product of two vectors in the plane and we use it prove that the determinant of a 2x2 matrix is (up to sign) equal to the change of area under the linear transformation corresponding to the matrix. We use this fact to explain the properties of the determinant.
9
We introduce eigenvalues and eigenvectors of a linear transformations. We find the eigenvalues and eigenvectors of a reflection in a line in the plane and we observe that most rotations have no eigenvectors. We derive a general method to find the eigenvectors and eigenvalues of a linear transformation when they exist.
10
Given a linear transformation and a vector in the plane we test to see if the vector is an eigenvector and if it is we determine the corresponding eigenvalue. We use eigenvalues and eigenvectors to find any power of a given matrix. As an application we find the steady state of a two state Markov process.
11
We introduce the determinant of a 3x3 matrix A. When the determinant is nonzero we obtain a formula for the inverse of A. We illustrate the process with an example.
12
In this lecture we consider more examples of using eigenvalues and eigenvectors to calculate powers of a matrix. We also solve some problems similar to the fourth online homework.
13
We introduce Euclid's algorithm and use it to find the greatest common divisor of two positive integers. We also use the algorithm to express the greatest common divisor as an integer (linear) combination of the two given positive integers.
14
We consider a class of problems (involving a fixed natural number n>1) where, in order to answer the problem, we treat all numbers that leave the same remainder when divided by n as "the same". Formalising this concept of "sameness" we introduce the set of integers modulo n. We learn how to add and multiply two elements in this set of integers modulo n.
15
In this lecture we learn how to subtract in the integers mod n and to find the inverse of an integer a mod n when the gcd(a,n)=1. We use modular arithmetic to find the missing digit in an ISBN.
16
In this lecture we use modular arithmetic to encipher text. We focus on sending the text one letter at a time using an affine encryption function. We discuss how to decrypt. We briefly consider encrypting two letters at a time. We also learn how to solve simultaneous congruences.
17

18

19

20

21

22

23

24