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Department of Mathematical Sciences MATH103 Introduction to Linear Algebra
Lecturer: Prof. Victor Goryunov Office: Room 519, Mathematical Sciences Building; email: goryunov@liv.ac.uk
Lectures: Monday 1pm and 2pm in Lecture Theatre B, Central Teaching Hub (no. 221 on the Campus Map), Tuesday 13.00 in Lecture Theatre (room 101), Electrical Engineering Building (no. 235 on the map). Attendance at lectures is compulsory. (See the Student Charter.) Make sure you scan your card. Tutorials: Normally on Tuesday afternoon. The tutorials will be in groups of about 20. They form an integral part of the course and attendance is compulsory. They start in Week 1. Set work Work will be set by the lecturer weekly on Monday at the 2pm lecture. It will be already available on VITAL from the previous Friday. The set work consist of two very similar parts: • set C – for solving and detailed discussion at the coming Tuesday tutorial, and • set H – written solutions to which you should hand to your MATH103 tutor by 4 pm next Thursday. All the problems are to be studied before the Tuesday tutorial. Your submitted solutions are marked by your tutor and handed back the following Tuesday; the lecturer’s solutions are available online the same day. Web publications All module materials will be available on VITAL as soon as they are needed in the class. Hard copies of some houndouts will also be distributed, but only once. Closer to the end of the semester, past papers will be put on VITAL. Class Tests There will be one class test during the module; the date will be announced at least a week in advance. Examinations The course will be assessed by a single 2 1 hour examination paper in January. Your perfor2 mance at the tutorial classes (including a mini-project consisting of 5 extra individual problems) contribute 10% to the final assessment (key skills). Another 10% of the final mark are coming from the homeworks. A resit examination is available in August-September if your mark in January is below 40%. (Detailed rules for passing the year are in the Handbook for Students.) Course Text: For complex numbers: Complex Numbers Made Simple by Verity Carr, Made Simple Books (Butterworth-Heinemann) (ISBN 0750625597) For linear algebra: Guide to Linear Algebra by David Towers, Macmillan Mathematical Guides (ISBN 033343627X). Linear Algebra by Steven A. Leduc (Wiley) (ISBN 0822053314) is more elementary. Another useful (but very expensive) book: H. Anton, Elementary Linear Algebra, 8th edition (or later), Wiley (ISBN 0471170550). The syllabus will include most of the complex numbers book and Chapters 1-4 and parts of 7-8 of Towers, or alternatively most of Leduc except for ‘Linear Transformations’. In Anton this is chapters 1-3, 4.1, 5.2-5.6 (excluding some of the more abstract parts), Chapters 7, 9.5-9.7 and a small amount of additional material but the treatment in this module will be more akin to that in Towers and Leduc. Aims: To develop techniques of complex numbers and linear algebra, including equation solving, matrix arithmetic and the computation of eigenvalues and eigenvectors; To develop geometrical intuition in 2 and 3 dimensions; To introduce students to the concept of subspace in a concrete situation; To provide a foundation for the study of linear problems both within mathematics and in other subjects; To introduce students to the concept of mathematical proof. Outline syllabus [Some topics may have to be omitted] Algebraic and geometric representation of complex numbers. Roots of unity and solution of simple equations with complex coefficients. 1

Vectors in 2-space and 3-space. Geometric vectors, addition, scalar multiplication. Coordinates (components). Rules of vector arithmetic. Dot product, cross product, scalar triple product, volume of a parallelepiped, condition for coplanarity of three vectors. Equations of planes and lines in 3-space, line of intersection of two planes, distance from a point to a plane. Idea of linear independence and dependence of vectors. Matrix algebra: addition, scalar multiplication, multiplication of matrices, transpose, orthogonal matrices, connection with rotations and reflections. Symmetric matrices. Systems of linear equations: consistent, inconsistent systems, solution by row reduction. Matrix form of equations. Invertible matrices, calculation of the inverse. Determinants. Eigenvalues and eigenvectors. Similar matrices and diagonalisation (including examples of non-diagonalisable matrices). Subspaces. Linear dependence and independence revisited. Basis of a subspace, coordinates. Basis theorem, dimension. Description of all subspaces of R3 . Here are some applications of linear algebra which you may meet later in your course: • geometry of lines and planes; • computer-aided geometric design; • input - output models (e.g. in economics); • linear programming; • data analysis in statistics; • population dynamics; • cryptography, game theory, graph theory. Basic Advice on Lectures • Arrive a few minutes before the lecture starts. Late arrivals disrupt others’ concentration and are altogether antisocial. • Make sure you have pen and paper! A pen of another colour (e.g. red) is very handy for marking crucial points. • If you are easily distracted then the solution is simple: sit near the front! • Chatting in lectures is extremely antisocial behaviour since it may prevent others from learning something they wish to know. Save the chat for between lectures. • Number the pages of your notes and put the date at the start of each lecture’s notes. Leave plenty of space, e.g. half of each page: it may be useful to annotate the notes later. Remember that you will need the notes when you try the homework questions, and when you revise for the exam. • Aim to take down what is written on the board or on an overhead projector slide, unless of course it is also given out to you on paper. • It isn’t easy, but is extremely worthwhile, to develop the art of noting in an abbreviated form extra comments made by the lecturer, aside from the work written on the board. • By all means ask questions in lectures. You will also have a chance to ask questions in tutorials. Discussing problems with your fellow-students can be very valuable. There is no objection to working with friends on your homework problems, but the sheets you actually hand in should be your own work, since how otherwise do we know how you are progressing?

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