CM222A LINEAR ALGEBRA
Lecturer :
Professor David Burns
Room :
418
Office hours :
Wednesday 11 – 12, Thursday 4 – 5.
E-mail :
david.burns@kcl.ac.uk
Other staff who can be consulted on this course are: Dr. Manuel Breuning (Room 408) and
Dr. John Silvester (Room 413a). However, since the content of this course is standard, most
lecturers in the Department could probably help you.
Lecture times. Wednesday 12– 1 in Room 2B08 and Thursday 2 – 4 in Room 2B08
Tutorials.
Thursday 12-1: Group 1 in Room GFSB2, Group 2 in Room 1B22 and Group 3 in Room. 436.
Thursday 1-2 : Group 4 in Room 423
Prerequisites.
CM113A or similar course giving familiarity with basic vectors and matrices in Rn (especially R2 and
R3); CM121A or some course containing abstract algebraic ideas.
Assessment.
There will be three class tests during the course, each of which will carry 5% of the final grade.
Students are warned that they put themselves at a grave disadvatage if they do not take the class tests
seriously. There will be no duplicate class tests for absentees. Students who miss class tests
without good reason will get zero marks. Students who miss class tests for good reason (e.g. can
supply medical certificates) will be given an appropriately higher weighting for the final examination.
Assignments.
Exercises will be set and marked weekly and the performance of each student will be recorded for
later reference. Students must realise that for successful progress in the course, the coursework
has to be done regularly, as it is handed out.
Aims.
Linear Algebra is a central topic in Mathematics and its use pervades the whole of the subject, both
pure and applied. The course sets concepts from the ‘methods' course CM113A in the more general
framework of abstract vector spaces. It also gives the precise definitions and proofs that are essential
for much of higher mathematics. Further concepts and methods are also introduced in the same
manner. Examples from Rn and Cn will be given where possible to illustrate the geometrical
meaning behind the algebraic ideas. The course will emphasize the interplay between abstract and
more concrete ideas.
Objectives.
It is expected that students will understand and be able to reproduce the proofs of the major theorems
of the subject. They should also appreciate the logical relationships between the different parts of the
subject and be able to use the ideas of the course in a variety of straightforward situations. Students
will be expected to carry out procedures and calculations with specific vector spaces and linear maps
(e.g. change of basis, diagonalisation etc.).
Syllabus.
General definition and properties of vector spaces, subspaces and linear maps. Linear independence,
basis and dimension. Rank and nullity for linear maps. The relation between linear maps and
matrices. Change of basis and similarity of matrices. Inverse matrices. Eigenvectors, eigenvalues and
diagonalisation of matrices. Inner product spaces and orthogonal diagonalisation.
Books.
The course will not follow one particular textbook. There is a vast array of books on linear algebra
that contain the material of the course (often also covering the preliminary material from linear
methods). Here is a small sample, listed in roughly increasing order of sophistication:
1. ‘Elementary Linear Algebra’, Howard Anton, 8th ed., Wiley, 2000.
2
‘Linear Algebra with Applications', W. Keith Nicholson, 3rd ed. PWS 1995.
3
‘Linear Algebra', RBJT Allenby, Edward Arnold Modular Mathematics, 1995.
4
‘Elementary Linear Algebra' W. Keith Nicholson,1st ed., McGraw-Hill, 2001.
5. ‘Introduction to Linear Algebra’ Thomas A. Whitelaw, Blackie 1991.
6. ‘Basic Linear Algebra’ T.S. Blyth and E.F. Robertson, Springer 1998.
7. ‘Linear Algebra’ S. Lang, Addison-Wesley, 1966.
8. ‘Linear Algebra’ A. Mary Tropper, Nelson 1969.