Linear Algebra III

5 credits

Syllabus, Bachelor's level, 1MA026

A revised version of the syllabus is available.

Code: 1MA026
Education cycle: First cycle
Main field(s) of study and in-depth level: Mathematics G1F
Grading system: Pass with distinction (5), Pass with credit (4), Pass (3), Fail (U)
Finalised by: The Faculty Board of Science and Technology, 15 March 2007
Responsible department: Department of Mathematics

What is a course syllabus?

Entry requirements

Algebra II, Linear Algebra II

Learning outcomes

In order to pass the course (grade 3) the student should be able to

give an account of important concepts and definitions in the area of the course;

exemplify and interpret important concepts in specific cases;

formulate important results and theorems covered by the course;

describe the main features of the proofs of important theorems;

express problems from relevant areas of applications in a mathematical form suitable for further analysis;

use the theory, methods and techniques of the course to solve mathematical problems;

present mathematical arguments to others.

Higher grades, 4 or 5, require a higher level of proficiency. The student should be able to solve problems of greater complexity, i.e. problems requiring a combination of ideas and methods for their solution, and be able to give a more detailed account of the proofs of important theorems and by examples and counter-examples be able to motivate the scope of various results. Requirements concerning the student's ability to present mathematical arguments and reasoning are greater.

Content

Linear spaces over arbitrary fields, sums and direct sums of subspaces, the dimension formula, quotient spaces. Linear transformations. Linear functionals, the dual space, dual bases. The canonical isomorphism between a linear space and its bidual. Forms: bilinear, Hermitian, symmetric, alternating, quadratic. Inner product spaces: unitary, Euclidean, orthogonal projection, the method of least squares. Linear operators: Hermitian, symmetric, unitary, orthogonal, normal, polynomial, the spectral theorem (complex and real), eigenspaces and generalised eigenspaces, the characteristic polynomial and the minimal polynomial, Jordan's normal form (complex and real). Polar decomposition.

Instruction

Lectures and problem solving sessions.

Assessment

Written and, possibly, oral examination at the end of the course. Moreover, compulsory assignments may be given during the course.