MA2008: LINEAR ALGEBRA I (2018-2019)

This is the first part of the two parts course on linear algebra. The whole course contains the following topics:

* Fields
* Vector spaces
* Linear maps
* Matrices
* Linear equations
* Eigenvalues and eigenvectors

Syllabus

• Basic set theory: Naive understanding of a set and of relations, in particular equivalence relations. Maps (injective/surjective/bijective). The principal of induction.
• Definitions, examples and elementary properties of groups, rings and fields. In particular integers mod n and fields of prime order.
• Solving a linear system over a field. Elementary row operations, row echelon form, Gaussian algorithm for solving a linear system over a field.
• Vector spaces and K-algebras. Definition of a vector space over a field. Examples.
• Subspaces of a vector space, intersection and sum of subspaces.
• Span, spanning sets. Linear independence. Basis, dimension. Elementary results about bases and dimension.
• Linear transformations. Definition, kernel, image, the matrix of a linear transformation with respect to bases. The rank of a matrix and the rank-nullity theorem.
• Invertible matrices and the Gauss-Jacobi method for finding the inverse of a matrix.
• A brief introduction to determinants of matrices with entries in arbitrary fields.

Course Aims

There are three main aims of the course. These are (1) to give an introduction to basic mathematical

and algebraic concepts like sets, relations, functions, groups, rings, fields, (2) to give a solid introduction to commonly used methods and results from linear algebra, and (3) to give the students an impression of how a mathematical theory is build up through definitions, lemmas/propositions/theorems and proofs.

Learning Objectives

By the completion of the course a student should be able to

• work with sets, relations, functions from the viewpoint of naive set theory;

• define the basic concepts of Linear algebra, such as a vector space over a field, a linear transformation etc., and understand the connections among linear transformations, matrices and systems of linear equations;

• define linear independence, span and related notions and learn how to determine independence;

• prove simple results about matrices;

• systematically solve a system of linear equations;

• describe row reduction in terms of elementary matrices, and prove simple results in terms of elementary matrices;

• find the inverse of a given matrix, if it exists, and know how to determine whether or not it does;

• define vector subspaces of a vector space, know how to determine if a given subset is a vector subspace, and apply to certain vector subspaces associated with a matrix and/or a linear transformation;

• prove elementary results about vector spaces and the connection with matrices and systems of linear equations;

• define a basis of a finite dimensional vector space, find a basis for a given vector space, define dimension, and know how to calculate it;

• prove elementary results about bases and dimension;

• define rank and nullity and know how to calculate them;

• define and calculate the determinant of a matrix;

Information on contact teaching time is available from the course guide.

More Information about Week Numbers

1st Attempt

1 two-hour written examination (80%); in-course assessment (20%).

There are no assessments for this course.