Linear Algebra and Geometry

Unit aims

Linear Algebra and Geometry constitute the bedrock of higher mathematics. They are indispensable and used in one form or another throughout every mathematical discipline. This unit aims to lay down foundational concepts for studying maths at the undergraduate level and enable students to develop clear mathematical thinking.

Unit description

Linear Algebra and Geometry begins with the complex plane, conics, and hyperplanes in n-space, which leads to the straightforward ideas of vectors and matrices, and develops the abstract notion of vector spaces. This is one of the basic structures of pure mathematics; yet the methods of the course are also fundamental for applied mathematics and statistics.

The unit has standard pure mathematical axiomatic structure, from definitions of abstract objects to proving their properties, as theorems. This is the mission of Lectures. Most of the Examples are presented as weekly Homework, a large part of which will be considered in Problem Classes.

• Not paying enough attention to homework will create a feeling that more examples should have been provided.

Relation to other units

This unit provides foundations for all other units in the Mathematics Honours programmes.

Learning objectives

At the end of the unit, students should:

• have developed some familiarity with abstract mathematical thinking;
• be familiar with geometric objects like lines, planes and hyperplanes, and their axiomatic generalisation into vector spaces and linear maps;
• be able to solve linear equations using elementary operations;
• be able to work with matrix algebra, including matrix inverses, determinants, and eigenvalues and eigenvectors.

Transferable skills

Logical thinking, mathematical writing, problem casting and solving, assimilation of abstract ideas, axiomatic approach to mathematical structures.

Syllabus

Note: topics may not appear in exactly this order.

1. Euclidean plane, cosine, sine, and the complex numbers, Euclidean three-space.
2. Distance, lines, and hyperplanes in n-space.
3. Systems of linear equations, matrices, Gaussian elimination, row echelon form. Matrix algebra. Gaussian elimination as matrix multiplication. Inverse of a square matrix.
4. Linear maps from n-space to m-space; surjectivity, injectivity, kernels and rank-nullity. Matrix representations of linear maps.
5. Determinants. Cross products in three dimensions.
6. Fields, abstract vector spaces and their basic properties.
7. Subspaces, linear combinations and span. Linear dependence and independence.
8. Bases and dimension.
9. Linear transformations from one vector space to another; using matrices to represent linear transformations from one finite-dimensional vector space to another.
10. Rank and nullity of a linear transformation, and the relationship between them.
11. Eigenvalues and eigenvectors; characteristic polynomial of a matrix. Determinant and trace.
12. Inner products and inner product spaces. Orthonormal bases. Linear operator matrices in orthonormal bases. Gramm-Schmidt orthogonalisation.
13. Adjoint and self-adjoint operators. Unitary and orthogonal operators. Hermitian, symmetric and orthogonal matrices.
14. Diagonalisation of a matrix; properties of real symmetric matrices.

There are many good linear algebra texts. They come in different styles, some follow a more abstract approach, others emphasise applications and computational aspects. Some students may prefer the style of one book more than another.

The following is a selection of textbooks which cover a variety of styles:

• G. Strang, Linear Algebra and its Applications
• R. Allenby, Linear Algebra
• H. Anton and C. Rorres, Elementary Linear Algebra
• S. Lang, Linear Algebra
• S. Lipschutz and M. Lipson, Linear Algebra

The lectures will present the material in a different order from most textbooks. There is no required text. Notes taken by students of mostly theoretical material taught during lectures and examples from homework and problem classes should suffice to master the material. Attendance of all contact hours is mandatory.

• A full version of written notes is availabe online on Blackboard. However, lectures aim to give a broader and more creative perspective of the material, focusing on the depth and meaning of studied concepts. Therefore, even though there is a natural correlation between lectures and written notes, it will be somewhat approximate and not without occasional deviation.

Unit code: MATH11005
Level of study: C/4 (Honours)
Credit points: 20
Teaching block (weeks): 1 and 2 (1-24)
Lecturers: Dr John Mackay Dr Farhad Babaee

Pre-requisites

An A in Mathematics A-level or equivalent.

None

Methods of teaching

Lectures and problem classes supported by lecture notes, problem sheets and small-group tutorials.

Methods of Assessment

The pass mark for this unit is 40.

Formative assessment:

• Problem sheets set by the lecturer and marked by the students’ tutors.

Summative assessment:

• Two 1.5h exams (45% each)
• Coursework (10%)

NOTE: Calculators are NOT allowed in the examination.

For information resit arrangements, please see the re-sit page on the intranet.

Please use these links for further information on relative weighting and marking criteria.

Further exam information can be found on the Maths Intranet.