Vectors, Geometry and Linear Algebra

This is a year one module.  It is taught intensively by Birmingham academics over eight weeks. There is continuous assessment and a final examination.

This module introduces a number of powerful ideas found in all area of mathematics and its applications that are broadly geometric in flavour. Complex numbers, which turn out to underpin a profound unification of many ideas in mathematics, are introduced.

Vectors, which have both magnitude and direction, provide a natural way to describe lines and planes and are the appropriate language with which to model physical systems in mechanics. Matrices provide both a convenient way to deal with large systems of linear equations and to transform vectors and coordinate systems.

This in turn leads to the theory of linear algebra and vector spaces.The abstraction of the notion of a vector space is another powerful unifying theory that is found across mathematics, with applications in abstract group theory, video games and signal processing. Coordinate systems for the Euclidean plane are discussed and the standard theory of conic sections is developed. The module also introduces the fundamental proof technique of Mathematical Induction.

By the end of the module students should be able to:

  • State the Principle of Mathematical Induction and demonstrate its use in typical proofs.
  • Work with complex numbers, perform standard calculations, appreciate the relationship to the roots of polynomials and trigonometric identities.
  • Perform vector calculations, including scalar and vector products, and describe lines and planes in terms of vectors.
  • Perform matrix calculations including reducing to echelon form and calculating inverses and determinants, use matrix methods to solve systems of linear equations. Understand the relationship between invertible matrices and systems of equations with solutions.
  • State the definition of a vector space and associated definitions of, for example, subspaces and bases. Calculate dimension and bases for given examples.
  • Understand the notion of a linear transformation and calculate the matrix of a linear transformation with respect to a given basis.
  • Use Cartesian and polar coordinate systems in the plane and find parametric expressions for simple curves.
  • Recognise, classify and express conic sections in various forms.
  • Typeset simple text and mathematics using the LaTeX package.