Your previous studies will have trained you to solve small sets of simultaneous linear equations in a number of unknowns – at least up to two!  This module provides a systematic study of general sets of linear equations which leads to methods for their solution.  It also gives an understanding of the properties of these solutions and answers such questions as whether a solution exists and if so how many solutions there are.

Sets of linear equations can best be written using matrices, and the module continues with an introduction to matrices and their basic algebraic properties.  The link with linear equations and their solutions is explored and the concept of inverting (or ‘dividing by’) a matrix investigated.  The determinant of a matrix is introduced (it measures the ‘size’ of a matrix in a sense) and the properties and calculation of determinants developed.

All of this material can be presented in a more abstract setting and this is known as Linear Algebra.  This approach is fundamental to virtually all branches of mathematics and plays an essential and increasing role in modules in later years of study.  In this module you will be introduced to some of the basic ideas of Linear Algebra.  This should both deepen your understanding of the earlier material of this module and prepare you for more advanced study.

Free Standing Module Requirements:  A pass in A-Level Mathematics of at least Grade A

Precursor Modules:  MA0122 Algebra I

• solve sets of linear equations and interpret the solutions
• manipulate matrices and determinants, including the inversion of matrices, both by row operations and by using determinants
• derive and interpret basic ideas of Euclidean vector spaces, including linear independence and basis.

27 - 50 minute lectures

5 - 50 minute tutorial classes

Some handouts will be provided in hard copy or via Learning Central, but students will be expected to take notes of lectures.

Students are also expected to undertake at least 50 hours private study including preparation of worked solutions for tutorial classes.

Skills:

Facility with matrices and array operations

Transferable Skills:

Linear modelling.  Ability to handle abstract concepts such as n-dimensional space.

Formative assessment is by means of regular tutorial exercises.  Feedback to students on their solutions and their progress towards learning outcomes is provided during lectures and tutorial classes.  

The in-course element of summative assessment is based on selected problems on the tutorial sheets.

The major component of summative assessment is the written examination at the end of the module.  This gives students the opportunity to demonstrate their overall achievement of learning outcomes.  It also allows them to give evidence of the higher levels of knowledge and understanding required for above average marks.

The examination paper has two sections of equal weight.  Section A contains a number of compulsory questions of variable length but normally short.  Section B has a choice of two from three equally weighted questions.

• Simultaneous linear equations
  • Systems of two linear equations in two unknowns. General systems of linear equations in n unknowns. Solution set.
  • Reduction of system to echelon form by elementary row operations (Gaussian elimination). Leading and non-leading unknowns. Echelon rank. Dimension of solution set.
  • Geometric interpretations of systems of equations and solutions sets for the cases of two and three unknowns.

• Matrices
  • Definitions and basic properties of addition, scalar multiplication and matrix multiplication.
  • Transpose of a matrix, inverse matrices and their basic properties.

• Matrices and simultaneous linear equations
  • Matrix form of a system of linear equations.  Finding solutions with the help of inverse matrices.
  • Elementary row operations and elementary matrices.  Elementary row operations method for finding inverse matrices.
  • Equivalence of Ax = 0 only for x = 0 to the invertibility of A.

• Euclidean spaces (Rⁿ)
  • Operations in Euclidean spaces. Linear independence of vectors.  Subspaces. Bases. Dimensions of subspaces.

• Determinants
  • Permutations, inversions. Definition of the determinant of a n x n matrix.
  • Properties of determinants. Minors and cofactors. Cofactor expansions.
  • Equivalence of invertibility of matrix to non-vanishing of the determinant. Formula for inverse of a matrix. Cramer's rule.

None given.

Elementary Linear Algebra, Anton H., Wiley

Hamilton, A. G., A first course in Linear Algebra, CUP