Linear Algebra

• Credit value: 15 credits at Level 4
• Convenor: Dan McVeagh
• Assessment: a short-problem set (20%), online quiz (10%) and two-hour examination (70%)

In this module we provide you with core linear algebra concepts including matrices, systems of linear equations, vectors, Markov chains and linear programming. You will develop the skills to perform matrix operations, solve systems of linear equations, explore vector spaces and orthogonality, and apply linear algebra techniques to areas like Markov chains and optimisation problems.

Indicative syllabus

• Matrices and systems of linear equations: operations on matrices, transposes, symmetric and antisymmetric matrices, invertible matrices, consistent and inconsistent equations, matrix form of a system of linear equations, elementary row operations, solving a system of linear equations, inverting a square matrix
• Determinants: cofactors, evaluating the determinant of a square matrix, properties of the determinant
• Real vectors: the dot product, the length of a vector, linear combinations, spanning subspaces, linearly independent vectors, bases, orthogonality, the angle between two vectors, orthogonal bases and the Gram-Schmidt process
• Eigenvalues and eigenvectors: finding eigenvalues and eigenvectors of a square matrix, the characteristic equation, diagonalisation and powers of square matrices
• Markov chains: transition matrices, state vectors, Markov matrices, regular transition matrices, steady state vectors
• Linear programming: linear inequalities, formulation of a linear programme, objective function and constraints, graphical solutions, introduction to the simplex method

By the end of this module you will be able to:

• solve systems of linear equations
• find an orthogonal basis of a subspace of n-dimensional real space
• evaluate the determinant, eigenvalues and eigenvectors of a square matrix
• show when a square matrix is diagonalisable, and diagonalise such matrices
• demonstrate knowledge of the properties of n-dimensional real space
• model a finite stochastic process using a Markov matrix, and find the solution
• model optimisation problems as a linear programme.