MA2002: Matrix Algebra

School Cardiff School of Mathematics
Department Code MATHS
Module Code MA2002
External Subject Code G100
Number of Credits 10
Level L5
Language of Delivery English
Module Leader Dr Andreas Artemiou
Semester Autumn Semester
Academic Year 2014/5

Outline Description of Module

A lecture based module, it provides an introduction to the manipulative parts of matrix algebra, and is essential for further work in all areas of mathematics.

On completion of the module a student should be able to

  • Interpret orthonormal bases and linear independence.
  • Determine the rank of a system of linear equations, to find kernels and spanning vectors for image spaces.
  • Explain what is meant by the rank of a matrix and apply this understanding.
  • Use orthonormalisation processes.
  • Find eigenvalues, eigenvectors and determine the corresponding eigenspaces.
  • Classify the various matrix types (symmetric, Hermitian, singular etc).
  • State and employ theories relating to the similarity of matrices, find canonical forms and use the Cayley-Hamilton theorem to derive the characteristic equation of a matrix.
  • Classify bilinear and quadratic forms and diagonalise them by completing the square.
  • Analyse quadrics by transforming them to canonical form in order to classify them.
  • Use Sylvester's test and simultaneously diagonalise two symmetric matrices (one positive definite) and their quadratic forms.

How the module will be delivered

27 - 50 minute lectures

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 solutions to given exercises.

Skills that will be practised and developed

Skills:
Obtain an orthogonal system of vectors. Find eigenvalues, eigenvectors and characteristic equations and use them to diagonalise a matrix. Diagonalise a quadratic form. Compute a vector or matrix norm.

Transferable Skills:
Manipulation of vectors and matrices. Calculation of eigenvalues, eigenvectors and characteristic equations. Transformation of coordinates, including isometries (especially as applied to quadratic forms). Employ vector and matrix norms.

How the module will be assessed

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

The 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.

Assessment Breakdown

Type % Title Duration(hrs)
Exam - Autumn Semester 100 Matrix Algebra 2

Syllabus content

  • Vectors, n-tuples and Euclidean spaces
    • rank, inner product, orthogonal vectors, orthonormal bases and orthogonal matrices
    • Linear dependence and independence
    • Change of basis
  • Linear transformations and matrices
    • products of transformations, isometries, change of basis and the orthonormalisation process
  • Eigenvalues and eigenvectors of matrices
    • the characteristic equation, properties of eigenvalues and eigenvectors (linear independence of eigenvectors corresponding to distinct eigenvalues)
    • Properties of eigenvalues and eigenvectors of symmetric matrices (eigenvalues are real numbers, orthogonality of eigenvectors corresponding to distinct eigenvalues)
  • Classification of matrices
    • real, symmetric, Hermitian, non-singular, similar, diagonal and orthogonal.
  • Diagonalisation of matrices:
    • necessary and sufficient conditions ensuring that a matrix can be diagonalised
    • the structure of the diagonalising matrix
    • calculation of powers of diagonisable matrices
    • orthogonal diagonalisation of symmetric matrices
  • The Cayley-Hamilton theorem (proof for the case of diagonalisable matrices)
    • application to finding the inverse of a matrix
  • Quadratic forms
    • the associated symmetric matrix of a quadratic form
    • diagonalisation via completion of the square
    • simultaneous diagonalisation of two quadratic forms
    • Sylvester's test
    • vector and matrix norms

Essential Reading and Resource List

Not applicable.

Background Reading and Resource List

 Elementary Linear Algebra, Anton, H., Wiley


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