MAU22E01 Engineering mathematics III

Module Code	MAU22E01
Module Title	Engineering mathematics III
Semester taught	Semester 1
ECTS Credits	5
Module Lecturer	Prof. Dmitri Zaitsev
Module Prerequisites	MAU11E02 Engineering mathematics II

11 weeks of teaching with 3 lectures and 1 tutorial per week.

On successful completion of this module, students will be able to

Relate linear systems, linear transformations and their matrices.
Check whether a system of vectors is linearly independent and/or a basis.
Calculate the dimension of a subspace.
Calculate the rank and the nullity of a matrix.
Construct a basis for the row, column and null spaces of a matrix.
Calculate the eigenvalues and the eigenvectors of a square matrix.
Apply the Gram-Schmidt process to transform a given basis into an orthogonal basis.
Solve ordinary differential equations using general and particular solutions.
Calculate the Fourier series of a given function and analyse its behaviour.
Apply Fourier series to solve ordinary differential equations.
Calculate the Fourier transformation of a given function.

Euclidean n-space and n-vectors.
Linear transformations and their matrices, subspaces, linear combinations of vectors, subspaces spanned by a set of vectors, linear independence of a set of vectors.
Basis and dimension, standard basis in n-space, coordinates of vectors relative to a basis.
General and particular solutions for a linear system.
Row, column and null space of a matrix, finding bases for them using elementary row operations, rank and nullity of a matrix.
Inner products, lengths, distances and angles.
Orthogonal and orthonormal bases relative to an inner product, orthogonal projections to subspaces, Gram-Schmidt process.
Eigenvalues and eigenvectors of square matrices.
Fourier series for periodic functions, Euler formulas for the Fourier coefficients, even and odd functions, Fourier cosine and Fourier sine series, Fourier integral and Fourier transform.