## Trinity College Dublin, The University of Dublin

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# MAU22E01 Engineering mathematics III

Module Code MAU22E01 Engineering mathematics III Semester 1 5 Prof. Dmitri Zaitsev MAU11E02 Engineering mathematics II

### Assessment Details

• This module is examined in a 2-hour examination at the end of Semester 1.
• Continuous assessment contributes 10% towards the overall mark.
• Re-assessment, if needed, consists of 100% exam.

### Contact Hours

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

### Learning Outcomes

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.

### Module Content

• 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.