Mathematical Studies
MA313: Linear Algebra (5 ECTS)
An advanced course in the theory and application of linear algebra, including the theory of vector spaces, linear independence, dimension and linear mappings.
Taught in Semester(s) 1. Examined in Semester(s) 1.
Workload: 126 hours (24 Lecture hours, 12 Tutorial hours, 90 Self study hours).
Module Learning Outcomes.
On successful completion of this module the learner should be able to:
- Identify and categorise examples of linear and nonlinear spaces.
- Decide whether or not a given set is a spanning set for a given vector space.
- Decide whether or not a given subset of $\mathbb{R}^n$ is linearly independent.
- Compute the rank of a matrix.
- Find a basis for the image and kernel of a linear transformation.
- Compute the matrix representation of a linear transformation on finite dimensional vector spaces.
- Use the Gram-Schmidt process to find an orthonormal basis for an inner product space.
- Prove the Cauchy-Schwarz inequality.
- Compute the Fourier coefficients of some simple periodic functions.
- Find the linear least squares fit to a given data set.
Indicative Content
- Vector Spaces and Linear Subspaces. Axioms and examples, linear combinations, spanning sets. (4 lectures)
- Linear Independence and Rank. Dependent and independent sets, bases, dimension, rank of a matrix.(5 lectures)
- Linear Transformations. Kernel, image, rank-nullity theorem, matrix representations. (5 lectures)
- Inner Product Spaces. Bilinear forms. Cauchy-Schwarz Inequality, Orthogonal sets, Gram-Schmidt process, function spaces and Fourier Series (some examples only), least squares approximation. (6 lectures)
- Applications. (4 lectures)
Module Resources
- "Linear Algebra" (4th ed.), by Lipschutz and Lipson (McGraw-Hill)
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