Module Descriptors

Mathematics

MA283: Linear Algebra (5 ECTS)

This course covers the theory and practice of Linear Algebra.

Taught in Semester(s) 2. Examined in Semester(s) 2.

Workload: 100 hours (24 Lecture hours, 12 Tutorial hours, 64 Self study hours).

Module Learning Outcomes. On successful completion of this module the learner should be able to:

  1. Perform matrix computations, solve linear systems of equations and determine bases of the related subspaces;
  2. Find the nullspace, row space and column space of a matrix; apply the rank-nullity theorem;
  3. Find bases for the canonical subspaces associated with a linear transformation;
  4. Use eigenvector bases to diagonalize a square matrix and use the diagonalization to analyze the properties of the matrix;
  5. Compute orthogonal projections and least squares solutions of overdetermined linear systems;
  6. Write proofs of facts about vector spaces and linear transformations;
  7. Identify practical situations where the techniques of Linear Algebra can be applied and carry out the application.

Indicative Content

  1. Linear Systems and Matrices: Gaussian Elimination. Elementary row operations. Invertible and singular matrices. Finding the inverse of a matrix by Gauss-Jordan Elimination. Elementary matrices. Determinants. Applications to traffic flow, networks and graphs.

  2. Vector Spaces and Linear Transformations: Vector spaces and subspaces. Linear combinations, spanning sets, linear independence. Bases. Finding bases for subspaces of $\mathbb{R}^n$. The kernel and image of a linear transformation and their dimensions. Finding the kernel (nullspace), image (column space) and the rank of a matrix. Fundamental Theorem of Linear Algebra. Applications to coding.

  3. Eigenvalues and Eigenvectors: Matrix representations. Eigenvalues and eigenvectors of a linear transformation. The characteristic polynomial. Diagonalization of square matrices and applications to coupled systems of linear differential equations and to page rank algorithms.

  4. Inner products: Inner product spaces. Orthogonality. The Cauchy-Schwarz Inequality. Orthogonal projections. Orthogonal diagonalization of symmetric matrices. The Spectral Theorem. Applications to least squares solution of overdetermined systems and regression.

Module Resources

  1. "Linear Algebra", S. Lipschutz & M. Lipson, Schaum Outline Series (Primary Textbook).
  2. "Linear Algebra and Its Applications", G. Strang, Brooks/Cole (Supplementary reading)

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