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MAU11100 Linear algebra

Module Code MAU11100
Module Title Linear algebra
Semester taught Semesters 1,2 (yearlong)
ECTS Credits 10
Module Lecturer Prof. Miriam Logan
Module Prerequisites N/A

Assessment Details

  • This module is examined in a 3-hour examination at the end of Semester 2.
  • Continuous assessment contributes 20% towards the final grade for this module.
  • Any failed components are reassessed, if necessary, by an exam in the reassessment session.

Contact Hours

11+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

  • Apply various standard methods to solve systems of linear equations.
  • Compute the sign of a given permutation and apply known results to compute the determinant of a given square matrix.
  • Demonstrate that a given set of vectors forms a basis for a vector space, compute the coordinates of a given vector relative to this basis, and find the matrix of a linear operator with respect to this basis.
  • Give examples of sets where some of the defining properties of vectors, matrices, vector spaces, subspaces and linear operators fail.
  • Find an explicit basis for the null space of a given matrix.
  • Solve linear recursive relations involving two or more terms.
  • Apply standard techniques to obtain the Jordan form and a Jordan basis for a given complex square matrix.
  • Compute the matrix of a bilinear form with respect to a given basis.
  • Apply various standard method to determine the signature of a given symmetric bilinear form.
  • Combine various results established in the module to either prove or disprove statements involving concepts introduced in the module.

Module Content

  • Lines, planes and vectors, dot and cross product.
  • Linear systems, Gauss-Jordan elimination, reduced row echelon form.
  • Matrix multiplication, elementary row operations, inverse matrix.
  • Odd and even permutations, determinants, transpose matrix.
  • Minors, cofactors, adjoint matrix, inverse matrix, Cramer's rule.
  • Vector spaces, linear independence and span, basis and dimension.
  • Linear operators, matrix of a linear operator with respect to a basis.
  • Diagonalisation: recursive relations, diagonalisable matrix, eigenvalues, eigenvectors, characteristic polynomial, null space, nullity.
  • Jordan forms: generalised eigenvectors, column space, rank, direct sum, invariant subspace, Jordan chain, Jordan block, Jordan form, Jordan basis, similar matrices, minimal polynomial.
  • Bilinear forms: matrix of a bilinear form, positive definite, symmetric, inner product, orthogonal and orthonormal basis, orthogonal matrix, quadratic form, signature, Sylvester's criterion.

    Recommended Reading

    • Algebra by Michael Artin.
    • Basic linear algebra by Blyth and Robertson.
    • Elementary linear algebra with applications by Anton and Rorres.