## Trinity College Dublin, The University of Dublin

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# Module MAU11001: Linear Algebra

Credit weighting (ECTS)
5 credits
Semester/term taught
Michaelmas term 2021-22
Contact Hours
11 weeks, 3 lectures including tutorials per week - per term
Michaelmas Term:

Lecturer
Prof. Miriam Logan,
Learning Outcomes
On successful completion of this module, students will be able to:
• operate with vectors in dimensions 2 and 3, and apply vectors to solve basic geometric problems;
• apply various standard methods (Gauss-Jordan elimination, inverse matrices, Cramer's rule) to solve systems of simultaneous linear equations;
• compute the sign of a given permutation, and apply theorems from the module to compute determinants of square matrices;
• demonstrate that a system of vectors forms a basis of the given vector space, compute coordinates of given vectors relative to the given basis, and calculate the matrix of a linear operator relative to the given bases;
• give examples of sets where some of the defining properties of vectors, matrices, vector spaces, subspaces, and linear operators fail;
• identify the above linear algebra problems in various settings (e.g. in the case of the vector space of polynomials, or the vector space of matrices of given size), and apply methods of the module to solve those problems.
Module Content

We will cover the following topics, yet not necessarily in the order listed.

• Lines, planes and vectors, dot and cross product.
• Linear systems, Gauss-Jordan elimination, reduced row echelon form.
• Matrix multiplication, elementary row operations, inverse matrix.
• Permutations, odd and even, determinants, transpose matrix.
• Minors, cofactors, adjoint matrix, inverse matrix, Cramer's rule.
• Vector spaces, linear independence and span, bases and dimension.
• Linear operators, matrix of a linear operator with respect to a basis.
• Change of basis, transition matrix, conjugate matrices.

Textbook We will not follow any particular textbook. Two typical references are

• Algebra by Michael Artin,
• Basic linear algebra by Blyth and Robertson.

Notes, homework assignments and solutions will be posted on Blackboard

Module Prerequisite
None for students admitted to the Mathematics, Theoretical Physics or Two-subject Moderatorships.

Assessment Detail
This module will be examined in a 2 hour examination in Michaelmas term. 20% homework, 80% final exam (based on homework and tutorials).

Re-assessments if required will consist of 100% exam.