## Trinity College Dublin, The University of Dublin

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# Module MAU11001: Linear Algebra

Credit weighting (ECTS)
10 credits
Semester/term taught
Michaelmas term vand Hilary Term2019-20
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

Hilary Term:

Lecturer
Prof Sergey Mozgovoy

Learning Outcomes
On successful completion of this module, students will be able to:
• Compute the rank of a given linear operator, and use proofs of theoretical results on ranks explained in the course to derive similar theoretical results;
• Compute the dimension and determine a basis for the intersection and the sum of two subspaces of a given space, determine a basis of a given vector space relative to a given subspace;
• Calculate the basis consisting of eigenvectors for a given matrix with different eigenvalues and, more generally, calculate the Jordan normal form and a Jordan basis for a given matrix;
• Apply Gram-Schmidt orthogonalisation to obtain an orthonormal basis of a given Euclidean space;
• Apply various methods (completing the squares, Sylvester's criterion, eigenvalues) to determine the signature of a given symmetric bilinear form;
• Identify the above linear algebra problems in various settings (e.g. in the case of the vector space of polynominals, or the vector space of matices of given size), and apply methods of the course to solve those problems
Module Content

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

• Kernel and image, rank and nullity, dimension formula.
• Characteristic polynomial, eigenvalues and eigenvectors, Jordan form.
• Cayley-Hamilton theorem, minimal polynomial of a linear operator.
• Invariant subspaces, orthogonal complements, direct sums.
• Inner product spaces, orthonormal bases, Gram-Schmidt, Bessel's inequality.
• Bilinear and quadratic forms, Sylvester's criterion, spectral theorem.
• Applications: recurrence relations, least squares approximation.

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 the web page https://www.maths.tcd.ie/~mozgovoy/2020H_MAU11100/

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

Assessment Detail
This module will be examined in a 3 hour examination in Trinity term. Homework assignments will be due every Thursday. 20% homework, 80% final exam (based on homework and tutorials).

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