University Of Limerick
Further Linear Algebra
MS 4122
Iowa State Course Substitution
Matrices and Linear Algebra
MATH 207
Course Info
International Credits:
6.0
Converted Credits:
3.5
Country:
Ireland
Language:
English
Course Description:
Module Code  Title:
MS4122  FURTHER LINEAR ALGEBRA
Year Last Offered:
2017/8
Hours Per Week:
Grading Type:
N
Prerequisite Modules:
MS4131
Rationale and Purpose of the Module:
Course restructuring in response to Project Maths.
The aim of this module is to build the student's understanding of Linear Algebra to a more advanced level.The
module includes a formal treatment of Vector Spaces and Inner Product Spaces followed by a careful treatment of
the properties of vectors and matrices on R^n and C^n.
12/5/2017 Book of Modules
https://bookofmodules.ul.ie/ 2/3
Syllabus:
Axiomatic treatment of Vector Spaces and Inner Product Spaces.
Linear Independence, spanning sets.
Bases & Dimension.
Inner products/norms.
Angles/orthogonality in Inner Product Spaces.
Orthonormal bases/Gram Schmidt Orthogonalisation.
Linear transformations/change of basis.
Properties of matrices.
Rank, row space, column space, null space.
Vector norms on R^n and C^n.
Existence and uniqueness of matrix inverse/relation to matrix rank.
Fredholm Alternative.
Unitary and Hermitian properties of matrices.
Eigenvalue & Eigenvector Topics.
Eigenvalue decomposition for Hermitian matrices.
Algebraic & Geometric Multiplicity.
Defective Eigenvalues and Matrices.
Similarity Transformations.
Diagonalisation/Unitary Diagonalisation.
Induced matrix norms.
Applications of the above topics.
Learning Outcomes:
Cognitive (Knowledge, Understanding, Application, Analysis, Evaluation, Synthesis)
The student will be able to understand and use notation and ideas from axiomatic Linear Algebra to prove general
results for Vector Spaces and Inner Product Spaces.
In particular the student should be able to:
Construct mathematical arguments in order to deduce or prove simple facts about vectors, matrices, vector
spaces and linear maps.
Understand the concept of vector space and concepts like subspace, spanning set, linearly independence and
basis.
Prove the uniqueness of the cardinality of a basis set for a vector space and how the dimension of a finite
dimensional vector space follows.
Define vector spaces, subspaces and inner product spaces.
Know how to test whether a subset of a vector space is a subspace.
Know how to decide if a product is an inner product.
Use the Gram Schmidt method to find an orthogonal basis in a subspace of an inner product space
be able to characterize orthogonal matrices and their relation to orthonormal bases.
The student will develop a deeper understanding of the properties of matrices with a particular focus on eigenvalue
problems and their relationship to other matrix properties.
In particular the student should be able to:
Understand and know how to determine the row space, column space and null space of a matrix.
Show that the dimensions of the row and column spaces are equal and define the rank of a matrix.
Calculate a matrix rank and demonstrate the connection between the rank and the dimension of the null space of a
matrix.
Determine the existence of the inverse of a matrix by using the rank of a matrix and the value of the determinant
of a matrix.
Find eigenvalues, eigenvectors and eigenspaces of a matrix and perform such calculations for matrices of small
dimensions.
Diagonalise a matrix when possible, e.g., diagonalise a real square matrix A with distinct eigenvalues.
Affective (Attitudes and Values)
None.
Psychomotor (Physical Skills)
12/5/2017 Book of Modules
https://bookofmodules.ul.ie/ 3/3
None.
How the Module will be Taught and what will be the Learning Experiences of the Students:
Conventional lecture format using presentation graphics where appropriate.
Students will develop an enhanced knowledge of the subject. They will be able to formulate and solve problems
using the concepts and techniques of Linear Algebra.
Research Findings Incorporated in to the Syllabus (If Relevant):
Prime Texts:
Gilbert Strang (2009) Introduction to Linear Algebra, Fourth Edition , Wellesley Cambridge Press; 4th edition
Howard Anton (2010) Elementary Linear Algebra , Wiley; 10th edition
Seymour Lipschutz (2012) Schaum's Outline of Linear Algebra, 5th Edition , McGrawHill; 5th edition
Other Relevant Texts:
Programme(s) in which this Module is Offered:
BSMSCIUFA  Mathematical Sciences
BSFIMAUFA  Financial Mathematics
BSMAPHUFA  Mathematics and Physics
BSECMSUFA  Economics and Mathematical Sciences
BSSCCHUFA  Science Choice
Semester  Year to be First Offered:
Module Leader:
Romina.Gaburro@ul.ie
Review
 Evaluation Date:
 December 6, 2017
 Evaluated:
 James Wilson
 Comments:

This is more then needed. May be too advanced for 1st course in linear algebra.