MS 4122

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 re-structuring 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 , McGraw-Hill; 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

Evaluation Date:
December 6, 2017
Evaluated:
James Wilson