| CODE | MAT3114 | ||||||||
| TITLE | Linear Algebra 2 | ||||||||
| UM LEVEL | 03 - Years 2, 3, 4 in Modular Undergraduate Course | ||||||||
| MQF LEVEL | 6 | ||||||||
| ECTS CREDITS | 2 | ||||||||
| DEPARTMENT | Mathematics | ||||||||
| DESCRIPTION | - Algebra of linear transformations; - Dual spaces and annihilators; - The Adjoint; - Normal, Hermitian Matrices and Isometries; - The minimum polynomial; - Primary decomposition theorem; - T-invariant spaces; - Schur’s theorem; and Cayley Hamilton Theorem; - Criteria for diagonalisation; - Simultaneous Diagonalization; - Jordan normal forms; - Companion Matrix; - Real symmetric matrices: - Positive semidefinite matrices; - Spectral decomposition. Study-unit Aims: This study-unit follows from the comprehensive MAT2112 which exposes the student to the linear algebra comprising vector spaces, direct sums, projections, linear functionals and linear transformations. The aim of this study-unit (MAT3114) is to familiarize the student with dual spaces and annihilators. The notion of the orthogonal eigenspaces of a normal operator, including the Hermitian operator and the isometry, which enables diagonalization, is emphasised. The student is expected to analyse a given operator and use similarity to reduce to the Jordan normal form or the companion matrix. These concepts are widely transferable not only to other study units, such as spectral graph theory, in mathematics but also to quantum mechanics in physics and machine learning in computer science. Learning Outcomes: ▪ Use the linear functionals of the dual space associated with an inner product space; ▪ Distinguish between Orthogonal complement and Annihilator; ▪ Relate the eigenspaces of commutative linear operators; ▪ Interpret the information coded in the minimum polynomial of a linear operator; ▪ Be able to obtain the JNF and the Companion matrix of a linear operator; ▪ Be able to prove and use Schur’s Theorem that a linear operator is isometrically similar to an upper triangular matrix; ▪ Be able to prove and use the spectral decomposition theorem for a real symmetric matrix in terms of the eigenvalues and orthogonal projections onto the respective eigenspaces; ▪ Relate the eigenvalues of AB and BA. Textbooks: - Lecture Notes. - Kaye R. and Wilson R., Linear Algebra, Oxford Science Publications, Oxford, 1998. - Herstein I.N., Topics in Algebra, John Wiley & Sons, 3rd Edition, 1996. - Nering E.D., Linear Algebra and Matrix Theory, John Wiley and Sons, 2nd Edition, 1970. - Nicholson R., Linear Algebra with Applications, McGraw-Hill, 2003. - Spence L.E., Insel A.J. and Friedberg S.H., Linear Algebra - a Matrix Approach, Pearson, 2nd Edition, 2007. |
||||||||
| ADDITIONAL NOTES | Follows from: MAT2112 Leads to: MAT3410 |
||||||||
| STUDY-UNIT TYPE | Lecture | ||||||||
| METHOD OF ASSESSMENT |
|
||||||||
| LECTURER/S | Irene Sciriha Aquilina |
||||||||
|
The University makes every effort to ensure that the published Courses Plans, Programmes of Study and Study-Unit information are complete and up-to-date at the time of publication. The University reserves the right to make changes in case errors are detected after publication.
The availability of optional units may be subject to timetabling constraints. Units not attracting a sufficient number of registrations may be withdrawn without notice. It should be noted that all the information in the description above applies to study-units available during the academic year 2024/5. It may be subject to change in subsequent years. |
|||||||||