| CODE | MAT2112 | ||||||||
| TITLE | Linear Algebra 1 | ||||||||
| UM LEVEL | 02 - Years 2, 3 in Modular Undergraduate Course | ||||||||
| MQF LEVEL | 5 | ||||||||
| ECTS CREDITS | 4 | ||||||||
| DEPARTMENT | Mathematics | ||||||||
| DESCRIPTION | - Vector spaces; - Dimension theorems; - Sum of Vector Spaces and Quotient Vector Spaces; - Direct sum of spaces; - Change of bases. - Inner product spaces; - Dual spaces; - Euclidean Inner product space over C; - Gram Schmidt orthogonalisation; - Orthogonal Complement; - Fundamental Theorem of Linear Algebra. - Linear Transformations - The rank of matrices; - Projection of a vector space onto a subspace; - Similar matrices; - Transition Matrices; - The Determinant of a linear operator. - Eigenvalues and eigenvectors; - The characteristic polynomial; - The Cayley-Hamilton theorem. - The Minimum Polynomial - Diagonalisation and applications; - Solution of first order DE and Recurrence Relations in R3 ; - The Jordan normal form. Study-Unit Aims: The aim of this study-unit is to introduce the students to basic notions of linear algebra which are fundamental notions in linear algebra and also needed in several study units they will follow in subsequent semesters in mathematics and related sciences. The structure of vector spaces is presented in axiomatic form. The concept of dimension plays a central role as inner product spaces are explored. Homomorphisms on inner product spaces are characterized as linear transformations, leading to the concept of the rank of a matrix representation. The climax of the course is the emphasis on the importance of the minimum polynomial of a linear operator which encodes crucial properties associated with its eigenspaces. Learning Outcomes: 1. Knowledge & Understanding: By the end of the study-unit the student will be able to: - Describe the main operations on vector spaces, orthogonality in inner product spaces. - Distinguish between diagonalizable and non--diagonalizable matrices. - Analyse basic definitions and results about inner product spaces and linear operators. - Use a variety of proof techniques to investigate the above topics, which are not only an end in themselves but also lead to more advanced work to be done in other study units. 2. Skills: By the end of the study-unit the student will be able to: - Use the definitions and properties of vector spaces and linear operators to prove fundamental results about their structure and to deduce simple applications to the natural sciences. - Investigate the basic properties of Euclidean inner products and linear operators. - Construct proofs by deduction, contradiction and induction in the contexts of vector spaces. Main Text/s and any supplementary readings: Main Texts: - Lecture Notes. - Nicholson R., Linear Algebra with Applications, McGraw-Hill, 2003. - Leon S., Linear Algebra with Applications, Prentice Hall, 6th Edition, 2002. - Kaye R. and Wilson R., Linear Algebra, Oxford Science Publications, Oxford, 1998. Supplementary Readings: - 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. - Moore H.G. and Yaqub A., A First Course in Linear Algebra with Applications, Academic Press, 3rd Edition,1998. - Lancaster P. and Tismenetsky M., The Theory of Matrices with Applications, 2nd Edition, 1985. |
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| ADDITIONAL NOTES | Follows from: MAT1116 Leads to: MAT3114 |
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| STUDY-UNIT TYPE | Lecture and Tutorial | ||||||||
| METHOD OF ASSESSMENT |
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| LECTURER/S | Irene Sciriha Aquilina |
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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 2023/4. It may be subject to change in subsequent years. |
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