| MATHEMATICS | |||||
| Bachelor | TR-NQF-HE: Level 6 | QF-EHEA: First Cycle | EQF-LLL: Level 6 | ||
| Course Code | Course Name | Semester | Theoretical | Practical | Credit | ECTS |
| MAT2004 | Linear Algebra and Analytic Geometry | Spring | 3 | 2 | 4 | 5 |
| The course opens with the approval of the Department at the beginning of each semester |
| Language of instruction: | En |
| Type of course: | Must Course |
| Course Level: | Bachelor |
| Mode of Delivery: | Face to face |
| Course Coordinator : | Instructor NERMINE AHMED EL SISSI |
| Course Lecturer(s): |
RA ESRA ADIYEKE |
| Course Objectives: | This course is the continuation of the first linear algebra course, MAT2003. The course will emphasize abstract vector spaces and linear maps. In this course, students will develop their ability to understand and manipulate the objects of linear algebra. In addition, students will get exposed to the versatility of linear algebra in different branches of mathematics. |
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The students who have succeeded in this course; • Define and analyze finite and infinite dimensional vector space and subspaces over a field. • Build new vector spaces from old ones using direct sums. • Define and use the properties of linear transformations. • Find the null space, range and matrix representation of a given linear transformation. • Determine whether a linear transformation is one-to-one and onto. • Determine the diagonalizability of a linear transformation using eigenvectors and eigenvalues. • Define inner product and use the notion of inner products to define orthogonal vectors. • Apply the Gram-Schmidt process to generate an orthonormal set of vectors. • Identify normal, self-adjoint and positive operators. • Find the singular value decomposition of a matrix. • Compute the characteristic and minimal polynomials of an operator. • Compute the polar decomposition of a matrix. • Find the Jordan canonical form of matrices. • Define and apply properties of the trace and determinant of an operator and of a matrix. |
| The course covers the following topics: complex vector spaces, direct sums, linear maps products and quotients of vector spaces, dual spaces, invariant subspaces, quotient spaces, inner product spaces, orthonormal bases and Gram-Schmidt orthogonalization, self-adjoint and normal operators, the spectral theorem, positive operators and isometries, polar decomposition, singular value decomposition, characteristic and minimal polynomials of an operator, canonical forms, and the trace and determinant of an operator. |
| Week | Subject | Related Preparation | |
| 1) | Vector Spaces, subspace and direct sum | ||
| 2) | The vector space of linear maps, null spaces and ranges, matrix representation of a linear map | ||
| 3) | Invertible linear map and isomorphic vector spaces, product of vector spaces | ||
| 4) | Quotients of vector space, duality, | ||
| 5) | Invariant subspaces, eigenvectors and upper-triangular matrices | ||
| 6) | Eigenspace and diagonal matrices, and Inner product spaces and norms | ||
| 7) | Orthonormal bases and Orthogonal complements | ||
| 8) | self-adjoint and normal operators and The spectral theorem | ||
| 9) | Positive operators and isometries, Polar Decomposition | ||
| 10) | Singular Value Decomposition and operators on Complex vector spaces | ||
| 11) | Decomposition of an operator, and The Cayley-Hamilton Theorem | ||
| 12) | The minimal polynomial, and the Jordan form | ||
| 13) | The determinant of an operator, the determinant of a matrix, volume | ||
| 14) | Review | ||
| Course Notes: | Linear Algebra Done Right, Sheldon Axler, Springer, Third Edition. |
| References: | . |
| Semester Requirements | Number of Activities | Level of Contribution |
| Attendance | 0 | % 0 |
| Laboratory | 0 | % 0 |
| Application | 0 | % 0 |
| Field Work | 0 | % 0 |
| Special Course Internship (Work Placement) | 0 | % 0 |
| Quizzes | 7 | % 10 |
| Homework Assignments | % 0 | |
| Presentation | 3 | % 5 |
| Project | 0 | % 0 |
| Seminar | 0 | % 0 |
| Midterms | 2 | % 40 |
| Preliminary Jury | 0 | % 0 |
| Final | 1 | % 40 |
| Paper Submission | 3 | % 5 |
| Jury | 0 | % 0 |
| Bütünleme | % 0 | |
| Total | % 100 | |
| PERCENTAGE OF SEMESTER WORK | % 60 | |
| PERCENTAGE OF FINAL WORK | % 40 | |
| Total | % 100 | |
| Activities | Number of Activities | Duration (Hours) | Workload |
| Course Hours | 14 | 3 | 42 |
| Laboratory | 0 | 0 | 0 |
| Application | 14 | 2 | 28 |
| Special Course Internship (Work Placement) | 0 | 0 | 0 |
| Field Work | 0 | 0 | 0 |
| Study Hours Out of Class | 14 | 3 | 42 |
| Presentations / Seminar | 3 | 3 | 9 |
| Project | 0 | 0 | 0 |
| Homework Assignments | 0 | 0 | 0 |
| Quizzes | 7 | 1 | 7 |
| Preliminary Jury | 0 | 0 | 0 |
| Midterms | 2 | 1 | 2 |
| Paper Submission | 3 | 3 | 9 |
| Jury | 0 | 0 | 0 |
| Final | 1 | 2 | 2 |
| Total Workload | 141 | ||
| No Effect | 1 Lowest | 2 Low | 3 Average | 4 High | 5 Highest |
| Program Outcomes | Level of Contribution |