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Syllabus ( MATH 511 )


   Basic information
Course title: Linear Algebra I
Course code: MATH 511
Lecturer: Assist. Prof. Roghayeh HAFEZIEH
ECTS credits: 7.5
GTU credits: 3 (3+0+0)
Year, Semester: 1/2, Fall and Spring
Level of course: Second Cycle (Master's)
Type of course: Area Elective
Language of instruction: English
Mode of delivery: Face to face
Pre- and co-requisites: None
Professional practice: No
Purpose of the course: The objective of this course is to teach the students how to formulate and solve problems related to linear algebra in an abstract, geometric setting. Then, the solution procedures obtained in this abstract setting are implemented in a PC by using MATLAB software in order to solve real life problems.
   Learning outcomes Up

Upon successful completion of this course, students will be able to:

  1. Formulate and solve problems related to linear algebra in an abstract, geometric setting.

    Contribution to Program Outcomes

    1. Define and manipulate advanced concepts of Mathematics
    2. Relate mathematics to other disciplines and develop mathematical models for multidisciplinary problems
    3. Review the literature critically pertaining to his/her research projects, and connect the earlier literature to his/her own results
    4. Acquire scientific knowledge and work independently
    5. Develop mathematical, communicative, problem-solving, brainstorming skills.

    Method of assessment

    1. Written exam
  2. Transfer the solution procedures obtained in abstract setting into a PC by using MATLAB software in order to solve real life problems.

    Contribution to Program Outcomes

    1. Define and manipulate advanced concepts of Mathematics
    2. Relate mathematics to other disciplines and develop mathematical models for multidisciplinary problems
    3. Review the literature critically pertaining to his/her research projects, and connect the earlier literature to his/her own results
    4. Acquire scientific knowledge and work independently
    5. Develop mathematical, communicative, problem-solving, brainstorming skills.

    Method of assessment

    1. Homework assignment
  3. Develop the skills to analyze various aspects in linear algebra.

    Contribution to Program Outcomes

    1. Define and manipulate advanced concepts of Mathematics
    2. Relate mathematics to other disciplines and develop mathematical models for multidisciplinary problems
    3. Review the literature critically pertaining to his/her research projects, and connect the earlier literature to his/her own results
    4. Acquire scientific knowledge and work independently

    Method of assessment

    1. Written exam
    2. Homework assignment
   Contents Up
Week 1: Algebraic preliminaries : Semi-groups, monoids,groups , rings,structure of rings, ideals,fields.
Week 2: Algebraic preliminaries : Vector spaces, modules and algebras.Linear independence,basis,subspaces and subspace operations, modular distributive rule,linear transformations and their representations. (HOMEWORK 1)
Week 3: Vector spaces : Matrices, geometry induced by linear transformations,norms in vector spaces.Permutations,elementary operations and determinants,minors and cofactors, inverse transformations.
Week 4: Vector spaces : Projections,insertion maps, factor spaces, invariant subspaces and induced maps, calculation of supremal and infimal invariant subspaces. (HOMEWORK 2)
Week 5: Generalized eigen spaces and their calculation, Jordan form for real eigenvalues and its calculation.
Week 6: Jordan form for complex eigenvalues and its calculation. (HOMEWORK 3)
Week 7: Minimal polynomial, cyclic transformations and subspaces, calculation of maximally cyclic subspaces and companion form.
Week 8: Cyclic index, invariant factors and the Rational Canonical form and its relation to Jordan Form.(MIDTERM EXAM)
Week 9: Linear functionals and dual spaces, annihilators and the geometry of the dual spaces,dual maps.
Week 10: Inner product spaces,Bessel and Schwartz inequalities, representation of linear functionals,singular value decomposition, Hermitian operators, quadratic forms. (HOMEWORK 4)
Week 11: Convex Analysis : Convex sets, convex and affine combinations, convex cones, dual cones.
Week 12: Convex Analysis : , Separation, Farkas Lemma. (HOMEWORK 5)
Week 13: Convex analysis: Extreme points and directions, Linear Programming problem, representation, duality.
Week 14: Factorization of matrices,LU-factorization, Cholesky factorization, Householder transformation and QR-factorization.(HOMEWORK 6)
Week 15*: Factorization of matrices,LU-factorization, Cholesky factorization, Householder transformation and QR-factorization.(HOMEWORK 6)
Week 16*: Final exam
Textbooks and materials:
Recommended readings: Ders Notları;Advanced Linear Algebra, S. Roman;Finite Dimensional Vector Spaces, P. Halmos;An In troduction To Linear Algebra, L. Mirsky;Matrix Analysis, R.A. Horn ve C.R.Johnson; Matrix Computations, G.H. Golub ve C.F. Van Loan.
  * Between 15th and 16th weeks is there a free week for students to prepare for final exam.
Assessment Up
Method of assessment Week number Weight (%)
Mid-terms: 8 30
Other in-term studies: 0
Project: 0
Homework: 1-15 10
Quiz: 0
Final exam: 16 60
  Total weight:
(%)
   Workload Up
Activity Duration (Hours per week) Total number of weeks Total hours in term
Courses (Face-to-face teaching): 3 14
Own studies outside class: 3 14
Practice, Recitation: 0 0
Homework: 10 6
Term project: 0 0
Term project presentation: 0 0
Quiz: 0 0
Own study for mid-term exam: 15 1
Mid-term: 1 1
Personal studies for final exam: 20 1
Final exam: 2 1
    Total workload:
    Total ECTS credits:
*
  * ECTS credit is calculated by dividing total workload by 25.
(1 ECTS = 25 work hours)
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