COURSE UNIT TITLE

: ADVANCED LINEAR ALGEBRA

DEGREE PROGRAMMES

Description of Individual Course Units

Course Unit Code Course Unit Title Type Of Course D U L ECTS
MAT 5065 ADVANCED LINEAR ALGEBRA ELECTIVE 3 0 0 7

Offered By

Graduate School of Natural and Applied Sciences

Level of Course Unit

Second Cycle Programmes (Master's Degree)

Course Coordinator

ASSOCIATE PROFESSOR ENGIN MERMUT

Offered to

Mathematics (English)
Mathematics (English)

Course Objective

The aim of this course is to introduce the fundamental concepts in linear and multilinear algebra, and show its usage in the whole body of mathematics.

Learning Outcomes of the Course Unit

1   Will be able to use the structure of linear transformations given by Rational Canonical Form or Jordan form.
2   Will be able to use multilinear algebra, bilinear forms and quadratic forms when studying affine and projective geometry.
3   Will be able to use the properties of orthogonal operators, unitary operators and self-adjoint operators on inner product spaces.
4   Will be able to use the tensor product of vector spaces and exterior algebras.
5   Will be able to use the theory of linear algebra in some main applications like matrix exponentials to solve differential equations and in Lie groups, computer graphics and numerical linear algebra.

Mode of Delivery

Face -to- Face

Prerequisites and Co-requisites

None

Recomended Optional Programme Components

None

Course Contents

Week Subject Description
1 Vector spaces and linear transformations; dual spaces; quotient spaces, direct sums and products, coordinates, change of basis, similarity. Algebras over a field. The endomorphism algebra of a vector space.
2 Determinants.
3 Eigenvalues, eigenvectors and diagonalizability.
4 Modules over a principal ideal domain: application to finitely generated abelian groups and canonical forms of a linear transformation.
5 Rational Canonical Form and Jordan Canonical Form.
6 Geometry of Spaces with an inner product. Euclidean spaces, unitary spaces.
7 Orthogonal and unitary operators.
8 Self-adjoint operators.
9 Multilinear Algebra: Bilinear forms, symmetric bilinear forms, quadratic forms, Hermitian forms.
10 The tensor prouct of vector spaces, tensor algebra, symmetric algebra, exterior algebra.
11 Affine and Projective Geometry. Applications to Computer Graphics and Computer-Aided Geometric Design.
12 Quadrics. Hyperbolic Geometry.
13 Matrix exponential and differential equations. Matrix groups as Lie groups. Representations of groups. Semisimple rings and the Wedderburn-Artin Theorem.
14 A few topics from numerical linear algebra: Singular Value Decomposition; QR factorization; Conditioning and stability.

Recomended or Required Reading

Textbook(s):
[1] Golan, J. S. The Linear Algebra a Beginning Graduate Student Ought to Know. Third edition. Springer, 2012.
[2] Weintraub, S. H. A Guide to Advanced Linear Algebra. The Mathematical Association of America, 2011.
[3] Serre, D. Matrices, Theory and Applications. Second Edition. Springer, 2010.
[4] Kostrikin, A. I. and Manin, Y. I. Linear Algebra and Geometry. Gordon and Breach, 1997.
[5] Igor R. Shafarevich, I. R. and Remizov, A. O. Linear Algebra and Geometry. Springer, 2013.
[6] Roman, S. Advanced Linear Algebra. 3rd edition. Springer, 2008.
[7] Kaplansky, I. Linear Algebra and Geometry, A Second Course. 2nd edition. CHELSEA Publishing Company, 1974.
[8] Prasolov, V. V. Problems and Theorems in Linear Algebra. American Mathematical Society, 1994.
[9] Trefethen, L. N. and Bau, D. Numerical Linear Algebra. SIAM, 1997.
[10] Marsh, D. Applied Geometry for Computer Graphics and CAD. Second edition. Springer, 2005.

Supplementary Book(s):
[11] Dym, H. Linear Algebra in Action. American Mathematical Society, 2007.
[12] Zhang, F. Matrix Theory, Basic Results and Techniques. Second Edition. Springer, 2011.
[13] Fuhrmann, P. A. A Polynomial Approach to Linear Algebra. Second Edition. Springer, 2012.
[14] Brown, W. C. A second course in Linear Algebra. John Wiley & Sons, 1988.
[15] Weintraub, S. H. Jordan Canonical Form, Theory and Practice. Morgan \& Claypool, 2009.
[16] Artin, M. Algebra. 2nd edition. Pearson, 2011.
[17] Adkins, W. A. and Weintraub, S. H. Algebra, An Approach via Module Theory. Springer, 1992.
[18] Meyer, C. D. Matrix Analysis and Applied Linear Algebra. SIAM, 2000.
[19] Edwards, H. M. Linear Algebra. Birkhauser, 1995.
[20] Axler, S. Linear Algebra Done Right. Third edition. Springer, 2015.

References:

Materials:
Instructor s notes and presentations

Planned Learning Activities and Teaching Methods

Lecture notes, presentation, problem solving, discussion.

Assessment Methods

SORTING NUMBER SHORT CODE LONG CODE FORMULA
1 MTE MIDTERM EXAM
2 ASG ASSIGNMENT
3 FIN FINAL EXAM
4 FCG FINAL COURSE GRADE MTE * 0.30 + ASG * 0.40 + FIN * 0.30
5 RST RESIT
6 FCGR FINAL COURSE GRADE (RESIT) MTE * 0.30 + MAKRASG * 0.40 + MAKRRST * 0.30


*** Resit Exam is Not Administered in Institutions Where Resit is not Applicable.

Further Notes About Assessment Methods

Homework
1 Midterm Exam
Final Exam

Assessment Criteria

%40 (Homework) + %30 (Midterm examination) +%30 (Final examination)

Language of Instruction

English

Course Policies and Rules

You can be successful in this course by studying from your textbooks and lecture notes on the topics to be covered every week, coming to class by solving the given problems, establishing the concepts by discussing the parts you do not understand with your questions, learning the methods, and actively participating in the course.

Contact Details for the Lecturer(s)

Engin Mermut
e-mail: engin.mermut@deu.edu.tr
Phone: (232) 301 85 82

Office Hours

To be announced later.

Work Placement(s)

None

Workload Calculation

Activities Number Time (hours) Total Work Load (hours)
Lectures 14 3 42
Preparations before/after weekly lectures 14 3 42
Preparation for midterm exam 1 15 15
Preparation for final exam 1 20 20
Preparing assignments 10 5 50
Final 1 3 3
Midterm 1 3 3
TOTAL WORKLOAD (hours) 175

Contribution of Learning Outcomes to Programme Outcomes

PO/LOPO.1PO.2PO.3PO.4PO.5PO.6PO.7PO.8PO.9PO.10PO.11
LO.14332224333
LO.24332224333
LO.34332224333
LO.44332224333
LO.54332224333

CONTACT INFORMATION
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Phone: +90(232) 412 12 12 - Fax: +90 (232) 464 81 35

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