LINEAR SYSTEM THEORY

AIM

The system concept, a key idea in engineering, can be used to assess and solve a wide range of engineering problems. Systems can be divided into two broad categories—linear and nonlinear—even though they may have a variety of different traits. Despite the fact that most systems are nonlinear, under specific circumstances, systems might be presumed to be linear. In this way, it is possible to analyze nonlinear systems using the perspective of linear systems. The purpose of this course is to give students the background they need to comprehend and solve engineering problems using the theory and techniques created for linear systems.

CONTENT

This course contains; Determinants and their Properties, Matrix Arithmetic. ,The Inverse of a Matrix and Cramer’s rule, Derivatives of Determinants.,Theory of Linear Equations, Vector Spaces, Subspaces , Span, Basis, Dimension, Rank. ,General Linear Systems, Least-squares Solutions. ,Elementary Row Operations, LU Decomposition.,Eigenvalues, Eigenvectors and Canonical Forms, Unitary Matrices. ,The Gram-Schmidt Process, Principal Axes of Ellipsoids, Hermitian Matrices.,Midterm Exam Study,Positive Definiteness, Unitary Triangularization, Normal Matrices.,The Jordan Canonical Form, Principal Vectors, Jordan’s Theorem.,Application of Matrix Analysis to Differential Equations, Exponential of a Matrix.,Solution of Differential Equations by Eigenvalues and Eigenvectors.,Variational Principles and Perturbation Theory, the Rayleigh Principle, the Courant Minimax Theorem,The Inclusion Principle, Criteria for Positive Definiteness, Hadamard’s Inequality, Weyl’s Inequality, Gershgorin’s Theorem.,Numerical Linear Algebra.

1. 1

   1. Understands the basic architecture of linear systems and solves a system of n equations in m variables describing the linear systems.

   Taught by: Question - Answer Technique, Experiential Learning, Lecture Method · Assessed by: Traditional Written Exam, Homework, Quiz

2. 2

   2. Determines the dimension of a vector space, rank of a matrix and basis for a vector space.

   Taught by: Question - Answer Technique, Experiential Learning, Lecture Method · Assessed by: Traditional Written Exam, Homework, Quiz

3. 3

   3. Applies the concepts of linear independence, linear transformation and determinants.

   Taught by: Question - Answer Technique, Experiential Learning, Lecture Method · Assessed by: Traditional Written Exam, Homework, Quiz

4. 4

   4. Determines matrix exponential, stability, and asymptotic behavior.

   Taught by: Question - Answer Technique, Experiential Learning, Lecture Method · Assessed by: Traditional Written Exam, Homework, Quiz

5. 5

   5. Finds the inverse of a matrix, eigen-values, eigenvectors of a matrix, SVD transformation and applies these to engineering problems.

   Taught by: Question - Answer Technique, Experiential Learning, Lecture Method · Assessed by: Traditional Written Exam, Homework, Quiz

6. 6

   6. Applies different LS techniques to engineering problems.

   Taught by: Question - Answer Technique, Experiential Learning, Lecture Method · Assessed by: Traditional Written Exam, Homework, Quiz

WEEKLY PLAN

1. WEEK 1

   Determinants and their Properties, Matrix Arithmetic.

   Preparation: The Textbook

2. WEEK 2

   The Inverse of a Matrix and Cramer’s rule, Derivatives of Determinants.

   Preparation: The Textbook

3. WEEK 3

   Theory of Linear Equations, Vector Spaces, Subspaces , Span, Basis, Dimension, Rank.

   Preparation: The Textbook

4. WEEK 4

   General Linear Systems, Least-squares Solutions.

   Preparation: The Textbook

5. WEEK 5

   Elementary Row Operations, LU Decomposition.

   Preparation: The Textbook

6. WEEK 6

   Eigenvalues, Eigenvectors and Canonical Forms, Unitary Matrices.

   Preparation: The Textbook

7. WEEK 7

   The Gram-Schmidt Process, Principal Axes of Ellipsoids, Hermitian Matrices.

   Preparation: The Textbook

8. WEEK 8

   Midterm Exam Study

   Preparation: The Textbook

9. WEEK 9

   Positive Definiteness, Unitary Triangularization, Normal Matrices.

   Preparation: The Textbook

10. WEEK 10

    The Jordan Canonical Form, Principal Vectors, Jordan’s Theorem.

    Preparation: The Textbook

11. WEEK 11

    Application of Matrix Analysis to Differential Equations, Exponential of a Matrix.

    Preparation: The Textbook

12. WEEK 12

    Solution of Differential Equations by Eigenvalues and Eigenvectors.

    Preparation: The Textbook

13. WEEK 13

    Variational Principles and Perturbation Theory, the Rayleigh Principle, the Courant Minimax Theorem

    Preparation: The Textbook

14. WEEK 14

    The Inclusion Principle, Criteria for Positive Definiteness, Hadamard’s Inequality, Weyl’s Inequality, Gershgorin’s Theorem.

    Preparation: The Textbook

15. WEEK 15

    Numerical Linear Algebra

    Preparation: The Textbook

ASSESSMENT

• Rate of Midterm Exam to Success50%
• Rate of Final Exam to Success50%

WORKLOAD

READING

• Applied Linear Algebra, Ben Noble and James W. Daniel

TEACHING STAFF

• Prof.Dr. Mehmet Kemal ÖZDEMİRCOORDINATOR
• Prof.Dr. İlteriş DEMİRKIRAN