LINEAR ALGEBRA

AIM

1. To provide the methods of solution of systems of linear equations and the applications of matrix and determinant. 2. To introduce the basic concepts of vector space, basis, dimension, linear dependency required to understand, construct, solve and interpret data spaces. 3. To give an ability to apply knowledge of mathematics on engineering problems

CONTENT

This course contains; Preliminaries: Matrices and Systems of Linear Algebraic Equations: Definitions and Notation,Matrix Algebra and Terminology and Notation for Systems of Linear Equations ,Elementary Row Operations, Row Echelon Matrices, Reduced Row Echelon Matrices and Solving Systems of Linear Algebraic Equations,Gaussian Elimination and Gauss Jordan Elimination Methods, and The Inverse of a Square Matrix ,Gauss Jordan Method, Determinants and Adjoint Method ,Elementary Matrices, LU Factorization, Cramer Rule ,Vector Spaces: Definition of a Vector Space, Subspaces and Spanning Sets ,Linear Dependency and Independency, Bases and Dimension ,Row and Column Spaces and The Rank-Nullity Theorem ,Inner Product Spaces and Orthogonality ,Eigenvalue/Eigenvector Problem: Eigenvalues and Eigenvectors and Eigenspaces ,Application of Eigenvalues and Eigenvectors Factorization ,Diagonalization and Singular Value Decomposition, Pseudo-inverse Calculation ,Linear Transformations, The Kernel and Range of a Linear Transformation and Further Properties of Linear Transformations .

1. 1

   1. Recognize arithmetic operations with matrices, properties of matrices, elementary row operations on matrices and determine row echelon form (REF) and reduced row echelon form (RREF) for matrices and rank of a matrix.

   Taught by: Problem Solving Method, Self Study Method, Lecture Method · Assessed by: Traditional Written Exam, Homework

2. 2

   2. Calculate the solutions to the systems of linear equations from: Gaussian and Gauss-Jordan elimination method, the inverse of a matrix, Gauss-Jordan method, and find the value of determinant of a matrix.

   Taught by: Problem Solving Method, Self Study Method, Lecture Method · Assessed by: Traditional Written Exam, Homework

3. 3

   4. Recognize the importance of the concepts of a vector space such as subspace, spanning set, linear dependency and independency, basis and dimension, row and column spaces, the Rank-Nullity theorem, inner product spaces and orthogonality.

   Taught by: Problem Solving Method, Self Study Method, Lecture Method · Assessed by: Traditional Written Exam, Homework

4. 4

   5. Analyze eigenvalues and the corresponding eigenvectors and eigenspaces of the matrix, diagonalization and singular value decomposition, and pseudo-inverse of a matrix, and linear transformations and apply on engineering problems.

   Taught by: Problem Solving Method, Self Study Method, Lecture Method · Assessed by: Traditional Written Exam, Homework

5. 5

   3. Analyze Adjoint Method to find the inverse matrix, elementary matrices, LU factorization and Cramer rule.

   Taught by: Problem Solving Method, Self Study Method, Lecture Method · Assessed by: Traditional Written Exam, Homework

WEEKLY PLAN

1. WEEK 1

   Preliminaries: Matrices and Systems of Linear Algebraic Equations: Definitions and Notation

   Preparation: Book Chapter 3.1

2. WEEK 2

   Matrix Algebra and Terminology and Notation for Systems of Linear Equations

   Preparation: Book Chapters 3.2, 3.3

3. WEEK 3

   Elementary Row Operations, Row Echelon Matrices, Reduced Row Echelon Matrices and Solving Systems of Linear Algebraic Equations

   Preparation: Book Chapter 3.4

4. WEEK 4

   Gaussian Elimination and Gauss Jordan Elimination Methods, and The Inverse of a Square Matrix

   Preparation: Book Chapters 3.5, 3.6

5. WEEK 5

   Gauss Jordan Method, Determinants and Adjoint Method

   Preparation: Book Chapters 3.6, 4

6. WEEK 6

   Elementary Matrices, LU Factorization, Cramer Rule

   Preparation: Book Chapters 3.7, 4.3

7. WEEK 7

   Vector Spaces: Definition of a Vector Space, Subspaces and Spanning Sets

   Preparation: Book Chapters 5.1, 5.2, 5.3, 5.4

8. WEEK 8

   Linear Dependency and Independency, Bases and Dimension

   Preparation: Book Chapters 5.5, 5.6

9. WEEK 9

   Row and Column Spaces and The Rank-Nullity Theorem

   Preparation: Book Chapters 5.7, 5.8

10. WEEK 10

    Inner Product Spaces and Orthogonality

    Preparation: Book Chapters 5.9, 5.10

11. WEEK 11

    Eigenvalue/Eigenvector Problem: Eigenvalues and Eigenvectors and Eigenspaces

    Preparation: Book Chapters 6.5, 6.6

12. WEEK 12

    Application of Eigenvalues and Eigenvectors Factorization

    Preparation: Book Chapters 6.7, other sources

13. WEEK 13

    Diagonalization and Singular Value Decomposition, Pseudo-inverse Calculation

    Preparation: Book Chapters 6.7, other sources

14. WEEK 14

    Linear Transformations, The Kernel and Range of a Linear Transformation and Further Properties of Linear Transformations

    Preparation: Book Chapters 6.1, 6.2, 6.3, 6.4

ASSESSMENT

• Rate of Midterm Exam to Success30%
• Rate of Final Exam to Success70%

WORKLOAD

READING

• Differential Equations & Linear Algebra Second Edition, Stephen W. Goode. Prentice-Hall, Inc. 2000,1991.

TEACHING STAFF

• Assist.Prof. Cihan Bilge GÜRBÜZCOORDINATOR
• Assist.Prof. Cem YETİŞMİŞOĞLU