Linear Algebra

Course Title: Linear Algebra

Course no: MTH-155                                                                      Full Marks: 80+20

Credit hours: 3                                                                                Pass Marks: 32+8

Nature of course: Theory

Course Synopsis:       Linear equations in linear algebra, Matrix algebra, Determinants,                                       Vector spaces, Eigen values and Eigen vectors. Orthogonality and                                             least squares. Symmetric matrices and Quadratic forms.

Goal:  This course provides students with the knowledge of fundamental of linear algebra and the theory of matrices. On completion of this course the student will master the basic concepts and acquires skills in solving problems in linear algebra.

Course Contents:

Unit 1: Linear equations in linear Algebra                                           10 Hrs.

1.1    Systems of linear equations                

1.2    Row reduction and Echelon Forms              

1.3    Vector equations                      

1.4    The matrix equations Ax = b              

1.5    Solution sets of linear systems             

1.6    Linear independence               

1.7    Introduction Linear Transformations            

1.8    The matrix of a Linear Transformations                   

Unit 2: Matrix Algebra                                                                               8 Hrs.

2.1    Matrix operations                      

2.2    The inverse of a matrix             

2.3    Characterization of invertible matrices                 

2.4    Partitioned Matrices                 

2.5    The Leontief Input-output model                 

2.6    Application to Computer graphics             

Unit 3: Determinants                                                                                   4 Hrs. 

3.1    Introduction to determinants                       

3.2    Properties of determinants                 

3.3    Cramer's rule value and linear transformations               

Unit 4: Vector Spaces                                                                                8 Hrs.

4.1    Vector spaces and sub polar             

4.2    Null spaces, Column spaces and linear transformations

4.3    Linearly Independent Sets; Bases     

4.4    Coordinate systems                  

4.5    The dimension of a vector space                 

4.6    Rank                      

4.7    Change of basis             

Unit 5: Eigen values and Eigen vectors                                                            7 Hrs.

5.1    Eigen vectors and Eigen values                    

5.2    The characteristics equations                       

5.3    Diagonalization              

5.4    Eigen vectors and Linear Transformations             

5.5    Complex Eigen values             

5.6    Discrete Dynamical System                

Unit 6: Orthogonality and Least Squares                                                         8 Hrs.

6.1    Linear product, length and Orthogonality            

6.2    Orthogonal sets             

6.3    Orthogonal Projections            

6.4    The Gram- Schmidt process               

6.5    Least square problems             

6.6    Applications to Linear models                       

Text books:               David C. lay: Linear Algebra and its applications, 3^rd edition, Pearson Education.

References:  1. Kolman, Bernard; Introductory Linear Algebra with Application.7^th   edition. Pearson.

                        2. Gilbert Strang; Linear Algebra and its Application.3^rd edition.

                        3. Kreszig, E. " Advanced Engineering Mathematics." 5^th edition. Wiley