Course of study

 

1.       Course Description

This is an introductory course in linear algebra and vector analysis which provides a foundation for the further study of mathematics. The algebraic structures dealt in this course are system of linear equations, matrices, determinants, vector spaces and theory of polynomial equations. It also provides fundamental knowledge of linear transformations, inner product spaces, eigen values , eigen vectors, product of vectors and vector differentiation.

2.       General Objectives

The general objectives of this course are as follows:

·   To help students to understand the general concept of linear algebra ,

·   To develop in the students the positive attitude towards linear algebra,

·   To familiarize students the basic algebraic structures,

·   To help students to develop the computational skills in matrices and determinants,

·   To provide computational skills to the students in solving linear and polynomial equations,

·   To enable students to relate theorems and problems of linear algebra,

·   To provide knowledge on vector spaces , subspaces and their properties ,

·   To familiarize students with eigen values and eigen vectors,

·   To help students in finding the product of three and four vectors,

·   To provide the students skills of vector differentiation and build up the concept of gradient of scalar function and divergence and curl of vector function

3.       Contents

Contents

Unit I: Introduction to Linear Algebra (15)       Brief historical information of linear algebra                                           Concept of set, relations and functions       Algebraic system and operations of sets and functions                                           Linear equations, non-linear equations.       Linear transformations, application of linear equations       Cartesian product, relations, functions and their properties       Equivalence  relation, quotient structures.

 

Contents

1.8    Composite functions, inverse function and their properties

Unit II: System of Linear Equations    (15)        System of homogeneous and non- homogeneous linear equations        Rank of a matrix and reduction to echelon form        Consistency and in-consistency of system of linear equations        Solution of homogeneous and non- homogeneous system of linear equations        Gaussian elimination method for solving linear equations

Unit III: Matrices and Determinants      (20)                       Matrices:                                          Introduction of matrices          Operations on matrices          Algebraic properties of matrix operation          Special types of matrices and partitioned matrices, echelon form of matrices          Elementary matrices                                      Inverse of matrices 3.1.6 Equivalent matrices 3.2 Determinants: 5.2.1 Properties of determinant co-factor expansion 5.1.2Use of determinants to solve Linear equations, Cramer’s rule and its profs and verifications.

Unit IV: Vectors in n-space (13) Definition and examples of points in n- space, algebraic operations (addition and multiplication) of points in n-space and their properties                              Norm, length, angle between two vectors and their properties Scalar and vector projections and their geometrical interpretations. Parallelogram     law,    Cauchy    Schwarz inequality                          Triangle inequality, Pythagorean identity.                          Orthogonality of vectors

    

Unit V: Vector Spaces                           (20)                           Vector spaces and sub spaces        Elementary properties of vector spaces and subspaces        Linear combination, linear hull, dependence and independence of vectors and their properties                              Direct sum of vector sub spaces                              Basis and dimension of vectors                              Co-ordinates and isomorphism                           Scalar Product of vectors.        Orthogonal and orthonormal vectors and their properties        Orthogonal and orthonormal bases and their properties        Gram Smith orthogonalization process of basis vectors

Unit VI: Linear Transformation            (15)             Definitions and examples of linear transformation                                          Algebra of linear transformations                                              Inverse of linear transformation.             Kernel      and     image      of      linear transformation                                      Geometry of linear transformations                                  Linear transformation and matrices                              Vector space of linear transformation                              Vector space of matrices                              Singular   and     non-singular      linear transformation Unit VIII: Products of Vectors               (13) 8.2 Scalar triple product   Geometrical meaning and properties    of scalar triple product                               Vector triple product          Geometrical meaning of vector triple product                               Scalar product of four vectors          Vector product of four vectors and linear relation of four vectors                              Reciprocal system of vectors and its properties Unit IX: Vector Differentiation              (11)               Continuity and successive derivative of vector functions             Geometrical interpretation of derivative of vector function with scalar variable                                           Gradient, divergence and curl   Unit X: Polynomial Equations of degree greater than two and its properties.                                                                   (13)                                  Properties of polynomial equations          Relation between roots and coefficients,   Application to the solution of an equation, symmetric function of roots         Transformation of equations, multiple roots, sum of the power of roots, reciprocal of  roots

 

Note: The figures within the parentheses indicate the approximate periods for respective units.

2.       Instructional Techniques

Because of the theoretical nature of the course, teacher-centered instructional techniques will lead the teaching learning process. The teacher will adopt the following techniques.

             General Instructional Techniques

The general instructional techniques are suggested as follows:

·       .Lecture with discussion

·       Use of software and math labs

·       Investigative approach in problem solving

 

            Specific Instructional Techniques

Unit-wise specific instructional techniques are given below .:

 

Units	Specific Instructional Techniques
I	Groupwork and individual assignment
II	Individual assignment and discussion
III	Individual and group work
IV	Group work and discussion
V	Assignment and discussion
VI	Assignment and presentation
VII	Group work and presentation
VIII	Individual assignment
IX	Project work
X	Individual assignment

 

3. Evaluation

Students will be evaluated on the basis of the written test. The Office of the Controller of the Examinations will conduct thefinal (annual ) examination at the end of the academic session to evaluate the students' performance. The types, number and marks of the subjective and objective questions will be as follows.

 

Types of questions	Total questions to be asked	Number of questions and marks allocated	Total marks
Group A: Multiple choice items	20 questions	20 × 1 mark	20
Group B: Short answer questions	8 with 3 'or' questions	8 × 7 marks	56