S.No.

Content of the topics

Learning Objectives

Teaching
Guidelines

Methodology

Time
(Hour))

1

 Vector Spaces
 Subspaces, sum and direct sum of subspaces
 Linear span
 Linearly dependent and independent subsets of vector space
 Finitely generated vector space &Existence theorem for basis of a finitely generated vector space
 Finite dimensional vector spaces
 Invariance of no. of elements of basis set, Dimensions
 Quotient space and its dimension

Student s will know about the vector Spaces, its subspaces, basis and quotient spaces 
Lecture should be effective so that student will be able to grasp the topics easily 
Assignments/ Seminar/ Class Tests/ Presentation

12

2

 Homomorphism and isomorphism of vector spaces
 Linear transformations and linear form of vector spaces
 Vector spaces of all the linear transformations
 Dual spaces, Bidual spaces
 Annihilator of subspaces of finite dimensional vector spaces
 Null space, Range space of a linear transformation, Rank & Nullity theorem
 Algebra of linear transformations
 Minimal polynomial of linear transformation
 Singular & Non –Singular linear transformation
 Matrix of a linear transformation
 Change of basis
 Eigen values and Eigen vector of linear transformations

Students will know about the homomorphism of vector spaces 
Lecture should be effective so that student will be able to grasp the topics easily 
Assignments/ Seminar/ Class Tests/ Presentation 
14

3

 Algebra of linear transformations
 Minimal polynomial of linear transformation
 Singular & Non –Singular linear transformation
 Matrix of a linear transformation
 Change of basis
 Eigen values and Eigen vector of linear transformations

Students will know the algebra of linear transformation 
Lecture should be effective so that student will be able to grasp the topics easily 
Assignments/ Seminar/ Class Tests/ Presentation 
8

4

 Inner product spaces
 Cauchy Schwarz inequality
 Orthogonal vectors, orthogonal complements
 Orthogonal sets and basis
 Bessel’s inequality for finite dimensional vector spaces
 GramSchmidt Orthogonalization process
 Adjoint of a linear transformation and its properties
 Unitary Linear Transformations

Students will know the inner Product Spaces 
Lecture should be effective so that student will be able to grasp the topics easily 
Assignments/ Seminar/ Class Tests/ Presentation 
10
