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#### Curriculum

 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, Bi-dual 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 Gram-Schmidt 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