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Description
Linear Algebra. Instructor: Dr. K. C. Sivakumar, Department of Mathematics, IIT Madras. Systems of linear equations, Matrices, Elementary row operations, and Rowreduced echelon matrices.
Vector spaces, Subspaces, Bases and dimension, Ordered bases and coordinates. Linear transformations, Ranknullity theorem, Algebra of linear transformations, Isomorphism, Matrix representation, Linear functionals, Annihilator, Double dual, Transpose of a linear transformation. Characteristic values and characteristic vectors of linear transformations, Diagonalizability, Minimal polynomial of a linear transformation, CayleyHamilton theorem, Invariant subspaces, Directsum decompositions, Invariant direct sums, The primary decomposition theorem, Cyclic subspaces and annihilators, Cyclic decomposition, Rational, Jordan forms. Inner product spaces, Orthonormal basis, GramSchmidt process. (from nptel.ac.in)
Course Curriculum

 Lecture 01 – Introduction to the Course Contents Unlimited
 Lecture 02 – Linear Equations Unlimited
 Lecture 03 – Equivalent Systems of Linear Equations I: Inverse Elementary Rowoperations Unlimited
 Lecture 03B – Equivalent Systems of Linear Equations II: Homogeneous Equations, Examples Unlimited
 Lecture 04 – Rowreduced Echelon Matrices Unlimited
 Lecture 05 – Rowreduced Echelon Matrices and Nonhomogeneous Equations Unlimited
 Lecture 06 – Elementary Matrices, Homogeneous Equations and Nonhomogeneous Equations Unlimited
 Lecture 07 – Invertible Matrices, Homogeneous Equations and Nonhomogeneous Equations Unlimited

 Lecture 08 – Vector Spaces Unlimited
 Lecture 09 – Elementary Properties in Vector Spaces, Subspaces Unlimited
 Lecture 10 – Subspaces, Spanning Sets, Linear Independence, Dependence Unlimited

 Lecture 11 – Basis for a Vector Space Unlimited
 Lecture 12 – Dimension of a Vector Space Unlimited
 Lecture 13 – Dimensions of Sums of Spaces Unlimited

 Lecture 14 – Linear Transformations Unlimited
 Lecture 15 – The Null Space and the Range Space of a Linear Transformation Unlimited
 Lecture 16 – The RankNullityDimension Theorem, Isomorphisms between Vector Spaces Unlimited
 Lecture 17 – Isomorphic Vector Spaces, Equality of the Rowrank and the Columnrank I Unlimited
 Lecture 18 – Equality of the Rowrank and the Columnrank II Unlimited

 Lecture 19 – The Matrix of a Linear Transformation Unlimited
 Lecture 20 – Matrix for the Composition and the Inverse, Similarity Transformation Unlimited

 Lecture 21 – Linear Functions, The Dual Space, Dual Basis Unlimited
 Lecture 22 – Dual Basis (cont.), Subspace Annihilators Unlimited
 Lecture 23 – Subspace Annihilators (cont.) Unlimited
 Lecture 24 – The Double Dual, The Double Annihilator Unlimited
 Lecture 25 – The Transpose of a Linear Transformation, Matrices of a Linear Transformation Unlimited

 Lecture 26 – Eigenvalues and Eigenvectors of Linear Operators Unlimited
 Lecture 27 – Diagonalization of Linear Operators, A Characterization Unlimited
 Lecture 28 – The Minimal Polynomial Unlimited
 Lecture 29 – The CayleyHamilton Theorem Unlimited

 Lecture 30 – Invariant Subspaces Unlimited
 Lecture 31 – Triangulability, Diagonalization in terms of Minimal Polynomial Unlimited
 Lecture 32 – Independent Subspaces and Projection Operators Unlimited

 Lecture 33 – Direct Sum Decompositions and Projection Operators I Unlimited
 Lecture 34 – Direct Sum Decompositions and Projection Operators II Unlimited

 Lecture 35 – The Primary Decomposition Theorem and Jordan Decomposition Unlimited
 Lecture 36 – Cyclic Subspaces and Annihilators Unlimited
 Lecture 37 – The Cyclic Decomposition Theorem I Unlimited
 Lecture 38 – The Cyclic Decomposition Theorem II, The Rational Form Unlimited

 Lecture 39 – Inner Product Spaces Unlimited
 Lecture 40 – Norms on Vector Spaces, The GramSchmidt Procedure Unlimited
 Lecture 41 – The GramSchmidt Procedure (cont.), The QR Decomposition Unlimited
 Lecture 42 – Bessel’s Inequality, Parseval’s Identity, Best Approximation Unlimited

 Lecture 43 – Best Approximation: Least Squares Solutions Unlimited
 Lecture 44 – Orthogonal Complementary Subspaces, Orthogonal Projections Unlimited
 Lecture 45 – Projection Theorem, Linear Functionals Unlimited

 Lecture 46 – The Adjoint Operator Unlimited
 Lecture 47 – Properties of the Adjoint Operation, Inner Product Space Isomorphism Unlimited

 Lecture 48 – Unitary Operators Unlimited
 Lecture 49 – Unitary Operators (cont.), SelfAdjoint Operators Unlimited
 Lecture 50 – SelfAdjoint Operators – Spectral Theorem Unlimited
 Lecture 51 – Normal Operators – Spectral Theorem Unlimited
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