Linear Algebra
Linear Algebra. Instructor: Dr. K. C. Sivakumar, Department of Mathematics, IIT Madras. Systems of linear equations, Matrices, Elementary row operations, 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)
Systems of Linear Equations 
Lecture 01  Introduction to the Course Contents 
Lecture 02  Linear Equations 
Lecture 03  Equivalent Systems of Linear Equations I: Inverse Elementary Rowoperations, Rowequivalent Matrices 
Lecture 03B  Equivalent Systems of Linear Equations II: Homogeneous Equations, Examples 
Lecture 04  Rowreduced Echelon Matrices 
Lecture 05  Rowreduced Echelon Matrices and Nonhomogeneous Equations 
Lecture 06  Elementary Matrices, Homogeneous Equations and Nonhomogeneous Equations 
Lecture 07  Invertible Matrices, Homogeneous Equations and Nonhomogeneous Equations 
Vector Spaces 
Lecture 08  Vector Spaces 
Lecture 09  Elementary Properties in Vector Spaces, Subspaces 
Lecture 10  Subspaces, Spanning Sets, Linear Independence, Dependence 
Basis and Dimension 
Lecture 11  Basis for a Vector Space 
Lecture 12  Dimension of a Vector Space 
Lecture 13  Dimensions of Sums of Spaces 
Linear Transformations 
Lecture 14  Linear Transformations 
Lecture 15  The Null Space and the Range Space of a Linear Transformation 
Lecture 16  The RankNullityDimension Theorem, Isomorphisms between Vector Spaces 
Lecture 17  Isomorphic Vector Spaces, Equality of the Rowrank and the Columnrank I 
Lecture 18  Equality of the Rowrank and the Columnrank II 
Matrix of a Linear Transformation 
Lecture 19  The Matrix of a Linear Transformation 
Lecture 20  Matrix for the Composition and the Inverse, Similarity Transformation 
The Dual Space 
Lecture 21  Linear Functions, The Dual Space, Dual Basis 
Lecture 22  Dual Basis (cont.), Subspace Annihilators 
Lecture 23  Subspace Annihilators (cont.) 
Lecture 24  The Double Dual, The Double Annihilator 
Lecture 25  The Transpose of a Linear Transformation, Matrices of a Linear Transformation and its Transpose 
Eigenvalues and Eigenvectors 
Lecture 26  Eigenvalues and Eigenvectors of Linear Operators 
Lecture 27  Diagonalization of Linear Operators, A Characterization 
Lecture 28  The Minimal Polynomial 
Lecture 29  The CayleyHamilton Theorem 
Invariant Subspaces and Triangulability 
Lecture 30  Invariant Subspaces 
Lecture 31  Triangulability, Diagonalization in terms of Minimal Polynomial 
Lecture 32  Independent Subspaces and Projection Operators 
Direct Sum Decompositions 
Lecture 33  Direct Sum Decompositions and Projection Operators I 
Lecture 34  Direct Sum Decompositions and Projection Operators II 
Primary and Cycle Decomposition Theorems 
Lecture 35  The Primary Decomposition Theorem and Jordan Decomposition 
Lecture 36  Cyclic Subspaces and Annihilators 
Lecture 37  The Cyclic Decomposition Theorem I 
Lecture 38  The Cyclic Decomposition Theorem II, The Rational Form 
Inner Product Spaces 
Lecture 39  Inner Product Spaces 
Lecture 40  Norms on Vector Spaces, The GramSchmidt Procedure 
Lecture 41  The GramSchmidt Procedure (cont.), The QR Decomposition 
Lecture 42  Bessel's Inequality, Parseval's Identity, Best Approximation 
Best Approximation 
Lecture 43  Best Approximation: Least Squares Solutions 
Lecture 44  Orthogonal Complementary Subspaces, Orthogonal Projections 
Lecture 45  Projection Theorem, Linear Functionals 
Adjoint of a Linear Operator 
Lecture 46  The Adjoint Operator 
Lecture 47  Properties of the Adjoint Operation, Inner Product Space Isomorphism 
SelfAdjoint, Normal and Unitary Operators 
Lecture 48  Unitary Operators 
Lecture 49  Unitary Operators (cont.), SelfAdjoint Operators 
Lecture 50  SelfAdjoint Operators  Spectral Theorem 
Lecture 51  Normal Operators  Spectral Theorem 
References 
Linear Algebra
Instructor: Dr. K. C. Sivakumar, Department of Mathematics, IIT Madras. Systems of linear equations, Vector spaces, Linear transformations, Eigenvalues and eigenvectors, Inner product spaces, Adjoint of a linear operator.
