# InfoCoBuild

## Linear Algebra

Linear Algebra. Instructor: Dr. K. C. Sivakumar, Department of Mathematics, IIT Madras. Systems of linear equations, Matrices, Elementary row operations, Row-reduced echelon matrices. Vector spaces, Subspaces, Bases and dimension, Ordered bases and coordinates. Linear transformations, Rank-nullity 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, Cayley-Hamilton theorem, Invariant subspaces, Direct-sum decompositions, Invariant direct sums, The primary decomposition theorem, Cyclic subspaces and annihilators, Cyclic decomposition, Rational, Jordan forms. Inner product spaces, Orthonormal basis, Gram-Schmidt process. (from nptel.ac.in)

 Introduction

 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 Row-operations, Row-equivalent Matrices Lecture 03B - Equivalent Systems of Linear Equations II: Homogeneous Equations, Examples Lecture 04 - Row-reduced Echelon Matrices Lecture 05 - Row-reduced Echelon Matrices and Non-homogeneous Equations Lecture 06 - Elementary Matrices, Homogeneous Equations and Non-homogeneous Equations Lecture 07 - Invertible Matrices, Homogeneous Equations and Non-homogeneous 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 Rank-Nullity-Dimension Theorem, Isomorphisms between Vector Spaces Lecture 17 - Isomorphic Vector Spaces, Equality of the Row-rank and the Column-rank I Lecture 18 - Equality of the Row-rank and the Column-rank 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 Cayley-Hamilton 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 Gram-Schmidt Procedure Lecture 41 - The Gram-Schmidt 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 Self-Adjoint, Normal and Unitary Operators Lecture 48 - Unitary Operators Lecture 49 - Unitary Operators (cont.), Self-Adjoint Operators Lecture 50 - Self-Adjoint 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.