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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.