IIT Madras Course , Prof. K.C. Sivakumar

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IIT Madras Course , Prof. K.C. Sivakumar

Introduction to the Course Contents - Linear Equations - Equivalent Systems of Linear Equations I: Inverses of Elementary Row-operations - Equivalent Systems of Linear Equations II: Homogeneous Equations, Examples - Row-reduced Echelon Matrices - Row-reduced Echelon Matrices and Non-homogeneous Equations - Elementary Matrices, Homogeneous Equations and Non-homogeneous Equations - Invertible matrices, Homogeneous Equations Non-homogeneous Equations - Elementary Properties in Vector Spaces. Subspaces - Subspaces (continued), Spanning Sets, Linear Independence, Dependence - Basis for a vector space - Dimension of a vector space - Dimensions of Sums of Subspaces - Linear Transformations - The Null Space and the Range Space of a Linear Transformation - The Rank-Nullity-Dimension Theorem. Isomorphisms Between Vector Spaces - Isomorphic Vector Spaces, Equality of the Row-rank and the Column-rank - The Matrix of a Linear Transformation - Matrix for the Composition and the Inverse. Similarity Transformation - Linear Functionals - The Dual Space - Dual Basis - Subspace Annihilators - Subspace Annihilators

The Transpose of a Linear Transformation. Matrices of a Linear - The Double Dual. The Double Annihilator - Eigenvalues and Eigenvectors of Linear Operators - Diagonalization of Linear Operators. A Characterization - The Minimal Polynomial - The Cayley-Hamilton Theorem - Invariant Subspaces - Triangulability, Diagonalization in Terms of the Minimal Polynomial - Independent Subspaces and Projection Operators - Direct Sum Decompositions and Projection Operators - The Primary Decomposition Theorem and Jordan Decomposition - Cyclic Subspaces and Annihilators - The Cyclic Decomposition Theorem - The Rational Form - Inner Product Spaces - Norms on Vector spaces - The Gram-Schmidt Procedure - The QR Decomposition - Bessel's Inequality, Parseval's Indentity, Best Approximation - Best Approximation: Least Squares Solutions - Orthogonal Complementary Subspaces, Orthogonal Projections - Projection Theorem. Linear Functionals - The Adjoint Operator - Properties of the Adjoint Operation. Inner Product Space Isomorphism - Unitary Operators - Self-Adjoint Operators - Spectral Theorem

The Transpose of a Linear Transformation. Matrices of a Linear - The Double Dual. The Double Annihilator - Eigenvalues and Eigenvectors of Linear Operators - Diagonalization of Linear Operators. A Characterization - The Minimal Polynomial - The Cayley-Hamilton Theorem - Invariant Subspaces - Triangulability, Diagonalization in Terms of the Minimal Polynomial - Independent Subspaces and Projection Operators - Direct Sum Decompositions and Projection Operators - The Primary Decomposition Theorem and Jordan Decomposition - Cyclic Subspaces and Annihilators - The Cyclic Decomposition Theorem - The Rational Form - Inner Product Spaces - Norms on Vector spaces - The Gram-Schmidt Procedure - The QR Decomposition - Bessel's Inequality, Parseval's Indentity, Best Approximation - Best Approximation: Least Squares Solutions - Orthogonal Complementary Subspaces, Orthogonal Projections - Projection Theorem. Linear Functionals - The Adjoint Operator - Properties of the Adjoint Operation. Inner Product Space Isomorphism - Unitary Operators - Self-Adjoint Operators - Spectral Theorem

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Linear Algebra by Dr. K.C. Sivakumar,Department of Mathematics,IIT Madras.For more details on NPTEL visit httpnptel.ac.in

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- 1.Introduction to the Course Contents.
- 2.Linear Equations
- 3.Equivalent Systems of Linear Equations I Inverses of Elementary Row-operations
- 4.Equivalent Systems of Linear Equations II Homogeneous Equations, Examples
- 5.Row-reduced Echelon Matrices
- 6.Row-reduced Echelon Matrices and Non-homogeneous Equations
- 7.Elementary Matrices, Homogeneous Equations and Non-homogeneous Equations
- 8.Invertible matrices, Homogeneous Equations Non-homogeneous Equations
- 9.Elementary Properties in Vector Spaces. Subspaces
- 10.Subspaces (continued), Spanning Sets, Linear Independence, Dependence
- 11.Basis for a vector space
- 12.Dimension of a vector space
- 13.Dimensions of Sums of Subspaces
- 14.Linear Transformations
- 15.The Null Space and the Range Space of a Linear Transformation
- 16.The Rank-Nullity-Dimension Theorem. Isomorphisms Between Vector Spaces
- 17.Isomorphic Vector Spaces, Equality of the Row-rank and the Column-rank I
- 18.Equality of the Row-rank and the Column-rank II
- 19.Mod-05 Lec19 The Matrix of a Linear Transformation
- 20.Matrix for the Composition and the Inverse. Similarity Transformation
- 21.Linear Functionals. The Dual Space. Dual Basis I
- 22.Dual Basis II. Subspace Annihilators I
- 23.Subspace Annihilators II
- 24.The Transpose of a Linear Transformation. Matrices of a Linear
- 25.The Double Dual. The Double Annihilator
- 26.Eigenvalues and Eigenvectors of Linear Operators
- 27.Diagonalization of Linear Operators. A Characterization
- 28.The Minimal Polynomial
- 29.The Cayley-Hamilton Theorem
- 30.Invariant Subspaces
- 31.Triangulability, Diagonalization in Terms of the Minimal Polynomial
- 32.Independent Subspaces and Projection Operators
- 33.Direct Sum Decompositions and Projection Operators I
- 34.Direct Sum Decomposition and Projection Operators II
- 35.The Primary Decomposition Theorem and Jordan Decomposition
- 36.Cyclic Subspaces and Annihilators
- 37.The Cyclic Decomposition Theorem I
- 38.The Cyclic Decomposition Theorem II. The Rational Form
- 39.Inner Product Spaces
- 40.Norms on Vector spaces. The Gram-Schmidt Procedure I
- 41.The Gram-Schmidt Procedure II. The QR Decomposition.
- 42.Bessels Inequality, Parsevals Indentity, Best Approximation
- 43.Best Approximation Least Squares Solutions
- 44.Orthogonal Complementary Subspaces, Orthogonal Projections
- 45.Projection Theorem. Linear Functionals
- 46.The Adjoint Operator
- 47.Properties of the Adjoint Operation. Inner Product Space Isomorphism
- 48.Unitary Operators
- 49.Unitary operators II. Self-Adjoint Operators I.
- 50.Self-Adjoint Operators II - Spectral Theorem

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