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

### Course Curriculum

 Introduction to the Course Contents. Details 26:47 Linear Equations Details 35:10 Equivalent Systems of Linear Equations I: Inverses of Elementary Row-operations Details 40:48 Equivalent Systems of Linear Equations II: Homogeneous Equations, Examples Details 43:58 Row-reduced Echelon Matrices Details 48:23 Row-reduced Echelon Matrices and Non-homogeneous Equations Details 47:19 Elementary Matrices, Homogeneous Equations and Non-homogeneous Equations Details 49:14 Invertible matrices, Homogeneous Equations Non-homogeneous Equations Details 50:58 Elementary Properties in Vector Spaces. Subspaces Details 48:16 Subspaces (continued), Spanning Sets, Linear Independence, Dependence Details 43:25 Basis for a vector space Details 48:48 Dimension of a vector space Details 48:31 Dimensions of Sums of Subspaces Details 52:11 Linear Transformations Details 50:10 The Null Space and the Range Space of a Linear Transformation Details 51:4 The Rank-Nullity-Dimension Theorem. Isomorphisms Between Vector Spaces Details 41:45 Isomorphic Vector Spaces, Equality of the Row-rank and the Column-rank I Details 47:34 Equality of the Row-rank and the Column-rank II Details 36:8 Mod-05 Lec19 The Matrix of a Linear Transformation Details 40:27 Matrix for the Composition and the Inverse. Similarity Transformation Details 47:4 Linear Functionals. The Dual Space. Dual Basis I Details 49:20 Dual Basis II. Subspace Annihilators I Details 38:53 Subspace Annihilators II Details 50:8 The Transpose of a Linear Transformation. Matrices of a Linear Details 45:22 The Double Dual. The Double Annihilator Details 47:34 Eigenvalues and Eigenvectors of Linear Operators Details 40:11 Diagonalization of Linear Operators. A Characterization Details 47:1 The Minimal Polynomial Details 42:38 The Cayley-Hamilton Theorem Details 47:21 Invariant Subspaces Details 39:19 Triangulability, Diagonalization in Terms of the Minimal Polynomial Details 51:30 Independent Subspaces and Projection Operators Details 48:42 Direct Sum Decompositions and Projection Operators I Details 48:49 Direct Sum Decomposition and Projection Operators II Details 46:40 The Primary Decomposition Theorem and Jordan Decomposition Details 38:51 Cyclic Subspaces and Annihilators Details 50:49 The Cyclic Decomposition Theorem I Details 49:56 The Cyclic Decomposition Theorem II. The Rational Form Details 46:12 Inner Product Spaces Details 44:44 Norms on Vector spaces. The Gram-Schmidt Procedure I Details 53:21 The Gram-Schmidt Procedure II. The QR Decomposition. Details 43:9 Bessel’s Inequality, Parseval’s Indentity, Best Approximation Details 41:53 Best Approximation: Least Squares Solutions Details 50:37 Orthogonal Complementary Subspaces, Orthogonal Projections Details 50:1 Projection Theorem. Linear Functionals Details 47:24 The Adjoint Operator Details 48:21 Properties of the Adjoint Operation. Inner Product Space Isomorphism Details 52:37 Unitary Operators Details 48:17 Unitary operators II. Self-Adjoint Operators I. Details 42:11 Self-Adjoint Operators II – Spectral Theorem Details 41:8

This course is part of NPTEL online courses, delivered by IIT Madras.

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