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)
| 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.
|