Advanced Matrix Theory and Linear Algebra for Engineers
Advanced Matrix Theory and Linear Algebra for Engineers. Instructor: Prof. Vittal Rao, Centre for Electronics Design and Technology, IISc Bangalore. Introduction to systems of linear equations, Vector spaces, Solutions of linear systems, Important subspaces associated with a matrix, Orthogonality, Eigenvalues and eigenvectors, Diagonalizable matrices, Hermitian and symmetric matrices, General matrices.
(from nptel.ac.in)
| Prologue |
| Lecture 01 - Prologue Part 1: Systems of Linear Equations, Matrix Notation |
| Lecture 02 - Prologue Part 2: Diagonalization of a Square Matrix |
| Lecture 03 - Prologue Part 3: Homogeneous Systems, Elementary Row Operations |
| Linear Systems |
| Lecture 04 - Linear Systems 1: Elementary Row Operations (EROs) |
| Lecture 05 - Linear Systems 2: Row Reduced Echelon Form, The Reduction Process |
| Lecture 06 - Linear Systems 3: The Reduction Process, Solution using EROs |
| Lecture 07 - Linear Systems 4: Solution using EROs: Non-homogeneous Systems |
| Vector Spaces |
| Lecture 08 - Vector Spaces Part 1 |
| Lecture 09 - Vector Spaces Part 2 |
| Linear Independence and Subspaces |
| Lecture 10 - Linear Combination, Linear Independence and Dependence |
| Lecture 11 - Linear Independence and Dependence, Subspaces |
| Lecture 12 - Subspace Spanned by a Finite Set of Vectors, The Basic Subspaces Associated with a Matrix |
| Lecture 13 - Subspace Spanned by an Infinite Set of Vectors, Linear Independence of an Infinite Set of Vectors |
| Basis |
| Lecture 14 - Basis, Basis as a Maximal Linearly Independent Set |
| Lecture 15 - Finite Dimensional Vector Spaces |
| Lecture 16 - Extension of a Linearly Independent Set to a Basis, Ordered Basis |
| Linear Transformations |
| Lecture 17 - Relation between Representation in Two Bases, Linear Transformations |
| Lecture 18 - Examples of Linear Transformations |
| Lecture 19 - Null Space and Range of a Linear Transformation |
| Lecture 20 - Rank Nullity Theorem, One-One Linear Transformation |
| Lecture 21 - One-One Linear Transformation, Onto Linear Transformations, Isomorphisms |
| Inner Product and Orthogonality |
| Lecture 22 - Inner Product and Orthogonality |
| Lecture 23 - Orthonormal Sets, Orthonormal Basis and Fourier Expansion |
| Lecture 24 - Fourier Expansion, Gram-Schmidt Orthonormalization, Orthogonal Complements |
| Lecture 25 - Orthogonal Complements, Decomposition of a Vector, Pythagoras Theorem |
| Lecture 26 - Orthogonal Complements in the context of Subspaces Associated with a Matrix |
| Lecture 27 - Best Approximation |
| Diagonalization |
| Lecture 28 - Diagonalization, Eigenvalues and Eigenvectors |
| Lecture 29 - Eigenvalues and Eigenvectors, Characteristic Polynomial |
| Lecture 30 - Algebraic Multiplicity, Eigenvectors, Eigenspaces and Geometric Multiplicity |
| Lecture 31 - Criterion for Diagonalization |
| Hermitian and Symmetric Matrices |
| Lecture 32 - Hermitian and Symmetric Matrices, Unitary Matrix |
| Lecture 33 - Unitary and Orthogonal Matrices, Eigen Properties of Hermitian Matrices, Unitary Diagonalization |
| Lecture 34 - Spectral Decomposition |
| Lecture 35 - Positive and Negative Definite and Semidefinite Matrices |
| Singular Value Decomposition (SVD) |
| Lecture 36 - Singular Value Decomposition (SVD) Part 1 |
| Lecture 37 - Singular Value Decomposition (SVD) Part 2 |
| Back to Linear Systems |
| Lecture 38 - Back to Linear Systems Part 1 |
| Lecture 39 - Back to Linear Systems Part 2 |
| Epilogue |
| Lecture 40 - Epilogue |