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Matrix Computation and its Applications

Matrix Computation and its Applications. Instructor: Prof. Vivek Kumar Aggarwal, Department of Mathematics, IIT Delhi. This course deals with applications of matrices to a wide range of areas of engineering and science. Some basics of linear algebra are discussed followed by matrix norms and sensitivity and condition number of the matrices. The course continues to discuss topics: linear systems, Jacobi, Gauss-Seidel and successive over relaxation methods, LU decompositions, Gaussian elimination with partial pivoting, Banded systems, positive definite systems, Cholesky decomposition - sensitivity analysis, Gram-Schmidt orthonormal process, Householder transformation, QR factorization, stability of QR factorization. Solution of linear least squares problems, normal equations, singular value decomposition (SVD), Moore-Penrose inverse, rank deficient least squares problems, sensitivity analysis of least squares problems, sensitivity of eigenvalues and eigenvectors. (from nptel.ac.in)

Lecture 43 - Inner Product Spaces


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Lecture 01 - Binary Operation and Groups
Lecture 02 - Vector Spaces
Lecture 03 - Some Examples of Vector Spaces
Lecture 04 - Some Examples of Vector Spaces (cont.)
Lecture 05 - Subspace of a Vector Space
Lecture 06 - Spanning Set
Lecture 07 - Properties of Subspace
Lecture 08 - Properties of Subspace (cont.)
Lecture 09 - Linearly Independent and Dependent Vectors
Lecture 10 - Linearly Independent and Dependent Vectors (cont.)
Lecture 11 - Properties of Linearly Independent and Dependent Vectors
Lecture 12 - Properties of Linearly Independent and Dependent Vectors (cont.)
Lecture 13 - Basis and Dimension of a Vector Space
Lecture 14 - Examples of Basis and Dimension of a Vector Space
Lecture 15 - Linear Functions
Lecture 16 - Range Space of a Matrix and Row Reduced Echelon Form
Lecture 17 - Row Equivalent Matrices
Lecture 18 - Row Equivalent Matrices (cont.)
Lecture 19 - Null Space of a Matrix
Lecture 20 - Four Subspaces associated with a Given Matrix
Lecture 21 - Four Subspaces associated with a Given Matrix (cont.)
Lecture 22 - Linear Independence of the Rows and Columns of a Matrix
Lecture 23 - Application of Diagonal Dominant Matrices
Lecture 24 - Application of Zero Null Space: Interpolating Polynomial and Wronskian Matrix
Lecture 25 - Characterization of Basis of a Vector Space and its Subspaces
Lecture 26 - Coordinate of a Vector with respect to Ordered Basis
Lecture 27 - Examples of Different Subspaces of a Vector Space of Polynomials
Lecture 28 - Linear Transformation
Lecture 29 - Properties of Linear Transformation
Lecture 30 - Determining Linear Transformation on a Vector Space by its Value on the Basis Element
Lecture 31 - Range Space and Null Space of a Linear Transformation
Lecture 32 - Rank and Nullity of a Linear Transformation
Lecture 33 - Rank-Nullity Theorem
Lecture 34 - Application of Rank-Nullity Theorem and Inverse of a Linear Transformation
Lecture 35 - Matrix Associated with Linear Transformation
Lecture 36 - Matrix Representation of a Linear Transformation Relative to Ordered Bases
Lecture 37 - Matrix Representation of a Linear Transformation Relative to Ordered Bases (cont.)
Lecture 38 - Linear Map associated with a Matrix
Lecture 39 - Similar Matrices and Diagonalization of Matrix
Lecture 40 - Orthonormal Bases of a Vector Space
Lecture 41 - Gram-Schmidt Orthogonalization Process
Lecture 42 - QR Factorization
Lecture 43 - Inner Product Spaces
Lecture 44 - Inner Product on Different Real Vector Spaces and Basis of Complex Vector Space
Lecture 45 - Inner Product on on Complex Vector Spaces and Cauchy-Schwarz Inequality
Lecture 46 - Norm of a Vector
Lecture 47 - Matrix Norm
Lecture 48 - Sensitivity Analysis of a System of Linear Equations
Lecture 49 - Orthogonality of the Four Spaces associated with a Matrix
Lecture 50 - Best Approximation: Least Square Method
Lecture 51 - Best Approximation: Least Square Method (cont.)
Lecture 52 - Jordan-Canonical Form
Lecture 53 - Some Examples on the Jordan Form of a Given Matrix and Generalized Eigenvectors
Lecture 54 - Singular Value Decomposition Theorem
Lecture 55 - MatLab/Octave Code for Solving SVD
Lecture 56 - Pseudo-Inverse/Moore-Penrose Inverse
Lecture 57 - Householder Transformation
Lecture 58 - MatLab/Octave Code for Householder Transformation