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Last updated:

September 23, 2023

Duration:

Unlimited Duration

FREE

This course includes:

Unlimited Duration

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Description

Linear Algebra. Instructor: Dr. K. C. Sivakumar, Department of Mathematics, IIT Madras. Systems of linear equations, Matrices, Elementary row operations, and 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)

Course Curriculum

    • Lecture 01 – Introduction to the Course Contents Unlimited
    • Lecture 02 – Linear Equations Unlimited
    • Lecture 03 – Equivalent Systems of Linear Equations I: Inverse Elementary Row-operations Unlimited
    • Lecture 03B – Equivalent Systems of Linear Equations II: Homogeneous Equations, Examples Unlimited
    • Lecture 04 – Row-reduced Echelon Matrices Unlimited
    • Lecture 05 – Row-reduced Echelon Matrices and Non-homogeneous Equations Unlimited
    • Lecture 06 – Elementary Matrices, Homogeneous Equations and Non-homogeneous Equations Unlimited
    • Lecture 07 – Invertible Matrices, Homogeneous Equations and Non-homogeneous Equations Unlimited
    • Lecture 08 – Vector Spaces Unlimited
    • Lecture 09 – Elementary Properties in Vector Spaces, Subspaces Unlimited
    • Lecture 10 – Subspaces, Spanning Sets, Linear Independence, Dependence Unlimited
    • Lecture 11 – Basis for a Vector Space Unlimited
    • Lecture 12 – Dimension of a Vector Space Unlimited
    • Lecture 13 – Dimensions of Sums of Spaces Unlimited
    • Lecture 14 – Linear Transformations Unlimited
    • Lecture 15 – The Null Space and the Range Space of a Linear Transformation Unlimited
    • Lecture 16 – The Rank-Nullity-Dimension Theorem, Isomorphisms between Vector Spaces Unlimited
    • Lecture 17 – Isomorphic Vector Spaces, Equality of the Row-rank and the Column-rank I Unlimited
    • Lecture 18 – Equality of the Row-rank and the Column-rank II Unlimited
    • Lecture 19 – The Matrix of a Linear Transformation Unlimited
    • Lecture 20 – Matrix for the Composition and the Inverse, Similarity Transformation Unlimited
    • Lecture 21 – Linear Functions, The Dual Space, Dual Basis Unlimited
    • Lecture 22 – Dual Basis (cont.), Subspace Annihilators Unlimited
    • Lecture 23 – Subspace Annihilators (cont.) Unlimited
    • Lecture 24 – The Double Dual, The Double Annihilator Unlimited
    • Lecture 25 – The Transpose of a Linear Transformation, Matrices of a Linear Transformation Unlimited
    • Lecture 26 – Eigenvalues and Eigenvectors of Linear Operators Unlimited
    • Lecture 27 – Diagonalization of Linear Operators, A Characterization Unlimited
    • Lecture 28 – The Minimal Polynomial Unlimited
    • Lecture 29 – The Cayley-Hamilton Theorem Unlimited
    • Lecture 30 – Invariant Subspaces Unlimited
    • Lecture 31 – Triangulability, Diagonalization in terms of Minimal Polynomial Unlimited
    • Lecture 32 – Independent Subspaces and Projection Operators Unlimited
    • Lecture 33 – Direct Sum Decompositions and Projection Operators I Unlimited
    • Lecture 34 – Direct Sum Decompositions and Projection Operators II Unlimited
    • Lecture 35 – The Primary Decomposition Theorem and Jordan Decomposition Unlimited
    • Lecture 36 – Cyclic Subspaces and Annihilators Unlimited
    • Lecture 37 – The Cyclic Decomposition Theorem I Unlimited
    • Lecture 38 – The Cyclic Decomposition Theorem II, The Rational Form Unlimited
    • Lecture 39 – Inner Product Spaces Unlimited
    • Lecture 40 – Norms on Vector Spaces, The Gram-Schmidt Procedure Unlimited
    • Lecture 41 – The Gram-Schmidt Procedure (cont.), The QR Decomposition Unlimited
    • Lecture 42 – Bessel’s Inequality, Parseval’s Identity, Best Approximation Unlimited
    • Lecture 43 – Best Approximation: Least Squares Solutions Unlimited
    • Lecture 44 – Orthogonal Complementary Subspaces, Orthogonal Projections Unlimited
    • Lecture 45 – Projection Theorem, Linear Functionals Unlimited
    • Lecture 46 – The Adjoint Operator Unlimited
    • Lecture 47 – Properties of the Adjoint Operation, Inner Product Space Isomorphism Unlimited
    • Lecture 48 – Unitary Operators Unlimited
    • Lecture 49 – Unitary Operators (cont.), Self-Adjoint Operators Unlimited
    • Lecture 50 – Self-Adjoint Operators – Spectral Theorem Unlimited
    • Lecture 51 – Normal Operators – Spectral Theorem Unlimited

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