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Last updated:
September 23, 2023
Duration:
Unlimited Duration
FREE
This course includes:
Unlimited Duration
Badge on Completion
Certificate of completion
Unlimited Duration
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
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- 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
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- 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 19 – The Matrix of a Linear Transformation Unlimited
- Lecture 20 – Matrix for the Composition and the Inverse, Similarity 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 33 – Direct Sum Decompositions and Projection Operators I Unlimited
- Lecture 34 – Direct Sum Decompositions and Projection Operators II 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 46 – The Adjoint Operator Unlimited
- Lecture 47 – Properties of the Adjoint Operation, Inner Product Space Isomorphism Unlimited
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