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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)

Lecture 47 - Properties of the Adjoint Operation, Inner Product Space Isomorphism

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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