Explore advanced topics in linear algebra through a comprehensive 5-hour video series. Delve into least squares approximation, including the fundamental theorem, fitting data to curves, and applications to more general functions. Learn about Hamming's error-correcting codes, their properties, and practical examples. Investigate the functional calculus, covering theorems, square roots of positive matrices, and polynomial interpolation. Study affine subspaces and transformations, including combinations, subspaces, and composition of transformations. Examine stochastic maps, conditional probabilities, and Bayes' theorem. Conclude with an introduction to finite-dimensional C*-algebras, exploring *-homomorphisms, positivity, and the Stochastic Gelfand-Naimark theorem.
Overview
Syllabus
Linear Algebra - Least Squares Approximation - 01 - Introduction.
Linear Algebra - Least Squares Approximation - 02 - Fundamental Theorem.
Linear Algebra - Least Squares Approximation - 03 - Fitting data to a straight curve Part 1.
Linear Algebra - Least Squares Approximation - 04 - Fitting data to a straight curve Part 2.
Linear Algebra - Least Squares Approximation - 05 - Fitting data to a straight curve Part 3.
Linear Algebra - Least Squares Approximation - 06 - Fitting data to a straight curve example.
Linear Algebra - Least Squares Approximation - 07 - Fitting data to more general functions.
Linear Algebra - Least Squares Approximation - 08 - The inverse of A transpose times A.
Linear Algebra - Hamming's error correcting codes - 01 - Hamming matrices.
Linear Algebra - Hamming's error correcting codes - 02 - Properties of Hamming matrices.
Linear Algebra - Hamming's error correcting codes - 03 - Example.
Linear Algebra - Hamming's error correcting codes - 04 - Parity bits.
Topics in Linear Algebra - The Functional Calculus - 01 - Theorem and Example.
Topics in Linear Algebra - The Functional Calculus - 02 - Square-root of a positive matrix.
Topics in Linear Algebra - The Functional Calculus - 03 - Polynomial interpolation.
Topics in Linear Algebra - The Functional Calculus - 04 - The determinant of a Vandermonde matrix.
Topics in Linear Algebra - The Functional Calculus - 05 - Proof of main theorem.
Affine subspaces and transformations - 01 - affine combinations.
Affine subspaces and transformations - 02 - affine subspaces.
Affine subspaces and transformations - 03 - affine transformations.
Affine subspaces and transformations - 04 - composition of affine transformations.
Stochastic maps - 01 - Conditional probabilities.
Stochastic maps - 02 - Composing conditional probabilities.
Stochastic maps - 03 - Products of conditional probabilities and a.e. equivalence.
Stochastic maps - 04 -Bayes' theorem.
Finite-dimensional C*-algebras - 01 - *-homomorphisms.
Finite-dimensional C*-algebras - 02 - positivity.
Finite-dimensional C*-algebras - 03 - Stochastic Gelfand--Naimark theorem.
Taught by
Arthur Parzygnat
