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Special Topics in Linear Algebra

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Overview

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This course covers topics in linear algebra such as least squares approximation, Hamming's error correcting codes, the functional calculus, affine subspaces and transformations, stochastic maps, and finite-dimensional C*-algebras. The learning outcomes include understanding fundamental theorems, fitting data to curves, working with Hamming matrices, polynomial interpolation, and affine transformations. Students will learn skills such as calculating inverses, working with conditional probabilities, and understanding *-homomorphisms and positivity in C*-algebras. The teaching method involves theoretical explanations, examples, and proofs to deepen understanding. This course is intended for learners interested in advanced linear algebra topics and their applications in various fields.

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

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