Explore an advanced lecture on Stirling numbers and Pascal triangles in linear algebra. Delve into the interpretation of polynomials as sequences, examining falling and rising powers, and their role in analyzing general sequences through difference operators. Discover the Stirling numbers of the first and second kind, using Karamata notation. Investigate the relationship between bases in linear spaces and coordinate vector transformations. Learn about combinatorial and number theoretic interpretations of Stirling numbers, as well as their recurrence relations. Study the Pascal matrix, binomial coefficients, and methods for inverting Stirling matrices. Apply these concepts to analyze polynomial sequences using Newton's formula. Engage with exercises to reinforce understanding of these advanced linear algebra topics.
Stirling Numbers and Pascal Triangles - Foundations of Linear Algebra
Insights into Mathematics via YouTube
Overview
Syllabus
CONTENT SUMMARY: pg 1: @00:08 Intro: Stirling numbers and Pascal triangles; sequences; change of terminology @00:44 ; falling power; rising power; list of rising powers; summation notation and Stirling numbers @;
pg 2: @04:55 James Stirling 1749, "Methodus Differentialis"; Stirling number notation warning @05:04 ; 'n bracket k' as Karamata notation Knuth; Stirling numbers of the first kind; Change of basis rewritten from pg 1 @05:29 ; Stirling matrix of the first kind; remark about unconventional indexing of Stirling numbers @;
pg 3: @ Calculating Stirling numbers; Theorem Recurrence relation: Stirling numbers; proof;
pg 4: @ Pascal's triangle and binomial coefficients; recurrence relation for binomial coefficients; Pascal matrix;
pg 5: @ Combinatorial interpretation of Sterling numbers;
pg 6: @17:34 Number theoretic interpretation of Sterling numbers; summary of Sterling number interpretation @;
pg 7: @ Sterling numbers of the 2nd kind; Inverting the Pascal matrices;
pg 8: @26:36 Inverting Stirling matrices; reintroduction of some ignored symmetry @ ; Sterling matrix of the 2nd kind;
pg 9: @ Definition of Stirling numbers of the second kind; 'n brace k' notation of Stirling numbers of the 2nd kind; Sterling matrix of the 2nd kind;
pg 10: @ Combinatorial interpretation of Sterling_numbers_2nd_kind ; Theorem Recurrence relation for Sterling_numbers_2nd_kind;
pg 11: @35:54 Statement of the importance of the Sterling numbers; important question @37:23 ; suggestion to review starting WLA1_pg7 @;
pg 12: @40:48 Of primary importance to problems of practical application; Non_standard ideas; This is at the heart of change of basis @;
pg 13: @ Transpose a matrix and vector;
pg 14: @50:11 Application of this effect of change of basis on coordinate vectors: analyse a polynomial sequence; Newtons formula; A very useful thing to be able to do @;
pg 15: @ General C: transpose of signed Stirling matrix of 1st kind;
pg 16: @ Exercises 23.1-3;
pg 17: @56:13 Exercises 23.4-5; closing remarks @; THANKS to EmptySpaceEnterprise
Introduction
James Stirling Methodus Differentialis
Pascal Matrix
Combinatorial interpretation
Number theoretic interpretation
Inverting Pascal matrices
Inverting Stirling matrices
Stirling numbers of the second kind
Combinatorial interpretation of Stirling numbers
Square pyramidal numbers
Transpose of signed Stirling matrix of first kind
Taught by
Insights into Mathematics
