Polynomials and Sequence Spaces - Wild Linear Algebra - NJ Wildberger

Explore polynomials and sequence spaces in this comprehensive lecture on Wild Linear Algebra. Delve into the interpretation of polynomials as functions and sequences, introducing concepts like falling and rising powers using D. Knuth's notation. Examine the connection between these powers and forward and backward difference operators. Investigate specific sequences, such as square pyramidal numbers, through the lens of difference calculus. Learn about various polynomial bases, including standard, lowering power, and raising power bases. Study important theorems like Newton's theorem and Archimedes' formula. Practice with exercises on finding the next term in sequences and applying the concepts learned throughout the lecture.

CONTENT SUMMARY: pg 1: @
polynomials and sequence spaces; remark about expressions versus objects @ ;
pg 2: @ Some polynomials and associated sequences; Ordinary powers; Factorial powers D. Knuth;
pg 3: @10:34 Lowering factorial power; Raising factorial power; connection between raising and lowering; all polynomials @;
pg 4: @ Why we want these raising and lowering factorial powers; general sequences; On-line encyclopedia of integer sequences N.Sloane; 'square pyramidal numbers'; Table of forward differences;
pg 5: @19:23 Forward and backward differences; forward/backward difference operators on polynomials; examples: operator on 1 @;
pg 6: @ Forward and backward differences on a sequence; difference below/above convention;
pg 7: @27:21 Forward and backward Differences of lowering powers; calculus reference @;
pg 8: @31:27 Forward and backward Differences of raising powers; operators act like derivative @ ; n equals 0 raising and lowering defined;
pg 9: @ Introduction of some new basis; standard/power basis, lowering power basis, raising power basis; proven to be bases;
pg 10: @ WLA22_pg10_Theorem Newton; proof;
pg 10b: @44:40 Lesson: it helps to start at n=0; example square pyramidal numbers;an important formula @;
pg 11: @50:00 formula of Archimedes; taking forward distances compared to summation @
pg 12: @ a simpler formula; example: sum of cubes;
pg 13: @ exercises 22.1-4;
pg 14: @59:06 exercise 22.5; find the next term; closing remarks @;
Introduction
Some polynomials and associated sequences
Lowering factorial powers
Forward and backward differences
Differences of lowering and raising powers are easy to compute!
Factorial power bases
A theorem of Newton
A formula of Archimedes
A formula for sum of cubes
Exercises 22.1-4;