Bases of Polynomial Spaces - Wild Linear Algebra - NJ Wildberger

Overview

Explore polynomial spaces from a linear algebra perspective in this comprehensive one-hour lecture. Delve into the four-dimensional space P^3, examining useful bases such as the standard (power) basis and the modified Factorial basis. Discover how translations of corresponding functions yield linear transformations, leading to Taylor bases and an algebraic definition of derivatives. Gain insights into the algebraic nature of fundamental calculus concepts, challenging the necessity of real numbers, limits, or tangent slopes. Investigate various bases, including the standard basis and factorial basis, and learn about basis isomorphism correspondence. Analyze the derivative as a linear transformation and explore change of basis matrices. Engage with practical exercises to reinforce your understanding of these advanced mathematical concepts.

Syllabus

CONTENT SUMMARY: pg 1: @00:08 Map of a space of polynomials to a space of vectors; Definition: linear/vector space; course distinction @ ;
pg 2: @ Definition: An ordered basis of a linear space; Example1 ;
pg 3: @ Example2 basis of a vector space; Example3 basis of a polynomial space; Definition: Dimension of a linear/vector space; examples;
pg 4: @09:25 The space of polynomials is richer than the isomorphic space of vectors; translating polynomials @ ; degree of the polynomial is preserved in translation;
pg 5: @13:20 Study01 of translation by 3 see previous page; A linear transformation @ ;
pg 6: @16:48 study01 continued; image and kernel of polynomial of degree 3 @;
pg 7: @ the derivative of a function appears in translation;
pg 8: @22:49 Definition: The derivative of a polynomial; calculus via linear algebra @;
pg 9: @ Calculus as algebra; pg9_Theorem product rule; proof Leibniz mentioned;
pg 10: @ Translating a polynomial and obtaining the derivatives; Taylor series mentioned;
pg 11: @ Importance of various bases; standard basis; factorial basis; Example ; coefficient vectors of a polynomial with respect to a basis;
pg 12: @42:10 Theorem Basis isomorphism correspondence; The standard vector space of column vectors @ ;
pg 13: @46:22 The derivative as a linear transformation; remark about formulas in calculus and combinatorics @ ;
pg 14: @ Another basis; the standard basis moved over by 3; example;
pg 15: @54:32 Change of basis matrix; How to get this matrix! @ ;
pg 16: @57:16 Exercises 20.1-3; closing remarks @; THANKS to EmptySpaceEnterprise
Introduction
Ordered basis of a linear space
Translation as a linear transformations
Derivative of a polynomial
Some bases of P³
Theorem Basis isomorphism correspondence
The derivative D as a linear transformation
Another basis
Change of basis matrix