Bases of Polynomial Spaces - Wild Linear Algebra - NJ Wildberger

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