This lecture studies spaces of polynomials from a linear algebra point of view. We are especially interested in useful bases of a four dimensional space like P^3: polynomials of degree three or less. We introduce the standard (or power) basis, also the modified Factorial basis. Translations of the corresponding functions yield linear transformations, giving Taylor bases and a purely algebraic definition of the derivative. We see that some basic calculus ideas are really algebraic in nature, not requiring `real numbers’, limits or slopes of tangents.
This course in Linear Algebra is given by N J Wildberger.
CONTENT SUMMARY: pg 1: @00:08 Map of a space of polynomials to a space of vectors; Definition: linear/vector space; course distinction @03:27 ;
pg 2: @04:46 Definition: An ordered basis of a linear space; Example1 ;
pg 3: @07:03 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 @09:50 ; degree of the polynomial is preserved in translation;
pg 5: @13:20 Study01 of translation by 3 (see previous page); A linear transformation @16:15 ;
pg 6: @16:48 study01 continued; image and kernel of polynomial of degree 3 @17:16;
pg 7: @19:04 the derivative of a function appears in translation;
pg 8: @22:49 Definition: The derivative of a polynomial; calculus via linear algebra @23:00;
pg8_Theorem; factorial notation;
pg 9: @28:05 Calculus as algebra; pg9_Theorem (product rule); proof (Leibniz mentioned);
pg 10: @33:46 Translating a polynomial and obtaining the derivatives; Taylor series mentioned;
pg 11: @37:58 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 @45:30 ;
pg 13: @46:22 The derivative as a linear transformation; remark about formulas in calculus and combinatorics @50:46 ;
pg 14: @51:13 Another basis; the standard basis moved over by 3; example;
pg 15: @54:32 Change of basis matrix; How to get this matrix! @54:56 ;
pg 16: @57:16 Exercises 20.1-3; closing remarks @58:27; (THANKS to EmptySpaceEnterprise)
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