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Linear Algebra

The University of New South Wales,

Updated On 02 Feb, 19

Overview

This course on Linear Algebra is meant for first year undergraduates or college students. It presents the subject in a visual geometric way, with special orientation to applications and understanding of key concepts. The subject naturally sits inside affine algebraic geometry. Flexibility in choosing coordinate frameworks is essential for understanding the subject. Determinants also play an important role, and these are introduced in the context of multi-vectors in the sense of Grassmann. NJ Wildberger is also the developer of Rational Trigonometry: a new and better way of learning and using trigonometry.

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Lecture 21: WildLinAlg21 More bases of polynomial spaces

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Lecture Details

Polynomial spaces are excellent examples of linear spaces. For example, the space of polynomials of degree three or less forms a linear or vector psace which we call P^3. In this lecture we look at some more interesting bases of this space the Lagrange, Chebyshev, Bernstein and Spread polynomial basis. The last comes from Rational Trigonometry.

This is one of a series on Linear Algebra given by N J Wildberger of UNSW.

CONTENT SUMMARY pg 1 @0008 Introduction review; polynomials of degree 3; Lagrange, Chebyshev, Bernstein, spread polynomials; basis standardpower, factorial, Taylor; Lagrange polynomials developed 0218;
pg 2 @0353 Lagrange developement continued; evaluation mapping;
pg 3 @0613 Lagrange developement continued; polynomials that map to the standard basis vectors e1,e2,e3,34 (Lagrange interpolation polynomials);
pg 4 @0912 Lagrange basis; Polynomial that goes through four desired points;
pg 5 @1139 Uniform approximation and Bernstein polynomials
pg 6 @1342 reference to Pascals triangle; Bernstein polynomials (named); Bernstein basis;
pg 7 @1637 view of Bernstein polynomials;
pg 8 @1806 Show that Bernstein polynomials of a certain degree do form a basis for that corresponding polynomial space; Pascals triangle; Unnormalized Bernstein polynomials; WLA21_pg8_theorem (Bernstein polynomial basis);
pg 9 @2104 How Bernstein polynomials are used to approximate a given continuous function on an interval;
pg 10 @2413 Chebyshev polynomials; using a recursive definition; Chebyshev polynomial diagram;
pg 14 @3656 Spread polynomials relation to Chebyshevs; Spread polynomials advantage over Chebyshev; Pascals array; Spread polynomials as a source of study @3917;
pg 15 @3943 Spread basis; change of basis matrices; moral @4215 ;
pg 16 @4240 exercises 21.1-4 ;
pg 17 @4344 exercises 21.5-7 ; closing remarks @4438 (THANKS to EmptySpaceEnterprise)