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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 22: WildLinAlg22 Polynomials and sequence spaces

4.1 ( 11 )

Lecture Details

Polynomials can be interpreted as functions, and also as sequences. In this lecture we move to considering sequences. Aside from the familiar powers, we introduce also falling and rising powers, using the notation of D. Knuth. These have an intimate connection to forward and backward difference operators. We look at some particular sequences, such as the square pyramidal numbers, from the view of this `difference calculus.

CONTENT SUMMARY pg 1 @0008
polynomials and sequence spaces; remark about expressions versus objects @0327 ;
pg 2 @0424 Some polynomials and associated sequences; Ordinary powers; Factorial powers (D. Knuth);
pg 3 @1034 Lowering (factorial) power; Raising (factorial) power; connection between raising and lowering; all polynomials @1328;
pg 4 @1352 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 @1923 Forward and backward differences; forwardbackward difference operators on polynomials; examples operator on 1 @2307;
pg 6 @2338 Forward and backward differences on a sequence; difference belowabove convention;
pg 7 @2721 Forward and backward Differences of lowering powers; calculus reference @2937;
pg 8 @3127 Forward and backward Differences of raising powers; operators act like derivative @3445 ; n equals 0 raising and lowering defined;
pg 9 @3617 Introduction of some new basis; standardpower basis, lowering power basis, raising power basis; proven to be bases;
pg 10 @3923 WLA22_pg10_Theorem (Newton); proof;
pg 10b @4440 Lesson it helps to start at n=0; example (square pyramidal numbers);an important formula @4747;
pg 11 @5000 formula of Archimedes; taking forward distances compared to summation @5246
pg 12 @5320 a simpler formula; example sum of cubes;
pg 13 @5738 exercises 22.1-4;
pg 14 @5906 exercise 22.5; find the next term; closing remarks @5950;