SignUpLinear Algebra
The University of New South Wales,
Updated On 02 Feb, 19
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The University of New South Wales,
Updated On 02 Feb, 19
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.
4.1 ( 11 )
When we interpret polynomials as sequences rather than as functions, new bases become important. The falling and rising powers play an important role in analysing general sequences through forward and backward difference operators.
The change from rising powers to ordinary powers, and from ordinary powers to falling powers give rise to two interesting families of numbers, called Stirling numbers of the first and second kind. We use Karamata notation, advocated by Knuth to describe these brackets and braces. Combinatorial and number theoretic interpretations are mentioned.
We discuss the important relation between two bases of a a linear space and the corresponding change from one kind of coordinate vector to another. This is applied to study general polynomial sequences.
This lecture is not easy, and represents a high point of this course in Linear Algebra. However it introduces powerful and common techniques which are actually quite useful in a variety of practical applications.
CONTENT SUMMARY pg 1 @0008 Intro (Stirling numbers and Pascal triangles); sequences; change of terminology @0044 ; falling power; rising power; list of rising powers; summation notation and Stirling numbers @0300;
pg 2 @0455 James Stirling (1749), "Methodus Differentialis"; Stirling number notation warning @0504 ; n bracket k as Karamata notation (Knuth); Stirling numbers of the first kind; Change of basis rewritten from pg 1 @0529 ; Stirling matrix of the first kind; remark about unconventional indexing of Stirling numbers @0636;
pg 3 @0713 Calculating Stirling numbers; Theorem (Recurrence relation Stirling numbers); proof;
pg 4 @1056 Pascals triangle and binomial coefficients; recurrence relation for binomial coefficients; Pascal matrix;
pg 5 @1408 Combinatorial interpretation of Sterling numbers;
pg 6 @1734 Number theoretic interpretation of Sterling numbers; summary of Sterling number interpretation @2150;
pg 7 @2309 Sterling numbers of the 2nd kind; Inverting the Pascal matrices;
pg 8 @2636 Inverting Stirling matrices; reintroduction of some ignored symmetry @2748 ; Sterling matrix of the 2nd kind;
pg 9 @3041 Definition of Stirling numbers of the second kind; n brace k notation of Stirling numbers of the 2nd kind; Sterling matrix of the 2nd kind;
pg 10 @3254 Combinatorial interpretation of Sterling_numbers_2nd_kind ; Theorem (Recurrence relation for Sterling_numbers_2nd_kind);
pg 11 @3554 Statement of the importance of the Sterling numbers; important question @3723 ; suggestion to review starting WLA1_pg7 @4027;
pg 12 @4048 Of primary importance to problems of practical application; Non_standard ideas; This is at the heart of change of basis @4708;
pg 13 @4726 Transpose a matrix and vector;
pg 14 @5011 Application of this (effect of change of basis on coordinate vectors) analyse a polynomial sequence; Newtons formula; A very useful thing to be able to do @5354;
pg 15 @5444 General C transpose of signed Stirling matrix fo 1st kind;
pg 16 @5530 Exercises 23.1-3;
pg 17 @5613 Exercises 23.4-5; closing remarks @5714; (THANKS to EmptySpaceEnterprise)
Sam
Sep 12, 2018
Excellent course helped me understand topic that i couldn't while attendinfg my college.
Dembe
March 29, 2019
Great course. Thank you very much.
