A First Course in Linear Algebra
A First Course in Linear Algebra (UNSW). Taught by Professor N. J. Wildberger, this course presents a geometrical view to Linear Algebra, with special orientation to applications and understanding of key concepts. The subject naturally sits inside affine geometry, which is the natural setting for vectors. Flexibility in choosing coordinate frameworks is important for understanding the subject. Determinants also play a key role, and these are introduced in the context of multi-vectors in the sense of Grassmann. The course features a careful treatment of polynomial spaces, with applications to Stirling numbers and cubic splines.
| Lecture 23 - Stirling Numbers and Pascal Triangles |
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.
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