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Published on Jan 18, 2013
In this lecture we put our previous discussion of coordinate vectors and change of basis into a more general and novel framework, and look at an important application to calculus.
We introduce the new idea of a vatrix: a matrix whose entries are themselves vectors. This concept allows us to encode the basis of a linear/vector space, such as P^3, as a vector, and linear combinations as products of row and column vatrices. Coefficient vectors now become row vectors, and change of basis matrices have a logical and intuitive labelling.
Our main example goes right back to our first lecture with Bob and Rachel's two bases for the affine plane. Then we look at Taylor coefficient vectors for a polynomial p in P^3. For every rational point c there is a Taylor basis, with an associated vatrix of powers of (alpha-c). The crucial change of basis matrices are generalized Pascal matrices which enjoy lovely algebraic properties--they form a one-parameter group of matrices.
In fact this whole theory has natural connections with the representation theory of sl(2), which we do not mention. This is the final lecture in this first half of this course on Linear Algebra.