Math 331 - February 26, 1999

Affine change of Coordinates

An affine change of coordinates in R^{n} is given by

x' = R x + r

where R is an invertible n \times n matrix, r is a column of length n, x is the column of coordinates of a general point in the ``old'' coordinate system, and x' is the column of coordinates of the same point in the ``new'' coordinate system. Of course, the rolls of ``old'' and ``new'' are reversible.

When r = 0 the coordinate change is said to be linear. Linear coordinate change is studied in second year ``linear algebra''.

Proposition. If an affine map is given by the formula y = A x + b relative to the old coordinate system, then in the new coordinate system one has

y' = A' x + b'

where

A' = R A R^{-1} and b' = r + R b - R A R^{-1} r .

Short Test: Monday, March 8

And please remember that the midterm test is scheduled for Wednesday, March 17.

Exercises due Monday, March 8

  1. If a function f is given by f(x) = M x, where M is a square matrix, relative to old coordinates, how is f defined after changing coordinates by a translation, i.e., when the new coordinate column x' is related to the old coordinate column x by x' = x + r ?

  2. With an change of affine coordinates

    x' = R x + r

    when is it true that the distance between two points computed relative to their new coordinates is equal to the distance computed relative to their old coordinates?

  3. What happens to the barycentric coordinates of a given point relative to given barycentrically independent points when the underlying Cartesian coordinates of all the given points undergo the same affine change?

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