Operator fundamentals

According to Merriam-Webster Online, an "operator" is (in this context) "something [...] that denotes or performs a mathematical or logical operation". That is, an operator is not a passive table of elements; rather it is something active like a function. This view is sometimes helpful in understanding BBTools.

Let us go back to basics: linear operators are functions that map a vector-space to another linearly. That is, T is a linear operator if it satisfies the following for all x,y in the input domain and scalars c:

  1. T(x+y)=T(x)+T(y)
  2. T(xc)=cT(x)

When the input and output domains of a linear operator have finite dimensions, then any element, say x, in the vector-space can be written in terms of a basis: x=c1b1+c2b2+.... As long as the basis is given, we may refer to x by giving the coefficients (or coordinates).

Traditionally these coefficients are collected in a column-vector. We recommend following this practice, although it is never enforced. (Statisticians often prefer to collect coefficients in a row-vector instead [14]).

It turns out that any linear operation that maps a column-vector to another, may be described uniquely in the form of a matrix. The transformation y=T(x) is computed as a matrix-vector product y=A*x.

It is a good idea to keep in mind that the object x is represented by the column-vector x in terms of a basis. In many cases the column vector is simply related to the basis, and is is easy to forget it exists. If an object is an image, for example, each element may refer to a pixel.


[14] Kanti V. Mardia, John T. Kent, and John M. Bibby. Multivariate Analysis. Academic Press, London, 1995. ISBN: 0-12-471252-5. 10th print. (Book)