Mathematical Definition
An affine map between two affine spaces is a map on the points that acts linearly on the vectors (that is, the vectors between points of the space). In symbols, f determines a linear transformation φ such that, for any pair of points :
or
- .
We can interpret this definition in a few other ways, as follows.
If an origin is chosen, and denotes its image, then this means that for any vector :
If an origin is also chosen, this can be decomposed as an affine transformation that sends, namely
followed by the translation by a vector .
The conclusion is that, intuitively, consists of a translation and a linear map.
Read more about this topic: Affine Transformation
Famous quotes containing the words mathematical and/or definition:
“It is by a mathematical point only that we are wise, as the sailor or the fugitive slave keeps the polestar in his eye; but that is sufficient guidance for all our life. We may not arrive at our port within a calculable period, but we would preserve the true course.”
—Henry David Thoreau (1817–1862)
“No man, not even a doctor, ever gives any other definition of what a nurse should be than this—”devoted and obedient.” This definition would do just as well for a porter. It might even do for a horse. It would not do for a policeman.”
—Florence Nightingale (1820–1910)