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:
“As we speak of poetical beauty, so ought we to speak of mathematical beauty and medical beauty. But we do not do so; and that reason is that we know well what is the object of mathematics, and that it consists in proofs, and what is the object of medicine, and that it consists in healing. But we do not know in what grace consists, which is the object of poetry.”
—Blaise Pascal (1623–1662)
“According to our social pyramid, all men who feel displaced racially, culturally, and/or because of economic hardships will turn on those whom they feel they can order and humiliate, usually women, children, and animals—just as they have been ordered and humiliated by those privileged few who are in power. However, this definition does not explain why there are privileged men who behave this way toward women.”
—Ana Castillo (b. 1953)