# Affine coordinate frame

A set of linearly-independent vectors () of -dimensional affine space , and a point . The point is called the initial point, while the vectors are the scale vectors. Any point is defined with respect to the affine coordinate frame by numbers — coordinates , occurring in the decomposition of the position vector by the scale vectors: (summation convention). The specification of two affine coordinate frames defines a unique affine transformation of the space which converts the first frame into the second (see also Affine coordinate system).

#### Comments

An equivalent, and more usual, definition is as follows. An affine coordinate frame in affine -space is a set of points which are linearly independent in the affine sense, i.e. the vectors , , are linearly independent in the corresponding vector space. Independence of the vectors in the definition should be understood as independence in a corresponding vector space.

#### References

[1] | E. Snapper, R.J. Troyer, "Metric affine geometry" , Acad. Press (1971) |

**How to Cite This Entry:**

Affine coordinate frame.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Affine_coordinate_frame&oldid=17248