Science Fair Project Encyclopedia
Affine group
In mathematics, the affine group of any affine space over a field K is the group of all invertible affine transformations from the space into itself. It is a Lie group if K is the real or complex field.
There is more than one convenient way to describe the structure of affine groups. There is the abstract result that it is a semidirect product: this is given on the affine space page. There is also a more down-to-earth matrix representation: represent a pair
- (M, v),
where M is an n×n matrix over K, and v a 1×n column vector, by the
- (n + 1)×(n + 1)
matrix
- (M*|v*).
Here M* is the (n + 1)×n matrix formed by adding a row of zeroes below M, and v* is the column matrix of size n + 1 formed by adding an entry 1 below v.
Last updated: 08-29-2005 01:58:54
10-26-2009 08:16:03
The contents of this article is licensed from www.wikipedia.org under the GNU Free Documentation License. Click here to see the transparent copy and copyright details
The contents of this article is licensed from www.wikipedia.org under the GNU Free Documentation License. Click here to see the transparent copy and copyright details