# Meaning of affine transformation

From Wikipedia, I learned that an affine transformation between two vector spaces is a linear mapping followed by a translation.

But in a book Multiple view geometry in computer vision by Hartley and Zisserman:

An affine transformation (or more simply an affinity) is a non-singular linear transformation followed by a translation.

I wonder if these are two different concepts, given that one does not require the linear transformation to be non-singular while the other does?

Thanks!

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Sure they are different concepts (and by asking it the way you ask it you have answered your question yourself already)... The map that takes all to $0$ is an affine transformation in the Wikipedia sense while it isn't in the second sense. – t.b. Dec 24 '11 at 21:13
You could think of it this way: an affine transformation in the sense of Wikipedia may or may not be an bijective (or invertible) function. However the book reserves the term for the bijective ones. [As t.b. points out, the Wikipedia definition is strictly more general.] – Srivatsan Dec 24 '11 at 21:19
An affine transformation preserves affine combinations, i.e. linear combinations in which the sum of the coefficients is $1$. Those are precisely the ones whose value does not depend on which point in the space is chosen to be the origin. If it's non-singular, then the image of a set of points will not have any affine relations not already present in the original set; otherwise it will. (But as to whether one or the other definition is correct, I have no opinion right now.) – Michael Hardy Dec 24 '11 at 21:31
Thanks, people! @t.b.: Can affine transformation be defined between two vector spaces with different dimensions? If yes, is it en.wikipedia.org/wiki/…, with affine spaces replaced by vector spaces? – Tim Dec 24 '11 at 22:53
I think the definition of an affine transformation between two vector spaces is as follows. A map $f:V\to W$ is affine if there exists a $w$ in $W$ such that the map $v\mapsto f(v)-w$ is linear. In words, an affine transformation is a linear transformation up to a translation. – Ohdur Dec 24 '11 at 23:08