# affine transformation

## Primary tabs

Defines:
IGL, translation, dilation, dilation map, homothetic transformation, affine property, affine isomorphism, associated linear transformation, affinely isomorphic, affinity
Synonym:
scaling
Type of Math Object:
Definition
Major Section:
Reference
Groups audience:

## Mathematics Subject Classification

### affine transformation requirement

I think the linear transformation in the definition of affine transformation is required to be invertible. Correct?

### Re: affine transformation requirement

It probably depends on the context.

In geometry, if one defines an affine map as
a map conserving straight lines, then L must
be invertible. For example, A(v)=w, maps
everything to a point. Also, in most practical
applications, L is probably invertible. Like

I took this definition from the top of my head.
However, you have editing rights so please feel free
to make modifications.

Oh.. in Java, an affine map does not need to be
invertible :-)
merganser.math.gvsu.edu/david/reed03/notes/chap4.pdf

### transformation of convex polygons

Hello,
If a point P is inside a convex polygon Q and f is an affine transformation, will f(P) be inside f(Q)? i think the answer is yes, but why?

### Re: transformation of convex polygons

Let P= sum_i {y_i v_i}, where the v_i are the vertices of Q, and the y_i are positive reals which sum to 1.

Let f be given by fx=Ax+w, for some matrix A and vector w.

Then

fP= AP + w
=A sum_i {y_i v_i} + w
= sum_i {y_i Av_i} + sum_i {y_i w}
= sum_i {y_i(Av_i +w)}
= sum_i {y_i fv_i}

Hence fP is in the convex hull of the f v_i. Therefore fP is in fQ.

### minimal number of pts. to determine that a transform is affi...

I have the following question:

Let $V$ be a finite-dimensional inner product space over a field $K$ with $\dim(V) = n$. Consider a finite set of points $v_1, \dots , v_m \in V$. Let $F:V\to V$. What is the lowest $m$ such that

if for all $i,j\in\{1, \dots, m\}$

$|F(v_i) - F(v_j)| = |v_i - v_j|$

then for all $v,w\in V$

$|F(v) - F(w)| = |v - w|$

???

I feel as though the answer is that the minimum such $m$ is $n+1$ based in Euclidean space intuition and I also feel that in this case the points should be affinely independent. Also, does one need to restrict the field $K$ in any way?

### Re: minimal number of pts. to determine that a transform is ...

Sorry, I do not think this question is well-posed, please disregard it!