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affine transformation

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

51A10 no label found51A15 no label found15A04 no label found

Comments

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

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
rotation about a point, etc.

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

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?

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.

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?

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

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