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In an affine transformation $x \mapsto Ax+b$, $b$ represents the translation; but what does the matrix $A$ represent exactly? In a 2D example, $A$ is a $2\times 2$ matrix, but what does each term represent?

Suppose that given an affine transformation $\operatorname{Aff}_1$ I want to introduce another affine transformation $\operatorname{Aff}_2$ that is "half" the transformation $\operatorname{Aff}_1$. For the translation $b$, I just need to divide each coefficient by $2$, but what about the matrix $A$?

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For $A \in GL(2, \mathbb{R})$, the map $x \mapsto Ax$ is an invertible linear transformation from $\mathbb{R}^2$ to itself. There are four types of such transformations:

  • rotations,
  • reflections,
  • expansions/compressions, and
  • shears.

So an affine transformation is a map which does one of the above four things, followed by a translation.

As for your second question, it depends what you mean by an affine transformation 'doing half' of another transformation. First of all, there is some sense in which you can 'do half' of some linear transformations (e.g. rotations - you can rotate by half the angle), but not others (e.g. reflections).

Furthermore, the translation by $b$ complicates matters. Let $f(x) = Ax + b$ be a given affine transformation and suppose $g(x) = Cx + d$ is the affine transformation which 'does half' of $f(x)$. What does this mean? Well, I'm going to take it to mean that if you do the affine transformation $g(x)$ twice, then you get $f(x)$. So we are trying to find $C \in GL(2, \mathbb{R})$ and $d \in \mathbb{R}^2$ such that $$Ax + b = f(x) = g(g(x)) = g(Cx + d) = C(Cx + d) + d = C^2x + (Cd + d).$$ That is, we want to find $C$ and $d$ such that $A = C^2$ and $b = Cd + d = (C+I)d$. Depending on what $A$ and $b$ are, there may be no such $C$ and $d$ (as is the case when $A$ defines a reflection).

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