In an affine transformation x ↦ A x + b, b represents the translation; but what does the matrix A represent exactly? In a 2D example, A is a 2 × 2 matrix, but what does each term represent?

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In an affine transformation   x  ↦  A  x  +  b,   b represents the translation; but what does the matrix A represent exactly? In a 2D example,   A is a   2  ×  2 matrix, but what does each term represent?

conveneau71 · Accepted Answer

For   A  ∈  G  L  (  2  ,  R  ), the map   x  ↦  A  x is an invertible linear transformation from         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 &#039;doing half&#039; of another transformation. First of all, there is some sense in which you can &#039;do half&#039; 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  )  =  A  x  +  b be a given affine transformation and suppose   g  (  x  )  =  C  x  +  d is the affine transformation which &#039;does half&#039; of   f  (  x  ). What does this mean? Well, I&#039;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  ∈  G  L  (  2  ,  R  ) and   d  ∈        R    2   such that   A  x  +  b  =  f  (  x  )  =  g  (  g  (  x  )  )  =  g  (  C  x  +  d  )  =  C  (  C  x  +  d  )  +  d  =      C    2    x  +  (  C  d  +  d  )  .That is, we want to find   C and   d such that   A  =      C    2   and   b  =  C  d  +  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).

In an affine transformation x ↦ A x + b , b represents the transla

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