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From Odwiki

Transformations

==Definition== Let X and Y each be a set; A transformation from X to Y is a rule f, which assigns every x element X stictly an element out of Y. This element gets referred to as f(x).
f:X →Y , x∈X ∧ y ∈ Y
The elements x of X are called preimages of the transformation f. The elements y out of Y for those at least a f(x) exists are the images. X is often referred to as the domain and Y as the range.

• Here are a few examples of transformations that everyone knows and make things a bit clearer:

1. A function such as
   f:R → R with f(x)=x²
   For every possible value of x we can assign a single value by evaluating the given function. I assume everyone will be familiar with this function and similar ones although not everyone might have thought of it as a transformation.
2. A transformation can be of a much more general nature such as the multiplication.
   R × R → R, (r,s) → r · s
   If we look at the above definition we see that all requirements of a transformation are met.
3. Finally here is an example of a more complex nature that shows how transformations can really simplify complex calculation: The Laplace Transformation. The Laplace Transformation is defined as
   f:R → C with L[f(x)]= ∫ exp(-st) f(t) dt
   The advantage of transforming a function from the time domain into the Laplace domain is that we can replace complex mathematical with simpler ones. It's most evident with differential equations: Finding a solution to a differential equation is usually quite complex; for a large group of differential equations however, the linear differential equations, we can solve this in the Laplace domain and the problem gets reduced to an algebraic one that usually is much easier to solve. There are many other special properties of the Laplace domain, but there is one other I want to highlight. The convolution of two function usually involves a complex convolution integral, however in the Laplace domain all this is reduced to the simple multiplication of these two functions.
   For now we'll stick to simpler Transformations, but hopefully this shows how verstatile the term transformation is and that due to this versatility it would be nice to find generic solutions for some frequently reoccuring problems that can be applied to whole groups of transformations.
   1. example A (ODE)
   2. example B (Convolution)

==Types==
In order to talk about transformations it is good to break them down into categories with associated properties. In this context we will often make use of the following terms

Let f denote a transformation from X to Y

1. injection We call f injective if there are no two (or more) elements of X that get mapped on a same element out of Y; in other words: there is a one to one mapping.
2. surjection We call f surjective if for each element y ∈ Y there exist an x ∈ X with f(x) = y. So to speak f is a mapping from X onto Y.
3. bijection We call f bijective if f is both injective and surjective

To get a better understand of these terms let's first look at some analytical examples and afterwards move on to examples relating closer to Houdini

1. f:R → R with f(x) = x² This transformation is neither injective nor surjective. Why?
   It's not injective since: f(-x) = (-x)² = (-1)²(x²) = (x²) = f(x)
   It's not surjective since we couldn't find any real number x ∈ X whose square is y = -1.
2. Have I seen some eye rousing there ? We know a number whose square is -1, however it's no real number, but a complex one: i. So if I would have written f:C → R with f(x) = x² it would have been an surjective mapping ... almost. Unfortunally we have got a new problem now: if x is a complex number it would have the form: x = a + bi. If both a ∧ b ≠ 0 x² would be a complex number, but without going into details, we'd run into new problems, even if we'd change to f:C → C.
   Let's not give up yet: Since changing the domain doesn't help much let's try to fix the range instead: f:R → R+ with f(x) = x² By ristricting Y to the set of positive real numbers we don't have the problem of worrying with complex numbers anymore: the mapping is surjective.
3. Let's see if we can get f even to be injective. We simply restrict the domain X to R+ just as we did with the range Y:
   f:R+ → R+ with f(x) = x²
   Now the transformation is at last both injective and surjective and thus bijective. For any valid element x ∈ X we can find the corresponding element y ∈ Y and vice versa. This leads on to the following usful lemma
   
   

• Invertibility of bijektive transformations