What is map, function and transformation?

Question

I confuse about the tranformation. There alot of tranformation types. I don't know that which one is general, which one is just a special case.

1. Map. What is the exactly map?

   • From wikipedia, there are also morphism and function. What exactly are they? Because when I read them, they seem to have same definition.

2. Function.

   • There are two definitions for "function" this one (transformation) and this one (function). So what exactly function?

3. Transformation. (main question)

   • Linear transformation, unitary transformation, orthogonal transformation, geometry transformation and coordinate transformation. Is there any transformation other than this? Which is the most general transformation? which on is a special case of some transformation? What is really definition of transformation?

Answer 1

To make it a bit easier try to think about functions as a Cartesian product between two sets. So

$$ f:A\to B $$

is a function or, mapping from a set $A$ to a set $B$, we can express this as

$$ f=\{(a,b)\mid a\in A, b\in B \} \subseteq A\times B $$

so the elements of $f$ are ordered pairs of elements $(a,b)$.

A morphism is a strucure retaining mapping, which means that some structural property in set $A$ carries over to $B$ in some form. To say an example let $\phi$ be a function mapping elements of $A$ to elemnts of $B$ that is

$$ \phi\subseteq A\times B $$

moreover

$$ \phi(a_1\circ a_2)=\phi(a_1)\circ\phi(a_2), \quad \forall a_1,a_2\in A $$

so the operation "$\circ$" carries through. It doesn't matter wether you use it before or after applying $\phi$.

As for the last question, there are many different types of transformations. A transformation is also a function that is a cartesian product between two sets. That is the most general form of a transformation.

Hope this helped a bit