What is the meaning of the notation of function

Question

More specifically, what is meant by the function $T: \mathbf V \to \mathbf W$? I saw it in the discussion of linear map in Axler's Linear Algebra Done Right but could not understand this notation. Does it mean something like 'take an object $a \in V$ then process it into an object $b \in W$? Could somebody please explain in detail to me?

Answer 1

Usually, when you write $T : V \to W$ in linear algebra to denote a linear transformation between vector spaces, this notation gives you three very important pieces of information.

1. It tells you the name of the map. In this case, the name is $T$. So when we talk about, for example, $T(v)$, where $v$ is a vector in the domain of the map, you know which map we are talking about.

2. It tells you the domain of the transformation $T$, i.e., the set of inputs. In this case, the domain is the vector space $V$.

3. It tells you the codomain of linear transformation $T$, i.e., the set/space to which the outputs belong. In this case, the codomain is the vector space $W$.

Note that the codomain is different in general than the range. This is because the range of any map is actually by definition equal to the set of outputs. But the codomain can be larger than this set. The codomain is the structure where the range lives. If the codomain equals the range (i.e., every element in the codomain has an element in the domain being mapped to it), then the transformation is said to be onto.

Answer 2

It means at the most fundamental level that $T$ is a set of ordered pairs $(v, w)$ where $v$ is an element of $V$ and $w$ of $W$, which has the additional constraint that there is only one pair with any given $v$: ie. it is one-to-one. It can be more useful to think about it in the sense of $T$ being an action that you do to v in order to produce $w$, but in terms of set theory, that's what the notation represents.