Image of function definition notation

Question

In my Linear Algebra and Geometry textbook, it defines the image of a linear transformation $T$ as:

$$\operatorname{Im}\, (T) := \{\; w \in W : \; w=Tv \;\;\text{ for some } v \in V \} $$

As far as I can see, this is just the same as:

$$\operatorname{Im} \, (T) := \{ \;Tv \in W : \;v \in V\}$$

Is there any difference in these definitions?

If not, why is the first one used?

Answer 1

Each statement defines the image of the linear tranformation, they just give different ways of describing the SAME EXACT set: the image of $T$.

Each definition uses Set-Builder Notation: notation which allows us to describe any given set of elements in any number of ways, and/or from different perspectives or for different purposes. E.g.:

• Let $E_1 = \{\; 2k | k \in \mathbb{Z}\;\}$;
  Let $E_2 = \{\;n \in \mathbb{Z} \mid n\equiv 0 \pmod{2}\};$
  Let $E_3 = \{\;n \in \mathbb{Z} : 2\mid n\;\}$.
  
  Each of $E_1$, $E_2$, and $E_3$ each define the same set of even integers. There is only one set being defined; which definition one chooses depends on context.

Back to your two definitions:

1. The first definition states "The set of all elements $w \in W$ such that $w$ is the value of $Tv$ where $v$ in some element in the domain $V$."
2. The second defines the set of all function values $Tv$ that end up in $W$ after $v \in V$ is transformed by $T$".

If both define the same set, as they do, then why use the first?

The first is often used to establish, e.g., surjectivity of a function $f: V\to W$. If $f$ is onto, then for every $w \in W$, there exists a $v\in V$ such that $f(v) = w.$ So it's not uncommon to define the image of a function as it is defined in the first case.

Answer 2

Both definitions are the same as you point out, they just give slightly different ways of thinking about the image.

The first definition views the image as "Things in $W$ reached by the function". The second takes a more constructive approach, it is saying that we can build the image by taking each element of $V$ and put the result after applying $T$ to it into the image.

Answer 3

In certain set-theoretic senses, the first one $$ \{\; w \in W : \; w=Tv \;\;\text{ for some } v \in V \} $$ is exactly what you get by the Axiom of Separation, while the second one $$ \{ \;Tv \in W : \;v \in V\} $$ is taken as a short-hand way of writing it. So perhaps the author thought the first one would be less confusing to some students.