Why should the image space be defined as “for some” but not “for all” input vectors? I read the following regarding the image space of a linear transformation $T$, denoted as $Im(T)$, in Chapter 6.1 of No Bullshit Guide on Linear Algebra by Ivan Savov that:
$$
Im(T) = \{w\in W | w=T(v) \color{red}{\text{ for some }} v \in V \}
$$
where,
$W$: the output space
$V$: the input space
$T$: the linear transformation from $V$ to $W$
But shouldn’t it be “for all” instead? I thought the image space should be the set of output vectors from a linear transformation $T$ mapped from $\color{red}{all}$ input vectors in the input vector space $V$.
 A: Well for $f:V\rightarrow W$ one way to write $\operatorname{Img} f$ is $f(V):=\{f(v)\vert v\in V\}$. This is the subset of $W$ of all the points that are the image of another point in $V$.
Now take some point $w \in W$. We have $w \in \operatorname{Img} f$ if and only if $w$ is hit by $f$. That is, there is some $v\in V$ such that $f(v)=w$. If you would require that $f(v)=w$ for all $v\in V$ then $f$ would just be the constant function taking the single value $w$ on all inputs $v\in V$.
So $f(V)$ is the set consisting of the images of all points in $V$ under $f$ or equivalently the set of all points in $W$ that are hit as the image of some point in $V$ under $f$. (This is $f(V)=\{f(v)\vert v\in V\}$ versus $\{w\in W\vert \exists v\in V:f(v)=w\}$). As a first introduction I find $f(V)$ much more intuitive, personally.
You hopefully now see that there are two different ways of characterizing $\operatorname{Img} f$ in words, where one uses "some" while the other one uses "all" ;)
A: The clarifications from others do help me know what I have misunderstood.  I interpreted (incorrectly) $\{ w=T(v) \text{ for some } v \in V\}$ in a computer science way as follows:
#List comprehension in Python
Im_T = [ T(v) for v in V[:10] ]

where V[:10] is to slice V to get the first 10 elements ,that is what I think (incorrectly) for some means - a subset of V.
And similarly, I interpreted (incorrectly) $\{ w=T(v) \text{ for all } v \in V\}$ as follows:
#List comprehension in Python
Im_T = [ T(v) for v in V ]

where V has not been sliced and is the entire V, that is what I think (incorrectly) for all is.
From what I have learnt from others, $\{ w=T(v) \text{ for all } v \in V\}$ should actually mean something like:
$$
w=T(v_1)=T(v_2)=...=T(v_n)
$$
where $v_i \in V$ for $i=1,...,n$.
Hope this can help those with a CS background.
