Quantifying a variable in a set after specifying the set My linear algebra instructor claims that the expression for a set of vectors 
$$ B = \left\{ \vec v \in \mathbb{R^2} : \vec v = a \begin{bmatrix}2 \\ 1\end{bmatrix} \right\}, \text{ where } a \in \mathbb{R} \tag{1}$$
is incorrect, whereas
$$ B = \left\{ \vec v \in \mathbb{R^2} : \vec v = a \begin{bmatrix}2 \\ 1\end{bmatrix}, \text{ where } a \in \mathbb{R} \right\} \tag{2}$$
is correct. 
His reasoning is that the variable $a$ doesn't exist anymore once the set is closed, so there is nothing to be quantified, and the first expression is meaningless.
On the other hand, a definition in the textbook we are using indicates that an  expression such as 
$$ \text{Let } a \in \mathbb{R}. \text{ Then, } B = \left\{ \vec v \in \mathbb{R^2} : \vec v = a \begin{bmatrix}2 \\ 1\end{bmatrix}\right\} \tag{3}$$
is correct.
I've also learned in another class, which tackled formal logic a little more deeply than this linear algebra class, that quantifying a variable before or after a statement involving the variable is mostly an arbitrary/stylistic decision on the part of the writer. If we can quantify a variable before a statement, we can quantify it after a statement as well. According to this other class, the first statement might be okay, although I don't remember tackling sets specifically when working with quantifiers.
I know there is such a thing as quantifier scope in formal logic, but I'm not sure whether set brackets delineate quantifier scope in mathematical statements such as the above, as my linear algebra instructor seems to suggest.
Since ultimately a statement such as this should make sense when expressed in natural language, I think the first statement is one where the speaker mentions, after specifying the set, that, "by the way, the $a$ in the preceding statement (which happens to contain a set in this case) is a real number, in case that wasn't clear", which is similar to how a speaker might mention the domain of the variable prior to specifying the set, as in statement $(3)$.
I was wondering whether anyone can confirm or deny my instructor's claim, and whether the first statement is truly a mathematically meaningless statement. 
 A: *

*First, let's simplify $$ B = \left\{ \vec v \in \mathbb{R^2} : \vec
    v = a \begin{bmatrix}2 \\ 1\end{bmatrix}, \text{ where } a \in
    \mathbb{R} \right\} \tag{2}$$ as $$ B = \left\{a \begin{bmatrix}2 \\
    1\end{bmatrix}:a \in \mathbb{R} \right\} \tag{2s},$$ which is read
as “$B$ is the set of vectors of the form $a\begin{bmatrix}2 \\
1\end{bmatrix},$ where $a$ is real”.
Consequently, each element of
set $B,$ for some real $a,$ equals $a\begin{bmatrix}2 \\
1\end{bmatrix}.$


*On the other hand, $$ \text{Let } a \in \mathbb{R}. \text{ Then, } B
= \left\{ \vec v \in \mathbb{R^2} : \vec v = a \begin{bmatrix}2 \\ 1\end{bmatrix}\right\} \tag{3}$$ implies that for each real $a,$
each element of set $B$ lies in $\mathbb R^2$ and equals
$a\begin{bmatrix}2 \\ 1\end{bmatrix};$ that is,
$B=\left\{\begin{bmatrix}3 \\
1.5\end{bmatrix}\right\}=\left\{\begin{bmatrix}2 \\ 1\end{bmatrix}\right\}=\left\{\begin{bmatrix}6 \\
3\end{bmatrix}\right\}=\ldots;$ so, $B$ does not exist.


*Finally, $$ B = \left\{ \vec v \in \mathbb{R^2} : \vec v = a
\begin{bmatrix}2 \\ 1\end{bmatrix} \right\}, \text{ where } a \in
\mathbb{R} \tag{1}$$ means $B = \left\{ \vec v \in \mathbb{R^2} : \vec v = a \begin{bmatrix}2 \\ 1\end{bmatrix} \right\}$ for some $a \in \mathbb{R},$ in which case $B$ is a single-element set.
Notice that in the above, there are altogether three different specifications of set $B,$ one of which is an ill definition.
