A transformation for independent variables to be dependent I ran into a confusing question.

If two variables are independent, maybe they will be dependent after linear transformation.

How it can happen? Is it possible for independent variables?
What is the operation that it makes variables to be dependent?

In my opinion the transformation that maps all variables to a point is true for this fact.

What is wrong with my answer?
The mean of Variables are features of data.
 A: It depends on what they mean by linear transformation. If the linear transform is a unary function, then by applying to each independent variable you end up with two new independent variables. If the linear transform is a binary function, i.e., it is a linear combination of both independent variables, then you will have created a new dependent variable.
For both cases your answer isn't correct because two variables that have the same constant value are still independent.
A: No, this cannot happen.

In my opinion the transformation that maps all variables to a point is true for this fact.


What is wrong with my answer?

Don't take this the wrong way, but most teachers to courses in this topic would probably first note that your opinion has nothing to do with it.
You should start with the definition of independence. There are multiple ways to do it, but most likely you work with a definition like,
$X$ and $Y$ are independent if for any $a$ and $b$, $P(X < a)\cdot P(Y < b) == P(X < a \text{ and } Y < b)$.
Now let's say you know $X$ and $Y$ are independent, but you want to prove that $c X  + d$ is independent of $eY + f$. Those are the general forms of a linear transformation. Then, given an arbitrary $a$ and $b$, how can you show that the above property also holds for these transformations?
Edit
Actually, to answer your question more directly instead of starting with a proof, if you map everything on a point $X$ and $Y$ are still independent.
Let's say you map both $X$ and $Y$ to a single point $c$. Then $P(X < a)$ is either zero or one, depending on if $c < a$ or not. The same goes for $P(Y < b)$, and if you plug those numbers into the definition, you'll see that the equation holds true for all $a$ and $b$, and therefore, these transformations of $X$ and $Y$ are independent.
This is a special case of the general situation, where, using the numbers above, $c = 0$ in $cX + d$ and $e = 0$.
