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: The converse statement is even more interesting: sometimes it is possible to transform dependent random variables into independent ones. For example, let $X$ and $Y$ be jointly normal, zero-mean, unit variance, with correlation coefficient $\rho$. Define $V=X+Y$ and $W=X-Y$. $V$ and $W$, are also jointly normal and zero-mean. As for correlation:
$$E\{VW\} = E\{X^2-Y^2\}=0$$
It is a simple and quite practical (in communication systems for instance) example of how a rotation can make it easier to deal with independent random variables instead of dealing with dependencies. Once you are done with the analysis, they can easily be reverted to the original variables. You can find many other examples like this, many of them with dramatic impacts. For instance, the main reason that the 4G cellular access links are way faster than 3G relies on a rather simple linear transform of signals, known as OFDMA. As fancy and complicated they might seem, the math behind them is quite simple: rotate the compound signals coming from hundreds of users so that they become independent, so you could deal with them separately. I am sure there are many abstract and practical examples that others can mention here. The bottom line is: statistical/probabilistic independence is subject to the representation of the phenomena and it is not necessarily inherent characteristics of the phenomena.
A: If you are asking about taking linear transformations of each variable separately, then they will remain independent- see here. If you are asking about linear transformation of both variables together, then the transformed variables may or may not be independent.
