linearly dependent in Q, linearly independent in R

I need help with this one. :(

Find such that is linearly dependent in and linearly independent in .

Here's what I did:

Solving this gets me:

and

.

This gets me:

which means that either or .

The first case gives me , which means that the set is linearly independent both in and so that is obviously not the I'm looking for.

Now, the other case, . This means that . The three equations I had are now two equations:

and

.

Solving that gets me:

.

Now, if , I get that are not necessarily zero, but if , the equation implies that , and so and .

Obviously, is the one we're looking for.

Is this correct?

Is this the only possible ?

Thank you!