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:
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:
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 ?