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!