Vector expressed in function of two other vectors.

Hi!

This is for a homework I had to submit yesterday. I think I managed to answer all the questions correctly, except the last one.

I had the following U set and v vector:

I had to prove that U is a subspace for V³, which I did.

Then I found:

Then I was asked to find a base for this set, and I found:

but I could also just have used the U specified above as a base. (I think)

I was then asked to find a base for , I got:

and:

found previously also belongs to .

Finally, I was asked to say if belonged to U, and if not to express in function of and :

I plugged the values of v in and it gave me 21, which is not equal to 0, so I said it wasn't in U. That's when my problems started, I was never able to express in function of and !

I tried:

Which gave me 3 equations and I tried to solve for m and n, but I wasn't able to. Now I know I have two orthogonal vectors (so they form a plane) and I know I need to use them to describe a third vector, so this third vector HAS to be on that plane too (that's why it doesn't work above). The problem I'm stuck with is that I don't know how to find and that also form a plane in which is!

Thanks in advance for helping me out.