I reformulate and detail just in case :

and

.

. I noticed that

.

I must find a vector

such that

.

My main question is : say I found a vector

and I want to check if it does span

along with

.

Is it enough to check that both

and

are linear combination of

and

? My intuition says yes because if

and

can be written as a comb. linear of

and

, so does any vector spanned by

and

. And I also must check that

is a comb. linear of

and

. If it is then I can conclude that

.

Is there a way to find

, other than having a mathematical eye?