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?