Is this what you mean?

The vector <1,-2,5> is the result of<3,4,0>+<-1,-6,2>+<-1,0,3>.

BUT ALSO, <1,-2,5>=<1,0,0>+<0,-2,0>+<0,0,5>.

Any given vector in R^3 is the linear sum of infinitely many three-element sets of vectors. So I think the answer to your question is no.

If you mean something else, please explain.