I don't understand the question.A set of vectors "spans" the subspace of all linear combinations of vectors in the set. {(2,1,03), (3,-1,2), (-1,0,2,1)} is a set of vectors, not a subspace so it makes no sense to ask if any set of vectors "spans" it. You could ask if a given set of vectors spans thesamesubspace that {(2,1,03), (3,-1,2), (-1,0,2,1)}, but then the answer is trivally no. Any "linear combination" of the single vector (0,0,0,0) is just (0,0,0,0) itself.

Again that makes no sense. Do you mean to say that for the zero vector to be expressed as a combination of these vectors (the reverse of what you said above), the coefficients must be 0? That would be saying that these vectors arei tried to use Gaussian elimination and i found that the the coefficients for these 3 vectors to be expressed as a combination of the zero vector have to be all 0.independent.

Once again, the zero vector does not span anything except itself. If you are asking if the span of this set of vectors includes the zero vector, the answer it trivially "yes". Just take the coefficients all equal to 0. That's a linear combination.this should mean that there exist no linear combi of these 3 vectors that will give the zero vector and hence the zero vector does not span {( 2,1,0,3), (3,-1,5,2), ( -1, 0 2,1)} right?

Please write the problembut my answer states that it spans {( 2,1,0,3), (3,-1,5,2), ( -1, 0 2,1)}....

did i do sth wrongly?exactlyas it is given.