How do you determine if a given set spans a vector space?

Question

For any given set, what am I suppose to look for. ie, if the set satisfies a subspace of V? Linear combinations of the set are in the vector space? or subspace?

Answer 1

You look to see if every vector in the vector space can be written as a linear combination of vectors in the set. Quick method for finite-dimensional spaces - if your space is n-dimensional and your set contains a subset of n vectors which are all linearly independent from each other, the set spans the entire vector space (and the aforementioned subset is a basis for that space).

Answer 2

Your question is a bit confused. If you have a set of vectors S in a vector space V, then S *does* span some subspace of V. That subspace is the collection of all linear combinations of vectors out of S.

There are two natural questions:

1) Is there a smaller collection of vectors from S that spans the same subspace?

The way to determine this is to figure out whether any of the elements from S can be written as linear combinations of *other* leements of S. If one can, throw it out. Then ask whether any of the remaining vectors can be written as linear combinations of other vectors from this smaller set. If so, throw one such out. Keep repeating until all the vectors are linearly independent. This can often be done by inspection.

There are other ways to do this same thing using matrix calculations, but they aren't as easy to explain in this forum.

2) Does S span all of V? In other words is the subspace spanned by S the same as the vector space V?

In this, you have to decide whether *every* vector in V can be written as a linear combination of vectors from S. If V is n-dimensional, this is equivalent to there being n linearly independent members of S. Again, this can be decided using matrix methods if necessary, but often it can also be done by inspection.