Subspaces and Spans

Subspaces and Spans?

Ok, I'm trying to teach myself linear algebra and having trouble understanding some the concepts...

If V is a subspace of, say R^3, then thats a collection of all vectors in some plane or some line going through the origin, correct? Then if we take the span of V, which a set of all linear combinations of the vectors in V, wouldn't that just give us the same plane or line?

Other Answers:

A one dimensional subspace of R^3 is spanned by a single, nonzero vector eminating from the origin. All multiples of this vector make up that subspace. A two dimensinal subspace is spanned by two linearly independent vectors (i.e. one is not a multiple of the other) eminating from the origin. All linear combinations of these vectors make up the subspace (in this case a plane through the origin). R^3 itself is spanned by three linearly independent (i.e. none of the three is a linear combination of the others) vectors. It cosists of all linear combinations of these three vectors. The only other subspace of R^3 consists of the 0 vector only (the null space). Spaces and subspaces are spanned by vectors. If the spanninjg vectors are linearly independent, they form a basis of the space.
I hope this helps.