Question

Is the Span of a given space the base for that space? Or what does W=Span((1,0,1),(2,3,1)) means?

Answer 1

"THE" Basis? Better to talk about "A" Basis.

Read the definition closely. Isn't SPAN(list of vectors) a way of indicating a basis?

Answer 2

I don't know this term was introduced in my linear algebra book and I cannot find any previous reference, but you are right I meant "A" basis.

Answer 3

We generally start with this:

1) Here is a vector space.
2) Let's see if we can find a Basis for it,

SPAN is backwards.

1) Here is a Basis.
2) Let's investigate the Vector Space it represents.

Same idea.

Answer 4

the span of a set of vectors is just all the possible linear combinations of those vectors.

Answer 5

..which may or may not be a Basis for a vector space that interests you.