1. ## Linear Algebra Question

Hi , I need clarification about something

Let's say V is a Vector space over the field F

A is a subspace of V, A={v1,v2,......,vn}

So my question is what is the difference between Span(A) and the subspace A? Isn't it kinda same thing? Span(A) is the linear combination of the vectors of A with the product of arbitrary scalars.

And subspace has axioms such as it is closed under addition and multiplication with scalar, so if there is a vector u that belongs to the Span(A) he belongs to the subspace A as well because A is closed under addition right?

So my questions is is it possible to have a vector u that belongs to Span(A) but not the subspace A={v1,v2,v3,.......,vn}?
I hope my question was clear, thanks!

2. ## Re: Linear Algebra Question

I forgot to ask one more question that I had on mind,
If u belongs to Span(A) does it mean that u is equal to vi when 1<i<k (including 1 and k) that Span(v1,v2,.....,vi,...vn)
or does it mean that with the linear combinations of vectors of A I can get the vector u?

In other words doest it mean u belongs to the set A or you can get u by the linear combinations of the vectors that belong to A

3. ## Re: Linear Algebra Question

If A is a subspace, it is spanned by any set of linearly independent vectors in A, or any set of vectors in A which contain the max number of linearly independent vectors; but that includes A itself, so I suppose you could say spanA = A, which I didn't see when I started this reply. Interesting. Thanks.

4. ## Re: Linear Algebra Question

Typically we talk about "span(A)" for some set of vectors, not, in general, a subspace. Span(A) can be defined as "the smallest vector space containing A". If A is itself a vector space, then it follows immediately that span(A)= A.

5. ## Re: Linear Algebra Question

By definition, given a finite set of vectors S, the span of S (spanS) is the vector space consisting of all linear combinations of the vectors of S.

Since A consists of an infinite set of vectors, spanA is undefined, contrary to my original speculation, ie, spanA is not A.

6. ## Re: Linear Algebra Question

But since S is closed under addition and multiplication with scalars, Span(S) has to be equal to S, isn't it ? Because linear combinations of the vectors of S are still in the subspace S

7. ## Re: Linear Algebra Question

S is a finite set of vectors. Span of S is an infinite set of vectors.

Let S = (0,1) and (1,0). Then span of S consists of all (x,y) and so is an infinite set.