Subspaces--Linear Algebra

dude15129 · #1 $Old$ March 2nd, 2009, 05:52 PM

I am trying to work on these proofs and I am having a hard time with this one.

Let V be a vector space and S={v1, v2, v3,…vk} be a set of vectors in V. Prove that span S is a subspace of V.

I am not really sure where to even start. I know what span and I know that in order to prove something is a subspace of something else you must show u+v is an element of W and c•u is an element of W. I just don't know what to do for this one.

TheEmptySet · #2 $Old$ March 2nd, 2009, 06:16 PM

Quote:

Originally Posted by dude15129 $View Post$

I am trying to work on these proofs and I am having a hard time with this one.

Let V be a vector space and S={v1, v2, v3,…vk} be a set of vectors in V. Prove that span S is a subspace of V.

I am not really sure where to even start. I know what span and I know that in order to prove something is a subspace of something else you must show u+v is an element of W and c•u is an element of W. I just don't know what to do for this one.

You have said excatly what you need to do.

First show it is non empty.

$\vec 0 \in span(S)$ becuase
$\vec 0= 0v_1+0v_2+... + 0v_k$

done

Now we need to show it is closed. So let $\vec a, \vec b \in span(S)$

Then $a=a_1v_1+...a_nv_n$ and
$b=b_1v_1+...+b_nv_n$
Then
$a+b=(a_1v_1+...a_nv_n)+(b_1v_1+...+b_nv_n)=(a_1+b_1)v_1+...+(a_n+b_n)v_n \in span(S)$

I think you can show the last one. Good luck