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Vector space, subspace, span

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veege
#1
Mar11-10, 09:01 PM
P: 4
The problem statement, all variables and given/known data

Suppose V is a vector space with operations + and * (under the usual operations) and W = {w1, w2, ... , wn} is a subset of V with n vectors. Show Span{W} is a subspace of V.



The attempt at a solution

I know that to show a set is a subspace, we need to show closure under addition and multiplication. I don't where to go from there. Any suggestions?
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Dick
#2
Mar11-10, 09:25 PM
Sci Advisor
HW Helper
Thanks
P: 25,228
Maybe start by reviewing what Span{W} means. Quote the definition in your next post, ok?
Dustinsfl
#3
Mar11-10, 09:30 PM
P: 629
Start with multiplication.

Span W = c*a*w1+...+c*an*wn

Does this exist in V?

For addition, add Span W to Span R or whatever you want to call it.

veege
#4
Mar11-10, 09:52 PM
P: 4
Vector space, subspace, span

Quote Quote by Dick View Post
Maybe start by reviewing what Span{W} means. Quote the definition in your next post, ok?


Quote Quote by Dustinsfl View Post
Start with multiplication.

Span W = c*a*w1+...+c*an*wn

Does this exist in V?

For addition, add Span W to Span R or whatever you want to call it.

The span is basically the set of all linear combinations of the vectors w1, w2, ... , wn. So then, I can define some vector S and some vector T in terms of w's:

S = c1*w1 + c2*w2 + ... + cn*wn

T = k1*w1 + k2*w2 + ... + kn*wn

I think I get it now. I can see how S + T will be closed, and some constant a*S will be closed.


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