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Vector space, subspace, span

  • Thread starter veege
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  • #1
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Homework Statement

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?
 

Answers and Replies

  • #2
Dick
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Maybe start by reviewing what Span{W} means. Quote the definition in your next post, ok?
 
  • #3
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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.
 
  • #4
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Maybe start by reviewing what Span{W} means. Quote the definition in your next post, ok?


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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