# vector space, subspace, span

by veege
Tags: linear algebra, vector spaces, vector subspace
 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?
 HW Helper Sci Advisor Thanks P: 24,460 Maybe start by reviewing what Span{W} means. Quote the definition in your next post, ok?
 P: 588 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.
P: 4

## vector space, subspace, span

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

 Quote by Dustinsfl 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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