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Vector space, subspace, span 
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#1
Mar1110, 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 = {w_{1}, w_{2}, ... , w_{n}} 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? 


#2
Mar1110, 09:25 PM

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Thanks
P: 25,235

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



#3
Mar1110, 09:30 PM

P: 629

Start with multiplication.
Span W = c*a*w_{1}+...+c*a_{n}*w_{n} Does this exist in V? For addition, add Span W to Span R or whatever you want to call it. 


#4
Mar1110, 09:52 PM

P: 4

Vector space, subspace, span
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 = c_{1}*w_{1} + c_{2}*w_{2} + ... + c_{n}*w_{n} T = k_{1}*w_{1} + k_{2}*w_{2} + ... + k_{n}*w_{n} 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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