vector space, subspace, span

veege

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, ... , 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?

Dick

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

Dustinsfl

Start with multiplication.

Span W = c*a*w₁+...+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.

veege

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 + ... + c_n*w_n

T = k₁*w₁ + k₂*w₂ + ... + 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.