Subset Span Proof: Proving W is a Subspace of V with Span(W) = W

[zmdeez]

show that a subset W of vector space V is a subspace of V iff span(W)=W

Can anyone help guide me along in this proof?

 

[matt grime]

Which of the directions of implication can you show, if any? The result follows from the definitions of all the words involved directly.

 

[zmdeez]

i have sketched a couple ideas but i don't think they are legit...so i guess i can't show either direction, I am sure one is far simpler than the other i just can't get a good start

 

[matt grime]

What is (your) definition of a vector subspace? Mine is that W is a subspace if W is a subset of V and for all x and y in W and s and t in R (or whatever the underlying field is, perhaps C) the sx+ty is in W and 0 is in W. (note this is redundant by settinf s=t=0)
What is the span of a subset? iti s the set of all combinations

t_1x_1+\ldots t_nx_n

for t_i in R (or the underlying field) and x_i in W
so W=span(W) means exactly that all finite combinations of elements of W are in W.

so we are trying to show that

W closed under combining (adding up) two elements of itself if and only if W is closed under combining a finite number of elements of itself.

Obvioulsy one way is simple: if i can add up any number of combinations of elements then i can in particular add up two of them. Conversely...?