# Subset span proof help

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?

Homework Helper
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

Homework Helper
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...?