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**Deveno** a non-empty subset W of V is a subspace if and only if for all w1,w2 in W, and a in F, w1+w2 and aw1 are in W.

these properties are called "closure of vector addition and scalar multiplication" and are enough to ensure that W is a

full-fledged vector space in its own right (using the same vector addition and scalar multiplication as in V).

so if {v1,v2,...,vr} are all in W, then so are the scalar multiples a1v1, a2v2,....arvr and thus also the sum:

a1v1 + a2v2 +....+ arvr, because of the closure properties listed above (properties any subspace of V has.

think about this: if W is to be a subspace, then it has to satisfy all of the vector space axioms, or else it

wouldn't be a subspace, just a subset).