How do I prove this is a subspace ?

Question

Prove that a subset W of a vector space V is a subspace of V if and only if W ≠empty and whenever a∈ F and x,y ∈W then ax∈W and x+y∈W

Answer 1

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given:   W is a subset AND a subspace of vector space V.

          Then by theorem, W is not empty and is closed under addition and scalar multiplication.

 

 

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  given:  W is a non empty subset of V equipped and endowed with the same operations as V;

             W is closed under addition and scalar multiplication.

 

  Then by the same theorem, W is a subspace.

 

http://linear.ups.edu/html/section-S.html

 

Note that since we are given the operations are closed on W, we do not

need to go through the rigorous proof of showing the operations are closed

on W. Otherwise we would have to let vector x and vector y be elements 

of W and show that x+y is in W. Likewise, for scalar k, kx is also in W.

However, this was nicely given. The proof is worded identically to

the theorem.