Thread: Vector Spaces

Let V be a vector space over a field F. W is a subspace of V if W is a subset of V and W is also a vector space over F.
Prove that if W is a subset of V such that:
1. for all a and b in W a+b in W and
2. for all k in F and for all a in W, ka in W.
Then W is a subspace of V.

Originally Posted by Coda202

Let V be a vector space over a field F. W is a subspace of V if W is a subset of V and W is also a vector space over F.
Prove that if W is a subset of V such that:
1. for all a and b in W a+b in W and
2. for all k in F and for all a in W, ka in W.
Then W is a subspace of V.

Basically all the other properties for subspace hold already for W since it is a subset of V. For example, + is associate since a+(b+c) = (a+b)+c already holds in V. Just show that all the properties of a vector subspace holds for W by using the fact V is already a vector space.