Linear algebra

Question

Given a vector space V and a subset of that vector space S, if you squint your eyes and only look at those vectors in
S, then S is a subspace of V if S itself is a vector space. If you add two vectors in S and they are no longer in S, then
S is not a subspace, and if you multiply any vector in S by any scalar and the result is not in S, then S is not a
subspace

Answer 1

You have to be a little bit careful about quantifiers:

"If there are any two vectors such that when you add them the result is not in S, then S is not a subspace" Notice I've added "there are". Another way to say this is by the contrapositive, "if S is a subspace then the sum of any two vectors in S is also in S"

 

"If THERE IS A VECTOR in S AND A scalar such that when you multiply them then the result is not in S, then S is not a subspace" Again, notice the all caps.

 

These two conditions can be succinctly stated as "S is closed under vector addition and scalar multiplication."

 

 

Finally, do you require that 0 be an element of the purported subspace? Or does it follow from one of your first two conditions? You'll want to be precise about whether you are talking about a vector space over the reals, the complex numbers, or an arbitrary field. Hope that helps.