Question

if you are testing a subset to see if it is a subspace of the vector space, does the scalar have to be part of the subset?

Answer 1

i am not sure this makes sense. a vector space comes with scalars, they are the same scalars for the subset of the vector space

Answer 2

usually you have a vector space over some field, like say the rational numbers or the real numbers or the complex numbers. it will be the same for the subset. so see if it is a subSPACE you have to check the usual vector space axioms
contains the zero vector
is closed
etc