Final answer:
H being a subspace of R3 means H is a vector space contained within R3, with possible forms like a line or plane through the origin. It is neither a scalar multiple nor necessarily perpendicular to R3, but it could form from the intersection with another vector space given it upholds vector space properties.
Step-by-step explanation:
When it is said that H is a subspace of R3, it means that H is a vector space contained within R3. This implies that H itself must satisfy the properties of a vector space, which include being closed under vector addition and scalar multiplication, containing the zero vector, and having additive inverses for all of its elements. As a subspace, H can be a line or a plane that passes through the origin within the three-dimensional space defined by R3.
A subspace is not a scalar multiple of R3, as a scalar multiple of R3 would still span the entire three-dimensional space, not a subset of it. It is not necessarily perpendicular to R3, as only certain subspaces, like planes or lines within R3, may exhibit perpendicularity to other subspaces in R3.
As for being an intersection with another vector space, it could be the result of such an intersection, but only if that intersection satisfies all the properties of a vector space.