In terms of vector spaces, what property does a subspace must fulfill?

A subspace of a vector space must fulfill several specific properties to qualify as a subspace. First and foremost, it must contain the zero vector. The zero vector is essential because it ensures that the subspace includes a point of reference; it also maintains closure under scalar multiplication, as any vector multiplied by zero will yield the zero vector itself.

Secondly, a subspace must include all linear combinations of its vectors. This means that if you take any vectors from the subspace and combine them through addition and scalar multiplication, the resulting vectors must still lie within the subspace. This property establishes that the subspace reflects the vector space's structure where linear combinations are essential.

Lastly, a subspace must be closed under both addition and scalar multiplication. Closure under addition means that if you take any two vectors within the subspace, their sum should also fall within that subspace. Closure under scalar multiplication means that if you take any vector from the subspace and multiply it by any scalar, the resulting vector must remain within the subspace.

Since all of these properties are collectively necessary for a subset of a vector space to be considered a subspace, the correct answer is that all of the listed requirements must be satisfied.