What crucial element must 'S' contain for it to be a subspace?

For a set ( S ) to be classified as a subspace of a vector space ( V ), it must satisfy three conditions:

1. The zero vector of ( V ) must be in ( S ).
2. ( S ) must be closed under addition, meaning if two vectors in ( S ) are added together, the result must also be in ( S ).

3. ( S ) must be closed under scalar multiplication, meaning if a vector in ( S ) is multiplied by a scalar, the result must also be in ( S ).

Among these crucial elements, the requirement for ( S ) to contain the zero vector is fundamental because the zero vector acts as an additive identity in the vector space. Without the zero vector, the closure properties required for a subspace cannot hold. For instance, if you take any vector ( v ) in ( S ) and consider ( 0 \cdot v ), this operation would need to yield the zero vector, which must also be in ( S ) for closure under scalar multiplication to be satisfied.

Those options that specify conditions not directly related to the foundational properties of subspaces, such as needing non-zero vectors, the scalar identity