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The zero vector must be in the set to ensure the set is nonempty. A subspace is a vector space in its own right, and every vector space must contain an additive identity, which is the zero vector.

Conditions: The set is a candidate for being a subspace of a vector space.; The definition of a subspace requires it to be a vector space under the inherited operations.

The zero vector must be in the set to ensure the set is nonempty. A subspace is a vector space in its own right, and every vector space must contain an additive identity, which is the zero vector.

Conditions: The set is a candidate for being a subspace of a vector space.; The definition of a subspace requires it to be a vector space under the inherited operations.