What happens to the axioms when a subspace inherits the vector space structure from a larger vector space?

When a subspace inherits the vector space structure from a larger vector space, it automatically satisfies the axioms of a vector space. This is because a subspace is defined specifically as a subset of a vector space that is closed under addition and scalar multiplication while also containing the zero vector. Therefore, since the operations of vector addition and scalar multiplication of the larger vector space are applicable to the subspace, all the properties that govern a vector space—such as associativity, commutativity of addition, existence of additive identity and inverses, and compatibility of scalar multiplication with field multiplication—are preserved.

Consequently, the axioms do not need to be proven separately for the subspace; they are inherently satisfied due to its relationship and shared structure with the larger vector space. This relationship ensures that the subspace retains the necessary properties to function as a vector space in its own right, reflecting the elegance of linear algebra where such structures are intricately connected.