The annihilator provides a precise duality between subspaces of and subspaces of .
Definition 7.0.1 (Annihilator). For a subspace , the annihilator of is
Theorem 7.0.1 (Dimension of the annihilator). If is finite-dimensional and is a subspace, then
Proof. The restriction map defined by is a linear map with . We claim is surjective: given , extend a basis of to a basis of , define to agree with on and be zero on the complementary basis vectors. Then .
By rank-nullity: Since and , we get .∎
Theorem 7.0.2 (Annihilator isomorphism). For a subspace , there is a natural isomorphism
Proof. Define by .
Well-defined: If , then , so since .
Injective: If , then for all , so .
Surjective: Given , define where is the quotient map. Then and .∎