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A subspace of a vector space V is a subset W of V that is also a vector space under the same operations of addition and scalar multiplication as V. In other words, W is a subspace if it is closed under addition and scalar multiplication, and it contains the zero vector.

The additive identity is an element that, when added to any other element in the vector space, doesn't change that other element. In the context of vector spaces, the additive identity is the zero vector.

The additive inverse of an element in the vector space V is an element that, when added to the original element, results in the additive identity (which is the zero vector in the case of vector spaces). In other words, for a vector v ∈ V, its additive inverse is -v, such that v + (-v) = 0.

To find the additive identity of subspaces of V, we need to determine the subspace W that, when added to any other subspace, doesn't change that other subspace. Let's consider the subspace W = {0}, which contains only the zero vector. The sum of this subspace with any other subspace U is defined as the set of all vectors that can be written as the sum of a vector from U and a vector from W. Since W only contains the zero vector, adding any vector u ∈ U to the zero vector results in the same vector u, i.e., u + 0 = u. So the sum of subspaces U and W would still be U. Thus, the additive identity of the operation of addition on subspaces of V is the subspace containing only the zero vector, denoted {0}.

To find which subspaces of V have additive inverses, we need to find subspaces such that when added to a given subspace U, they result in the additive identity, {0}. Let's consider a subspace U of V, and let W be the additive inverse of U, so the sum U + W = {0}. For any vector u ∈ U, there must be a vector w ∈ W such that u + w = 0. But since U is a subspace, it must contain the zero vector, so u + (-u) = 0. This implies that -u ∈ U, and thus, U itself contains all the additive inverses of its vectors. Therefore, the only subspace that has an additive inverse is the subspace {0}. Adding {0} to itself results in {0}, and there is no other subspace for which the sum of itself and another subspace would result in {0}. To conclude, the additive identity of the operation of addition on subspaces of V is the subspace {0}, and the only subspace having an additive inverse is also {0}.