For a subspace S of a finite-dimensional vector space V over a field F, the annihilator S^0 is the subspace of the dual V* consisting of all linear functionals φ with φ(s) = 0 for every s ∈ S.

Principle

The annihilator is the orthogonal complement of S inside the dual space V*, capturing all linear constraints that vanish on S and producing an exact sequence relating dimensions of S and S^0.

Demonstration

If V = R^3 with basis e1,e2,e3 and S = span{e1,e2}, then S^0 = {φ ∈ (R^3)* : φ(e1)=φ(e2)=0} is one-dimensional, spanned by the functional picking the e3-coordinate.

Misapplication

Confusing the annihilator S^0 ⊂ V* with the orthogonal complement S^⊥ ⊂ V in an inner-product space; while related via identification V ≅ V*, equality holds only after choosing an inner product.

Consequence

In finite dimensions dim S + dim S^0 = dim V; annihilators classify subspaces via their vanishing linear forms and are used to describe kernels of induced quotient maps and dual exact sequences.

Reversal

The kernel of a set of linear functionals A ⊂ V* is the subspace of V annihilated by A; this inverts the annihilator construction by producing a primal subspace from a set of dual constraints.

Boundary

Definition uses the algebraic dual V*; in infinite-dimensional contexts the algebraic dual is large and topological duals are often preferable; annihilator properties depend on whether dual means algebraic or continuous dual.

Semantic Tension

Annihilator versus orthogonal complement: the annihilator lives in the dual and is coordinate-free; the orthogonal complement is defined by an inner product in V and coincides with the annihilator only after an inner-product identification.

Synthesis

The annihilator of S is the subspace of linear functionals vanishing on S; it encodes constraints orthogonal to S in the dual, satisfies dim S + dim S^0 = dim V in finite dimensions, and inverts under passage from subsets of V* back to kernels in V.