annihilator
The annihilator of a subspace U of V, written U^0, is the set of functionals that kill everything in U: all f in V* with f(u) = 0 for every u in U. It lives in the dual space V*, not in V, which is the first thing to keep straight.
Think of it as the dual notion of 'orthogonal complement', but without needing an inner product. An ordinary orthogonal complement uses angles; the annihilator only uses the bare pairing f(u) = 0. That makes it more primitive and available in any vector space whatsoever.
The dimension count is beautifully clean: dim U + dim U^0 = dim V. The bigger the subspace you must vanish on, the fewer functionals survive, and the two dimensions always add up to the whole. This is the duality analogue of the rank-nullity theorem, and it is the engine behind 'row rank equals column rank'.
Annihilators reverse inclusions: if U is inside W, then W^0 is inside U^0. Together with the double-annihilator theorem (taking the annihilator twice, inside the canonical V = (V*)*, returns U), this sets up an order-reversing dictionary between subspaces of V and subspaces of V*.
Vanishing on a bigger subspace leaves fewer surviving functionals; the dimensions complement.
Do not confuse U^0 (lives in V*, no inner product needed) with the orthogonal complement U-perp (lives in V, needs an inner product). With an inner product they correspond under v -> <v, ->, but the annihilator is the more fundamental object.