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Annihilators: Subspaces Seen From the Dual Side

Given a subspace, which measurements vanish on all of it? That collection — the annihilator — translates subspace geometry into the dual space, with a clean dimension law.

The annihilator of a subspace

Let W be a subspace of V. Its annihilator, written W^0, is the set of all functionals that kill every vector of W: W^0 = { f in V* : f(w) = 0 for all w in W }. It is itself a subspace — of V*, not of V. Think of it as 'all the measurements that can't see W'.

The dimension law

Here is the clean theorem: dim W + dim W^0 = dim V. The bigger your subspace, the fewer functionals can blindly annihilate it. If W is a line in R^3 (dim 1), its annihilator is a plane of functionals (dim 2). This is the annihilator duality theorem, and it is the dual-space twin of the rank-nullity theorem.

  1. Pick a basis w1, ..., wk of W and extend it to a basis w1, ..., wk, u1, ..., um of all of V.
  2. Take the dual basis. The functionals dual to the u's vanish on every w — so they lie in W^0.
  3. Those m functionals are a basis of W^0, so dim W^0 = m = dim V - dim W.

Worked example and a glimpse ahead

V = R^3.  W = span{ (1,2,0) }  (a line, dim 1).
W^0 = { rows [a b c] : a*1 + b*2 + c*0 = 0 } = { a + 2b = 0 }.

Solve: a = -2b, c free.  Basis of W^0:
   f1 = [-2  1  0]   (b=1, c=0)
   f2 = [ 0  0  1]   (b=0, c=1)

dim W^0 = 2,  dim W = 1,  sum = 3 = dim V.  ✓
Geometrically: f1=0 and f2=0 are two planes whose
intersection is exactly the original line W.
A 1-D line is cut out as the common zero set of a 2-D space of functionals.

Two related facts you'll meet soon: a single nonzero functional cuts out a hyperplane (a separating functional that puts a subspace on one side), and the annihilator W^0 is naturally the dual of the quotient V/W. Annihilators are the dictionary that translates 'subspace of V' into 'subspace of V*'.