Recipe

Find a basis for an annihilator W^{\circ}

Use when: you're given a spanning set for a subspace W and need an explicit basis for its annihilator W^{\circ}

Goal: a set of explicit functionals that both annihilate W and span all of W^{\circ}

Taught in Annihilators

1. 1

   Write \varphi(x,y)=ax+by in \mathbb R^2 (one coefficient per coordinate), with every coefficient still unknown.

   Example

   For W=\operatorname{span}\{(2,1)\} in \mathbb R^2, write \varphi(x,y)=ax+by.

   Why: an annihilator functional is a linear combination of coordinates just like any other functional, so the unknowns have to be named before they can be solved for.

2. 2

   Substitute each spanning vector of W into \varphi's general form and set the result equal to zero, one equation per spanning vector.

   Example

   \varphi(2,1)=2a+b=0, the single equation coming from W's one spanning vector.

   Why: vanishing on a spanning set forces vanishing on all of W, since every vector in W is a combination of that spanning set.

3. 3

   Solve for the dependent coefficients in terms of whichever ones are left free; the number of free coefficients equals \dim(W^{\circ}).

   Example

   From 2a+b=0, b=-2a, leaving a free. So every \varphi\in W^{\circ} has the form \varphi(x,y)=a(x-2y).

   Why: a homogeneous system's free coefficients directly count how many independent functionals span the solution set.

4. 4

   For each free coefficient in turn, set it to 1 and every other free coefficient to 0, then substitute back to get one basis functional.

   Example

   Setting a=1 above gives the single basis functional \psi(x,y)=x-2y.

   • exactly one free coefficient — W^{\circ} is one-dimensional, spanned by the single functional that comes from setting it to 1.
   • more than one free coefficient, such as one vector spanning W inside \mathbb R^3 — build one basis functional per free coefficient, each by isolating a different one at a time.

   Why: setting exactly one free coefficient to 1 at a time is what turns an infinite family of solutions into a finite, genuinely independent basis.