---
tag: 0R6M158SQ9
kind: definition
name: Annihilator of a subspace
msc: [15A04]
defines: 0ASHNYF7AG
uses: [0M856Q1AMZ, 0EPTJA5QV4]
sources:
  - url: https://en.wikipedia.org/wiki/Dual_space
    locator: Quotient spaces and annihilators
---

## Definition

Let $F$ be a field, let $V$ be a vector space over $F$, and let $W$ be a [[subspace|00YC1D652K]] of $V$.
The *annihilator* of $W$ is the subset of the [[dual space|07JBSZE0V4]]
$$W^0 = \{f \in V^* : f(w) = 0 \text{ for all } w \in W\}.$$

## Remarks

The annihilator $W^0$ is a subspace of $V^*$.
If $W_1 \subseteq W_2$, then $W_2^0 \subseteq W_1^0$.
In finite dimensions, $\dim W + \dim W^0 = \dim V$, see [[the dimension of the annihilator|06Q1EFVH0Q]].
The annihilator of the line spanned by $(1, 1)$ in $F^2$ is spanned by the form $(x, y) \mapsto x - y$.

## Explanations

### L1 human

The annihilator of a subspace collects the linear forms that are zero on all of it.
In $\R^3$, the forms that vanish on a plane through the origin are the multiples of one form, whose kernel is that plane.
A larger subspace has a smaller annihilator.