Chapter 29. Annihilators

An annihilator is a subspace of the dual space consisting of all linear functionals that vanish on a given set of vectors. If $V$ is a vector space over a field $F$, and $S\subseteq V$, then the annihilator of $S$ is

The elements of $S^0$ are linear measurements that give zero on every vector in $S$. When $S$ is a subspace, its annihilator records all linear equations that are satisfied by every vector in $S$. The annihilator is always a subspace of $V^*$.

29.1 Definition

Let $V$ be a vector space over $F$. Let $S\subseteq V$. The annihilator of $S$ is

The notation $S^0$ is common. Some books use $S^\circ$ or $\operatorname{Ann}(S)$.

If $U\subseteq V$ is a subspace, then $U^0$ is the set of all linear functionals on $V$ that vanish on $U$.

Thus

The annihilator lives in the dual space, not in the original vector space.

29.2 First Example

Let

and let

Then $U$ is the $x$-axis.

A linear functional on $\mathbb{R}^2$ has the form

For $\varphi$ to vanish on $U$, we need

for every $t\in\mathbb{R}$. But

This is zero for every $t$ exactly when

Therefore

So the annihilator is one-dimensional. It consists of all scalar multiples of the functional that reads the second coordinate.

29.3 Annihilators Are Subspaces

For every subset $S\subseteq V$, the annihilator $S^0$ is a subspace of $V^*$.

First, the zero functional belongs to $S^0$, because

for every $s\in S$.

Next, let

Then for every $s\in S$,

Thus

For a scalar $c$,

Thus

Therefore $S^0$ is a subspace of $V^*$.

29.4 Extreme Cases

There are two useful extreme cases.

First,

Every linear functional sends the zero vector to zero, so every functional vanishes on $\{0\}$.

Second,

The only linear functional that vanishes on every vector of $V$ is the zero functional.

These identities show the reversal built into annihilators: a smaller subset has a larger annihilator, and a larger subset has a smaller annihilator.

29.5 Inclusion Reversal

If

then

Indeed, if a functional vanishes on all of $T$, then it certainly vanishes on all of $S$. Thus it belongs to $S^0$.

This reversal is important. Annihilators turn containment around.

A large subspace imposes many conditions on a functional, so fewer functionals vanish on it. A small subspace imposes fewer conditions, so more functionals vanish on it.

29.6 Annihilator of a Span

The annihilator of a set equals the annihilator of its span:

If a functional vanishes on every vector of $S$, then by linearity it vanishes on every linear combination of vectors from $S$. Hence it vanishes on $\operatorname{span} S$.

Conversely, if it vanishes on $\operatorname{span} S$, then it vanishes on $S$, since

Therefore the annihilator depends only on the subspace generated by the set.

29.7 Computing an Annihilator in Coordinates

Let $U\subseteq \mathbb{R}^n$ be spanned by columns of a matrix

A functional $\varphi\in(\mathbb{R}^n)^*$ can be represented by a row vector

The condition $\varphi(u_i)=0$ for every $i$ becomes

Equivalently,

Transposing,

Thus the annihilator of the column space of $A$ corresponds to the null space of $A^T$:

This is the coordinate form of the left null space.

29.8 Example in $\mathbb{R}^3$

Let

A linear functional has the form

We require

and

These give

Hence

Let

Then

Therefore

The annihilator is one-dimensional. This agrees with the dimension formula:

29.9 Dimension Formula

If $V$ is finite-dimensional and $U\subseteq V$ is a subspace, then

The number on the right is the codimension of $U$ in $V$. Thus

An annihilator contains one independent linear condition for each direction missing from $U$. Equivalently, it measures how many independent functionals can vanish on $U$.

29.10 Proof of the Dimension Formula

Let

Choose a basis of $U$:

Extend it to a basis of $V$:

Let the dual basis be

A functional in $U^0$ must vanish on

The dual basis functionals

vanish on all of $U$, and they form a basis of $U^0$. There are $n-k$ of them.

Therefore

29.11 Annihilator of a Sum

If $U$ and $W$ are subspaces of $V$, then

A functional vanishes on $U+W$ exactly when it vanishes on every vector of $U$ and every vector of $W$.

Indeed, if

then $U\subseteq U+W$ and $W\subseteq U+W$, so

Conversely, if

then for any

we have

Thus

29.12 Annihilator of an Intersection

For finite-dimensional spaces,

This identity is dual to the previous one. The annihilator changes intersections into sums and sums into intersections.

The inclusion

is immediate: if a functional vanishes on $U$, and another vanishes on $W$, then their sum vanishes on every vector belonging to both.

The reverse inclusion is deeper and follows from the dimension formula.

29.13 Double Annihilator

If $V$ is finite-dimensional and $U\subseteq V$, then

after identifying $V$ with its double dual $V^{**}$.

Here

means the annihilator of $U^0$ inside $V^{**}$. Under the natural identification

it returns the original subspace $U$.

This says that in finite dimensions, a subspace is completely determined by the linear functionals that vanish on it.

29.14 Annihilators and Quotient Spaces

There is a natural isomorphism

A functional on the quotient $V/U$ is the same as a functional on $V$ that vanishes on $U$.

To see this, let

be linear. Define

Then $\varphi$ is a linear functional on $V$. If $u\in U$, then

so

Thus

Conversely, if $\varphi\in U^0$, define

This is well-defined because $\varphi$ vanishes on $U$.

29.15 Annihilators and Linear Equations

A subspace can often be described as the common zero set of linear functionals.

For example, in $\mathbb{R}^3$,

is the kernel of the functional

Thus

The annihilator $U^0$ is the set of all linear functionals that vanish on this plane. Since the plane has dimension $2$, its annihilator has dimension $1$. Hence

This shows that annihilators encode systems of homogeneous linear equations.

29.16 Orthogonal Complements and Annihilators

In a finite-dimensional inner product space, a vector $a\in V$ defines a functional

Under this identification, the annihilator of a subspace corresponds to its orthogonal complement:

The annihilator is more general because it does not require an inner product. Orthogonal complements depend on a chosen inner product. Annihilators depend only on the vector space and its dual.

29.17 Row Space and Null Space

For a matrix $A$, the null space is the annihilator of the row space.

Let the rows of $A$ be

Then $x\in \operatorname{Null}(A)$ means

for every row $r_i$.

Thus $x$ is annihilated by every row functional. In dual language,

after identifying row vectors with linear functionals on $F^n$.

This is one of the fundamental relationships between the four matrix subspaces.

29.18 Column Space and Left Null Space

Similarly, the left null space of $A$ is the annihilator of the column space.

The left null space is

A vector $y\in F^m$ lies in $\operatorname{Null}(A^T)$ exactly when

Equivalently,

for every column $a$ of $A$.

Thus $y$ defines a functional that vanishes on $\operatorname{Col}(A)$. Hence

29.19 Practical Computation

To compute $U^0$ for a subspace $U\subseteq F^n$:

This procedure reduces annihilator computation to a null space computation.

29.20 Summary

An annihilator is the set of all linear functionals that vanish on a given set or subspace. It is a subspace of the dual space.

The key ideas are:

Annihilators express subspaces through the linear functionals that vanish on them. They provide a coordinate-free way to describe linear equations, quotient duals, orthogonal complements, and the relationships among row spaces, column spaces, and null spaces.