# Chapter 29. Annihilators # 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 $$ S^0=\{\varphi\in V^*:\varphi(s)=0 \text{ for all } s\in S\}. $$ 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 $$ S^0=\{\varphi\in V^*:\varphi(s)=0 \text{ for every } s\in S\}. $$ 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 $$ U^0 \subseteq V^*. $$ The annihilator lives in the dual space, not in the original vector space. ## 29.2 First Example Let $$ V=\mathbb{R}^2 $$ and let $$ U=\operatorname{span} \left( \begin{bmatrix} 1\\ 0 \end{bmatrix} \right). $$ Then \(U\) is the \(x\)-axis. A linear functional on \(\mathbb{R}^2\) has the form $$ \varphi(x,y)=ax+by. $$ For \(\varphi\) to vanish on \(U\), we need $$ \varphi(t,0)=0 $$ for every \(t\in\mathbb{R}\). But $$ \varphi(t,0)=at. $$ This is zero for every \(t\) exactly when $$ a=0. $$ Therefore $$ U^0=\{\varphi(x,y)=by:b\in\mathbb{R}\}. $$ 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 $$ 0(s)=0 $$ for every \(s\in S\). Next, let $$ \varphi,\psi\in S^0. $$ Then for every \(s\in S\), $$ (\varphi+\psi)(s)=\varphi(s)+\psi(s)=0+0=0. $$ Thus $$ \varphi+\psi\in S^0. $$ For a scalar \(c\), $$ (c\varphi)(s)=c\varphi(s)=c0=0. $$ Thus $$ c\varphi\in S^0. $$ Therefore \(S^0\) is a subspace of \(V^*\). ## 29.4 Extreme Cases There are two useful extreme cases. First, $$ \{0\}^0=V^*. $$ Every linear functional sends the zero vector to zero, so every functional vanishes on \(\{0\}\). Second, $$ V^0=\{0\}. $$ 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 $$ S\subseteq T\subseteq V, $$ then $$ T^0\subseteq S^0. $$ 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: $$ S^0=(\operatorname{span} S)^0. $$ 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 $$ S\subseteq \operatorname{span} S. $$ 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 $$ A= \begin{bmatrix} |&|&&|\\ u_1&u_2&\cdots&u_k\\ |&|&&| \end{bmatrix}. $$ A functional \(\varphi\in(\mathbb{R}^n)^*\) can be represented by a row vector $$ y^T. $$ The condition \(\varphi(u_i)=0\) for every \(i\) becomes $$ y^Tu_i=0. $$ Equivalently, $$ y^TA=0. $$ Transposing, $$ A^Ty=0. $$ Thus the annihilator of the column space of \(A\) corresponds to the null space of \(A^T\): $$ (\operatorname{Col}(A))^0 \cong \operatorname{Null}(A^T). $$ This is the coordinate form of the left null space. ## 29.8 Example in \(\mathbb{R}^3\) Let $$ U=\operatorname{span} \left( \begin{bmatrix} 1\\ 1\\ 0 \end{bmatrix}, \begin{bmatrix} 0\\ 1\\ 1 \end{bmatrix} \right) \subseteq \mathbb{R}^3. $$ A linear functional has the form $$ \varphi(x,y,z)=ax+by+cz. $$ We require $$ \varphi(1,1,0)=0 $$ and $$ \varphi(0,1,1)=0. $$ These give $$ a+b=0, $$ $$ b+c=0. $$ Hence $$ a=-b, \qquad c=-b. $$ Let $$ b=t. $$ Then $$ (a,b,c)=(-t,t,-t)=t(-1,1,-1). $$ Therefore $$ U^0= \operatorname{span} \{\varphi(x,y,z)=-x+y-z\}. $$ The annihilator is one-dimensional. This agrees with the dimension formula: $$ \dim U^0=3-2=1. $$ ## 29.9 Dimension Formula If \(V\) is finite-dimensional and \(U\subseteq V\) is a subspace, then $$ \dim U^0=\dim V-\dim U. $$ The number on the right is the codimension of \(U\) in \(V\). Thus $$ \dim U^0=\operatorname{codim} U. $$ 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 $$ \dim V=n, \qquad \dim U=k. $$ Choose a basis of \(U\): $$ u_1,\ldots,u_k. $$ Extend it to a basis of \(V\): $$ u_1,\ldots,u_k,v_{k+1},\ldots,v_n. $$ Let the dual basis be $$ u^1,\ldots,u^k,v^{k+1},\ldots,v^n. $$ A functional in \(U^0\) must vanish on $$ u_1,\ldots,u_k. $$ The dual basis functionals $$ v^{k+1},\ldots,v^n $$ vanish on all of \(U\), and they form a basis of \(U^0\). There are \(n-k\) of them. Therefore $$ \dim U^0=n-k=\dim V-\dim U. $$ ## 29.11 Annihilator of a Sum If \(U\) and \(W\) are subspaces of \(V\), then $$ (U+W)^0=U^0\cap W^0. $$ A functional vanishes on \(U+W\) exactly when it vanishes on every vector of \(U\) and every vector of \(W\). Indeed, if $$ \varphi\in (U+W)^0, $$ then \(U\subseteq U+W\) and \(W\subseteq U+W\), so $$ \varphi\in U^0\cap W^0. $$ Conversely, if $$ \varphi\in U^0\cap W^0, $$ then for any $$ u+w\in U+W, $$ we have $$ \varphi(u+w)=\varphi(u)+\varphi(w)=0. $$ Thus $$ \varphi\in (U+W)^0. $$ ## 29.12 Annihilator of an Intersection For finite-dimensional spaces, $$ (U\cap W)^0=U^0+W^0. $$ This identity is dual to the previous one. The annihilator changes intersections into sums and sums into intersections. The inclusion $$ U^0+W^0\subseteq (U\cap W)^0 $$ 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 $$ U^{00}=U, $$ after identifying \(V\) with its double dual \(V^{**}\). Here $$ U^{00} $$ means the annihilator of \(U^0\) inside \(V^{**}\). Under the natural identification $$ V\cong V^{**}, $$ 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 $$ (V/U)^*\cong U^0. $$ A functional on the quotient \(V/U\) is the same as a functional on \(V\) that vanishes on \(U\). To see this, let $$ \lambda:V/U\to F $$ be linear. Define $$ \varphi(v)=\lambda(v+U). $$ Then \(\varphi\) is a linear functional on \(V\). If \(u\in U\), then $$ u+U=U, $$ so $$ \varphi(u)=\lambda(U)=0. $$ Thus $$ \varphi\in U^0. $$ Conversely, if \(\varphi\in U^0\), define $$ \lambda(v+U)=\varphi(v). $$ 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 \(\mathbb{R}^3\), $$ U=\{(x,y,z):x+y+z=0\} $$ is the kernel of the functional $$ \varphi(x,y,z)=x+y+z. $$ Thus $$ U=\ker \varphi. $$ 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 $$ U^0=\operatorname{span}(\varphi). $$ 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 $$ \varphi_a(v)=\langle v,a\rangle. $$ Under this identification, the annihilator of a subspace corresponds to its orthogonal complement: $$ U^0 \cong U^\perp. $$ 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 $$ r_1,\ldots,r_m. $$ Then \(x\in \operatorname{Null}(A)\) means $$ r_i x=0 $$ for every row \(r_i\). Thus \(x\) is annihilated by every row functional. In dual language, $$ \operatorname{Null}(A) = \operatorname{Row}(A)^0, $$ 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 $$ \operatorname{Null}(A^T). $$ A vector \(y\in F^m\) lies in \(\operatorname{Null}(A^T)\) exactly when $$ A^Ty=0. $$ Equivalently, $$ y^Ta=0 $$ for every column \(a\) of \(A\). Thus \(y\) defines a functional that vanishes on \(\operatorname{Col}(A)\). Hence $$ \operatorname{Null}(A^T) \cong \operatorname{Col}(A)^0. $$ ## 29.19 Practical Computation To compute \(U^0\) for a subspace \(U\subseteq F^n\): | Step | Operation | |---|---| | 1 | Put spanning vectors of \(U\) as columns of a matrix \(A\) | | 2 | Write a general functional as \(y^T\) | | 3 | Solve \(y^TA=0\), equivalently \(A^Ty=0\) | | 4 | Convert the solution vectors \(y\) into functionals \(y^Tx\) | | 5 | The resulting functionals form a basis for \(U^0\) | 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: | Concept | Meaning | |---|---| | Annihilator | \(S^0=\{\varphi:\varphi(s)=0\text{ for all }s\in S\}\) | | Ambient space | \(S^0\subseteq V^*\) | | Inclusion reversal | \(S\subseteq T\) implies \(T^0\subseteq S^0\) | | Dimension formula | \(\dim U^0=\dim V-\dim U\) | | Sum rule | \((U+W)^0=U^0\cap W^0\) | | Intersection rule | \((U\cap W)^0=U^0+W^0\) in finite dimensions | | Double annihilator | \(U^{00}=U\) in finite dimensions | | Quotient relation | \((V/U)^*\cong U^0\) | | Matrix computation | \(U^0\) is found by solving \(A^Ty=0\) | 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.