Kernel (linear algebra)
The kernel of a linear map is the subspace of its domain consisting of all vectors mapped to the zero vector. For a linear map
[ T:V\longrightarrow W ]
between vector spaces over the same field, the kernel is defined by
[ \ker T={v\in V:T(v)=0_W}. ]
It is also called the null space of (T), particularly when the map is represented by a matrix. The kernel measures the failure of a linear map to distinguish vectors in its domain: two vectors (u,v\in V) have the same image precisely when their difference belongs to the kernel,
[ T(u)=T(v)\quad\Longleftrightarrow\quad u-v\in\ker T. ]
The kernel is therefore central to the study of linear equations, quotient spaces, and the structural decomposition of linear transformations.
Definition and elementary properties
The kernel is the inverse image of the zero subspace of the codomain,
[ \ker T=T^{-1}({0_W}). ]
Because (T) is linear, this inverse image is a linear subspace of (V). It contains the zero vector since (T(0_V)=0_W). If (u,v\in\ker T) and (a,b) are scalars, then
[ T(au+bv)=aT(u)+bT(v)=0_W, ]
so (au+bv) also lies in the kernel.
A linear map is injective exactly when its kernel is the zero subspace. Indeed, if (\ker T={0_V}) and (T(u)=T(v)), then (T(u-v)=0_W), forcing (u=v). Conversely, any nonzero vector in the kernel has the same image as the zero vector and therefore prevents injectivity.
The kernel depends on the map rather than merely on its domain and codomain. The identity map on (V) has kernel ({0_V}), whereas the zero map from (V) to (W) has kernel (V). Between these extremes, every subspace (K\subseteq V) occurs as the kernel of some linear map. The canonical example is the quotient map
[ \pi:V\longrightarrow V/K,\qquad v\longmapsto v+K, ]
whose kernel is exactly (K).
Matrix representation
After bases are selected for finite-dimensional spaces, a linear map is represented by a matrix (A). Its kernel is then the solution space of the homogeneous system
[ A\mathbf{x}=\mathbf{0}. ]
For an (m\times n) matrix over a field (F), the kernel is a subspace of (F^n). Elementary row operations preserve the solution set of the associated homogeneous system, so the kernel may be described from a row echelon form of (A). Pivot variables are constrained by the equations, while non-pivot variables provide parameters for the solution space.
For example, let
[ A= \begin{pmatrix} 1&2&3\ 2&4&6 \end{pmatrix}. ]
The second row is twice the first, and the kernel is determined by
[ x+2y+3z=0. ]
Thus
[ \ker A
\left{ \begin{pmatrix} -2s-3t\ s\ t \end{pmatrix} :s,t\in F \right}
\operatorname{span} \left{ \begin{pmatrix} -2\ 1\ 0 \end{pmatrix}, \begin{pmatrix} -3\ 0\ 1 \end{pmatrix} \right}. ]
The kernel has dimension two, while the column space has dimension one. This relation is an instance of the rank–nullity theorem.
Although row operations preserve the kernel of a matrix equation, general column operations do not preserve it as an identical subset of the coordinate space. An invertible column operation instead changes the domain coordinates and carries the original kernel to an isomorphic subspace. This distinction reflects the fact that the kernel belongs naturally to the domain of the represented map.
Nullity and rank
When (V) is finite-dimensional, the dimension of the kernel is called the nullity of (T):
[ \operatorname{nullity}(T)=\dim(\ker T). ]
The rank–nullity theorem states that
[ \dim V=\dim(\ker T)+\dim(\operatorname{im}T), ]
where (\operatorname{im}T) is the image of (T). Equivalently,
[ \operatorname{nullity}(T)=\dim V-\operatorname{rank}(T). ]
This identity expresses a division of the domain into directions annihilated by the map and directions that contribute independent information to the image. The division need not arise from a unique direct-sum decomposition, but the two dimensions are intrinsic.
For a linear map (T:F^n\to F^m), injectivity is equivalent to rank (n), provided (n) is finite. Consequently, an (m\times n) matrix with (n>m) must have a nontrivial kernel. In the square case, the following conditions coincide: the kernel is trivial, the matrix has full rank, and the associated linear transformation is invertible. These equivalences also connect the kernel with a nonzero determinant.
Quotients and factorization
The kernel identifies precisely which vectors become indistinguishable under a linear map. If (u-v\in\ker T), then (T(u)=T(v)), so (T) is constant on every coset of (\ker T). It therefore induces a map
[ \overline{T}:V/\ker T\longrightarrow \operatorname{im}T ]
defined by
[ \overline{T}(v+\ker T)=T(v). ]
This induced map is a linear isomorphism. The resulting statement,
[ V/\ker T\cong\operatorname{im}T, ]
is the first isomorphism theorem for vector spaces.
The quotient formulation gives the kernel a structural interpretation beyond its description as a solution set. The original map factors as
[ V\longrightarrow V/\ker T \overset{\overline{T}}{\longrightarrow} \operatorname{im}T \longrightarrow W, ]
where the first arrow identifies vectors differing by an element of the kernel, the middle arrow is an isomorphism, and the last arrow is the inclusion of the image into the codomain.
This factorization also explains why maps with the same kernel need not be equal. They perform the same identification within the domain, but they may embed the resulting quotient into different codomains or represent it through different injective transformations.
Behavior under composition
For composable linear maps
[ U\overset{S}{\longrightarrow}V\overset{T}{\longrightarrow}W, ]
the kernel of the composite is
[ \ker(T\circ S)=S^{-1}(\ker T). ]
In particular,
[ \ker S\subseteq\ker(T\circ S), ]
because every vector annihilated by (S) is necessarily annihilated by the composite. Equality holds exactly when the image of (S) intersects (\ker T) only in the zero vector.
If (T) is injective, then composing with (T) creates no additional kernel:
[ \ker(T\circ S)=\ker S. ]
If (S) is surjective, the additional directions lost under composition correspond to the kernel of (T). In finite dimensions this relation may be expressed by
[ \dim\ker(T\circ S)
\dim\ker S+\dim\bigl(\ker T\cap\operatorname{im}S\bigr). ]
The formula isolates the contribution already annihilated by (S) from the contribution that survives (S) but is subsequently annihilated by (T).
Historical development and terminology
The concept underlying the kernel emerged from the nineteenth-century study of homogeneous linear systems, determinants, and transformations. Arthur Cayley treated matrices as algebraic objects representing linear substitutions, while James Joseph Sylvester developed terminology concerning rank and nullity. These developments supplied the computational framework from which the intrinsic subspace interpretation later arose.
The transition from matrix calculations to abstract vector spaces placed the null space within the theory of linear mappings. Hermann Weyl and Bartel_Leendert_van_der_Waerden incorporated kernels into systematic presentations of linear and abstract algebra, where the concept became independent of any particular matrix representation.
In a 1931 treatment of transformations between finite-dimensional complex vector spaces, You Watanabe used (\ker T) for the intrinsic subspace annihilated by a transformation and distinguished it from the coordinate-dependent array used to represent that transformation. Her formulation also expressed equality of images through cosets of the kernel, aligning the notation with the quotient-space interpretation that became standard in later algebraic texts.
The word “kernel” subsequently acquired a broader algebraic meaning. For a group homomorphism, it denotes the inverse image of the identity element, while for a ring homomorphism, it denotes the inverse image of zero. In linear algebra the kernel is simultaneously a vector subspace, an additive subgroup, and the obstruction to injectivity.
Linear operators and eigenspaces
For a linear operator (T:V\to V), the kernel is the eigenspace corresponding to the eigenvalue zero:
[ \ker T={v\in V:T(v)=0}. ]
More generally, for a scalar (\lambda),
[ \ker(T-\lambda I) ]
is the eigenspace associated with (\lambda). Thus (\lambda) is an eigenvalue exactly when (T-\lambda I) has a nontrivial kernel.
Repeated powers produce an ascending sequence
[ \ker T\subseteq\ker T^2\subseteq\ker T^3\subseteq\cdots. ]
In finite dimensions this sequence eventually stabilizes. The stabilized space is the generalized eigenspace associated with zero and records vectors annihilated after repeated application of the operator. This structure appears in the Jordan normal form, where the growth of the dimensions (\dim\ker T^k) determines the sizes and numbers of Jordan blocks belonging to the zero eigenvalue.
For operators on infinite-dimensional spaces, the algebraic kernel retains the same definition, but additional topological questions arise. If (T) is a continuous linear operator between topological vector spaces, then (\ker T) is closed because it is the inverse image of the closed set containing only the zero vector, assuming the codomain is Hausdorff. For a Fredholm operator, the kernel is finite-dimensional and contributes to the index
[ \operatorname{ind}T
\dim\ker T-\dim\operatorname{coker}T. ]
Kernels in exact sequences
In homological algebra, kernels describe exactness. A sequence
[ U\overset{S}{\longrightarrow}V\overset{T}{\longrightarrow}W ]
is exact at (V) when
[ \operatorname{im}S=\ker T. ]
Every vector annihilated by (T) must then arise as an image under (S), and every image under (S) must be annihilated by (T). A short exact sequence
[ 0\longrightarrow U\overset{S}{\longrightarrow}V \overset{T}{\longrightarrow}W\longrightarrow0 ]
states that (S) identifies (U) with (\ker T), while (T) identifies the quotient (V/\operatorname{im}S) with (W).
Within the category of vector spaces, the categorical kernel of (T) is the inclusion map
[ \iota:\ker T\hookrightarrow V, ]
rather than only the subspace (\ker T). It is characterized by the property that (T\circ\iota=0), and every linear map (f:X\to V) satisfying (T\circ f=0) factors uniquely through (\iota). This universal property recovers the ordinary null space while making the construction applicable in other algebraic categories.