Consider a linear map represented as a matrix with coefficients in a
field (typically \mathbb{R} or \mathbb{C}), that is operating on column vectors with components over . The kernel of this linear map is the set of solutions to the equation , where is understood as the
zero vector. The
dimension of the kernel of
A is called the
nullity of
A. In
set-builder notation, \operatorname{N}(A) = \operatorname{Null}(A) = \operatorname{ker}(A) = \left\{ \mathbf{x}\in K^n \mid A\mathbf{x} = \mathbf{0} \right\}. The matrix equation is equivalent to a homogeneous
system of linear equations: A\mathbf{x}=\mathbf{0} \;\;\Leftrightarrow\;\; \begin{alignat}{7} a_{11} x_1 &&\; + \;&& a_{12} x_2 &&\; + \;\cdots\; + \;&& a_{1n} x_n &&\; = \;&&& 0 \\ a_{21} x_1 &&\; + \;&& a_{22} x_2 &&\; + \;\cdots\; + \;&& a_{2n} x_n &&\; = \;&&& 0 \\ && && && && &&\vdots\ \;&&& \\ a_{m1} x_1 &&\; + \;&& a_{m2} x_2 &&\; + \;\cdots\; + \;&& a_{mn} x_n &&\; = \;&&& 0\text{.} \\ \end{alignat} Thus the kernel of
A is the same as the solution set to the above homogeneous equations.
Subspace properties The kernel of a matrix over a field is a
linear subspace of . That is, the kernel of , the set , has the following three properties: • always contains the
zero vector, since . • If and , then . This follows from the distributivity of
matrix multiplication over addition. • If and is a
scalar , then , since .
The row space of a matrix The product
Ax can be written in terms of the
dot product of vectors as follows: A\mathbf{x} = \begin{bmatrix} \mathbf{a}_1 \cdot \mathbf{x} \\ \mathbf{a}_2 \cdot \mathbf{x} \\ \vdots \\ \mathbf{a}_m \cdot \mathbf{x} \end{bmatrix}. Here, denote the rows of the matrix . It follows that is in the kernel of , if and only if is
orthogonal (or perpendicular) to each of the row vectors of (since orthogonality is defined as having a dot product of 0). The
row space, or coimage, of a matrix is the
span of the row vectors of . By the above reasoning, the kernel of is the
orthogonal complement to the row space. That is, a vector lies in the kernel of , if and only if it is perpendicular to every vector in the row space of . The dimension of the row space of is called the
rank of
A, and the dimension of the kernel of is called the
nullity of . These quantities are related by the
rank–nullity theorem \operatorname{rank}(A) + \operatorname{nullity}(A) = n.
Left null space The
left null space, or
cokernel, of a matrix consists of all column vectors such that , where T denotes the
transpose of a matrix. The left null space of is the same as the kernel of . The left null space of is the orthogonal complement to the
column space of , and is dual to the
cokernel of the associated linear transformation. The kernel, the row space, the column space, and the left null space of are the
four fundamental subspaces associated with the matrix .
Nonhomogeneous systems of linear equations The kernel also plays a role in the solution to a nonhomogeneous system of linear equations: A\mathbf{x} = \mathbf{b}\quad \text{or} \quad \begin{alignat}{7} a_{11} x_1 &&\; + \;&& a_{12} x_2 &&\; + \;\cdots\; + \;&& a_{1n} x_n &&\; = \;&&& b_1 \\ a_{21} x_1 &&\; + \;&& a_{22} x_2 &&\; + \;\cdots\; + \;&& a_{2n} x_n &&\; = \;&&& b_2 \\ && && && && &&\vdots\ \;&&& \\ a_{m1} x_1 &&\; + \;&& a_{m2} x_2 &&\; + \;\cdots\; + \;&& a_{mn} x_n &&\; = \;&&& b_m \\ \end{alignat} If and are two possible solutions to the above equation, then A(\mathbf{u} - \mathbf{v}) = A\mathbf{u} - A\mathbf{v} = \mathbf{b} - \mathbf{b} = \mathbf{0} Thus, the difference of any two solutions to the equation lies in the kernel of . It follows that any solution to the equation can be expressed as the sum of a fixed solution and an arbitrary element of the kernel. That is, the solution set to the equation is \left\{ \mathbf{v}+\mathbf{x} \mid A \mathbf{v}=\mathbf{b} \land \mathbf{x}\in\operatorname{Null}(A) \right\}, Geometrically, this says that the solution set to is the
translation of the kernel of by the vector . See also
Fredholm alternative and
flat (geometry). ==Illustration==