homogeneous equations. Subspace properties The kernel of a matrix A over a field K is a linear subspace of Kn. That is, the kernel of A, the set Null(A), has the following three properties: Null(A) always contains the zero vector, since . If and , then . This follows from the distributivity of matrix multiplication over addition. If and c 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: Here, a1, ... , am denote the rows of the matrix A. It follows that x is in the kernel of A, if and only if x is orthogonal (or perpendicular) to each of the row vectors of A (since orthogonality is defined as having a dot product of 0). The row space, or coimage, of a matrix A is the span of the row vectors of A. By the above reasoning, the kernel of A is the orthogonal complement to the row space. That is, a vector x lies in the kernel of A, if and only if it is perpendicular to every vector in the row space of A. The dimension of the row space of A is called the rank of A, and the dimension of the kernel of A is called the nullity of A. These quantities are related by the rank–nullity theorem Left null space The left null space, or cokernel, of a matrix A consists of all column vectors x such that xTA = 0T, where T denotes the transpose of a matrix. The left null space of A is the same as the kernel of AT. The left null space of A is the orthogonal complement to the column space of A, 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 A are the four fundamental subspaces associated to the matrix A. Nonhomogeneous systems of linear equations The kernel also plays a role in the solution to a nonhomogeneous system of linear equations: If u and v are two possible solutions to the above equation, then Thus, the difference of any two solutions to the equation Ax = b lies in the kernel of A. It follows that any solution to the equation Ax = b can