be expressed as the sum of a fixed solution v and an arbitrary element of the kernel. That is, the solution set to the equation Ax = b is Geometrically, this says that the solution set to Ax = b is the translation of the kernel of A by the vector v. See also Fredholm alternative and flat (geometry). Illustration The following is a simple illustration of the computation of the kernel of a matrix (see , below for methods better suited to more complex calculations). The illustration also touches on the row space and its relation to the kernel. Consider the matrix The kernel of this matrix consists of all vectors for which which can be expressed as a homogeneous system of linear equations involving x, y, and z: The same linear equations can also be written in matrix form as: Through Gauss–Jordan elimination, the matrix can be reduced to: Rewriting the matrix in equation form yields: The elements of the kernel can be further expressed in parametric vector form, as follows: Since c is a free variable ranging over all real numbers, this can be expressed equally well as: The kernel of A is precisely the solution set to these equations (in this case, a line through the origin in R3). Here, since the vector (−1,−26,16)T constitutes a basis of the kernel of A. The nullity of A is 1. The following dot products are zero: which illustrates that vectors in the kernel of A are orthogonal to each of the row vectors of A. These two (linearly independent) row vectors span the row space of A—a plane orthogonal to the vector (−1,−26,16)T. With the rank 2 of A, the nullity 1 of A, and the dimension 3 of A, we have an illustration of the rank-nullity theorem. Examples If , then the kernel of L is the solution set to a homogeneous system of linear equations. As in the above illustration, if L is the operator: then the kernel of L is the set of solutions to the equations Let C[0,1] denote the vector space of all continuous real-valued functions on the interval [0,1], and define by the rule Then the kernel of L consists of all functions for which . Let C∞(R) be the vector space of all infinitely differentiable functions , and let be the differentiation operator: Then the kernel of D consists of all