An important example is the kernel of a linear map x ↦ ↦ A x for some fixed matrix A . The kernel of this map is the subspace of vectors x such that A x = 0 , which is precisely the set of solutions to the system of homogeneous linear equations belonging to A . This concept also extends to linear differential equations a 0 f + a 1 d f d x + a 2 d 2 f d x 2 + ⋯ ⋯ + a n d n f d x n = 0 , where the coefficients a i are functions in x , too. In the corresponding map f ↦ ↦ D ( f ) = ∑ ∑ i = 0 n a i d i f d x i , the derivatives of the function f appear linearly (as opposed to f ′ ′ ′ ′ ( x ) 2 , for example). Since differentiation is a linear procedure (that is, ( f + g ) ′ ′ = f ′ ′ + g ′ ′ and ( c ⋅ ⋅ f ) ′ ′ = c ⋅ ⋅ f ′ ′ for a constant c ) this assignment is linear, called a linear differential operator. In particular, the solutions to the differential equation D ( f ) = 0 form a vector space (over R or C).