Not necessarily. We can think of a homogeneous system of n equations as a matrix equation Ax= 0 where A is the matrix of coefficients of the system of equations. The solution set of that homogeneous system of equations is the kernel of the matrix A and its dimension is the "nullity", k, of A. by the rank-nullity property, if A maps to , then th e sum of the rank and nullity of A is n. That is, A maps all of to a n- k dimensional subspace of .

We can write a non-homogeneous system as Ax= b where b is the vector containing the right side of the equations.Ifbe happens to lie in the n- k dimensional subspace that A maps all of into (the "image of under A") then the solution set has dimension k, the same as the kernel. But if b is not in that subspace there isnosolution. That is what is sometimes called the "Fredholm alternative".