Common theory for Linear equations, linear ODEs and linear recurrence relations In Linear equations, linear ODEs and linear recurrence relations, when solving homogeneous equations, there are a subspace of solutions, and when solving inhomogeneous equations, a particular inhomogeneous solution plus the subspace of homogeneous solutions will give the set of inhomogeneous solutions. I was wondering what the general common theory for Linear equations, linear ODEs and linear recurrence relations in this aspect? Thanks and regards!
 A: This is simply a result of the fact that you are dealing with a linear operator, so you could say this is all subsumed in the study of linear transformations, which is one of the central concerns of Linear Algebra.
If you are dealing with a linear transformation $T$, so that $T(\alpha\mathbf{x}) = \alpha T(\mathbf{x})$ and $T(\mathbf{x}+\mathbf{y}) = T(\mathbf{x})+T(\mathbf{y})$, and you are considering the solution to an equation
$$T(\mathbf{x}) = \mathbf{a},$$
then the difference between any two solutions is a solution to $T(\mathbf{x})=\mathbf{0}$. This is simply the fact that if $T(\mathbf{x}_1) = \mathbf{a}= T(\mathbf{x}_2)$, then 
$$T(\mathbf{x}_1-\mathbf{x}_2) = T(\mathbf{x}_1) - T(\mathbf{x}_2) = \mathbf{a}-\mathbf{a} = \mathbf{0}.$$
This means that if you can find all solutions to $T(\mathbf{x}) = \mathbf{0}$, and a single solution to $T(\mathbf{x}) = \mathbf{a}$, then you can generate all solutions to the latter.
This comes down to an application of the First Isomorphism Theorem. The solutions to $T(\mathbf{x})=\mathbf{0}$ form the kernel or nullspace of the transformation, so the image of $T$ is isomorphic to $\mathbf{V}/\mathrm{ker}(T)$. The "target" $\mathbf{a}$ corresponds to a single element of this quotient, which is of the form $\mathbf{v}+\mathrm{ker}(T)$, where $\mathbf{v}$ is any particular solution.
A: Perhaps you are looking for the notion of an affine space, or more generally, a torsor?
A: You need look up Linear Algebra.
