TITLE: An exercise for eigenvalues and eigenvectors
QUESTION [6 upvotes]: The following is from an exercise in Gilbert Strang's Linear Algebra and its Applications:

Suppose $A$ has eigenvalues $0,3,5$ with independent eigenvectors $u,v,w$.
  Find a particular solution to $Ax = v+w$. Find all solutions.

It is not difficult to find that the particular solution can be $\frac{1}{3}v+\frac{1}{5}w$. 
Here is my question:  

How should I find all solutions for the equation?

If the equation is $Ax = 0$, one needs to find a basis for the null space of $A$. However in this case, the right hand side is $v+w$.

REPLY [6 votes]: Thanks to Geoff Robinson's hint, one should first find a basis for the null space. In this case, $\{u\}$ can be the basis. Then any solution can be written as
$$x=ku+\frac{1}{3}v+\frac{1}{5}w\qquad k\in {\mathbb R}.$$
This may be the standard way to solve the non-homogeneous linear equation.