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 $v+w$.