Let A be a linear operator. If has a complex-valued solution, then it also has a real-valued solution.
How do I prove this?
What is the overall context for this problem? Is it from a textbook? If so, how did the book define a linear operator?
In particular, I'm interested in knowing if A always maps real-valued vectors/functions to real-valued vectors/functions. If it does, then you can choose as your real-valued solution. If not, your problem is a bit more complicated.
Note, that, since the real numbers are a subset of the complex numbers, saying there is a complex solution could mean that there is a real number solution. However, the problem says "it also has a real solution" and that is not true.