I'm having trouble with this:
Let's say we have a homogeneous system of linear equations (HSLE), where all the coefficients are real numbers. Let $f_1, f_2, \dots , f_m$ be the fundamental system of solutions (FSS) for all the real solutions.
Question: How do you prove that $f_1, f_2, \dots , f_m$ is also the fundamental system of solutions for all the complex solutions.
I know that the solutions of the HSLE should form a vector space whose basis is the (FSS), but in this case the FSS is the basis for only the real solutions vector space. I somehow need to show that this FSS is the basis for both real and complex solutions vector space, but how?