I'm not sure that this will make a lot of sense, but I guess that I'm trying to decide which course of study or action to persue.
I have a system of equations that seems to be linear, if the complex numbers are treated as scalars.
In other words, I have a system of equations with complex coefficients. I'm wondering how to go about solving these, and if linear algebra may somehow suffice. The variables are complex themselves, but I wonder whether scalar variables (instead of complex variables) would change anything.
The question/Summary
To make more sense of this question and to sum up what I'm looking for, I'd like to know the best way to solve a system of equations with complex coefficients and variables. Which branch(es) of math/science does this involve? An example showing how to solve a system of equations of this form would be particularly helpful, but not necessary.
An Example I'm interested In I have a set of values, and I multiply them by complex values taken from the unit circle. So I may have a set${a,b,c}$ of complex values and I take this set and multiply it by $e^{i\pi d}$, which gives me three coefficients to use for three variables, say $x_1$, $x_2$, and $x_3$. I repeat this process several times (for different $d$ to create PART of several equations. So far, I can create euqations with these values, but they are complex multiples of one another.
These values (of the $x$'s) are added to another set of values that come from powers of complex numbers - this essentially ensures that the system of equations aren't "scalar" multiples of one another. I calculate corresponding sums of the variables. Then I have a system of equations.
What I'm trying to say is that I get a system of equations that aren't complex multiples of one another. I'm really thrilled at the possibility that I can solve this by linear algebra methods.