Elementary Linear Algebra Question

Given $a_{ij},x_1,x_2\in{}\mathbb{R}$, it is reasonable to interpret the system

$$a_{11}x_1+a_{12}x_2=0$$ $$\vdots{}$$ $$a_{m1}x_1+a_{m2}x_2=0$$

as describing $m$ lines in the plane, all of which pass through the origin.

Question: If instead $a_{ij},x_1,x_2\in{}\mathbb{C}$, what is the visual interpretation, if any?

Follow up: Does an equation $a_{11}x_1+a_{12}x_2=0$ with $a_{1},a_{12},x_1,x_2\in{}\mathbb{C}$ always describe a line in the complex plane? Example:

$$ix_1+(2-3i)x_2=0$$

How do I plot the graph of this equation in the complex plane? Basically it feels like two dimensions are not enough.

Solution 1:

In terms of visualization, two complex variables would require 4 axes, which we can't do in our 3-d world. The only alternative I can see is to make a series of 3-axis charts in which e.g. each one has a different fixed value for the imaginary of x1 ; then the real of x1 is on one axis and x2 is plotted on the other two axes (one for real, one for imaginary).