Jonathan wrote:

> Similarly, \\(\mathbb{C}\\) is known to have a representation in terms of \\(\mathbb{R}^2\\), via the \\(\Re\\) and \\(\Im\\) projections: \\(\Re(1 + i) = 1\\), and \\(\Im(1 + i) = 1\\).

Right, it's just like that. The only difference is that here you are using real linear combinations of \\(1\\) and \\(i\\), while in my lecture I was using natural number linear combinations of elements of any set \\(S\\).