I'm confused, too. Can you sort this out? Where's the quotation from? Is ##F## meant to be a subfield of ##\mathbb{R}##? And is the scalar field of ##V## now ##F## or ##\mathbb{R}\,##? Or ##\mathbb{C}## as that's where the vector operations are defined for?

I can only guess, that the statement is: Complex and real vector spaces are different and both differ from vector spaces with a scalar field like e.g. ##F=\mathbb{Q}(\pi,\sqrt{2},e,\log 2)## or whatever. To me this statement is a total mess.

It is written ##F## is a field of ##R##. I don't think it matters if the scalar field is ##R## or ##F## since both are same. It is certainly not ##C##.

So we have ##F \subseteq \mathbb{R} \subset \mathbb{C}##. Next we have a vector space ##V_\mathbb{K}## of finite dimension ##n## and a field of scalars ##\mathbb{K} \in \{F,\mathbb{R},\mathbb{C}\}##. All versions of ##V_\mathbb{K}## lead to different vector spaces, despite the fact that they all are ##n-##dimensional.

I do not understand the remark ##\alpha = (x_1,\ldots ,x_n) \in \mathbb{C}^n##.
Essential to the vector space ##V_\mathbb{K}## is where ##c## is from, and that is ##c \in \mathbb{K}##.

E.g. let's consider ##V := \mathbb{C}^2_\mathbb{R}## and ##W:= \mathbb{C}^2_\mathbb{C}##. Then ##(i,2i)## and ##(1,2)## are two different vectors in both, but in ##V## they point in two different directions (i.e. they are linearly independent), whereas in ##W## there is an equation ##-i \cdot (i,2i) = (1,2)## which means one is a multiple of the other and thus point in the same direction (and they are linearly dependent). All because we have ##c= -i ## available for ##W## which is not available for ##V##.

I've chosen the most general case, because the author(s) introduced a field ##F## of real numbers, e.g. ##\mathbb{Q}(\pi)##, then also spoke about the entire real number field ##\mathbb{R}## as well as of complex numbers ##\mathbb{C}##. I thought a neutral ##\mathbb{K}## would split this Gordian knot of fields.

I guess it is because of the example I added in my previous post.

Nevertheless I think I got it.
I think that ##\alpha \in \Bbb C^n## is to show that there can be different spaces for the same vector set ##V## like ##\Bbb C^n## and vector space of this example.