$\mathbb{C}$ and $\mathbb{R}$ vector spaces. $\mathbb{C}^n$ is a $\mathbb{C}$-vector space with a natural structure of $\mathbb{R}$-vector space. Then, what is the dimension of $\mathbb{C}^n$ as $\mathbb{R}$-vector space? I'm struggling so hard with this question, I do not even know how to start. Can somebody help me please.
 A: If $\{e_1,\dots,e_n\}$ is the standard basis of $\mathbb{C}^n$, consider
$$
\{e_1,\dots,e_n,ie_1,\dots,ie_n\}
$$
A: The dimension is $2n$. Think of $\mathbb{C}$. It can be generated by just one non-zero complex number (if multiplication by complex scalars is allowed) or, alternatively, by two independent complex numbers like $1$ and $i$ if only multiplication by reals is allowed (any complex number is $a+bi$ where $a,b$ are reals). You can easily generalize this to the $n$-dimensional case.
A: Hint: Note that $$\dim(\mathbb{C}^{n},\mathbb{C},\oplus, \odot)=n$$
and $$\dim(\mathbb{C}^{n},\mathbb{R},\oplus,\odot)=2n$$
This is very intuitive if you take $\mathbb{C}^{1}$ or $\mathbb{C}^{2}$ examples on $\mathbb{R}$ and $\mathbb{C}$ scalar fields, for example.
A: Every element $v\in\mathbb C^n$ is uniquely expressed using $2n$ real numbers $a_1,\dots,a_n,b_1,\dots,b_n\in\mathbb R$ as
$$
v=
\begin{pmatrix} a_1+\mathrm i b_1 \\ a_2+\mathrm i b_2 \\ \vdots \\ a_n+\mathrm i b_n\end{pmatrix}
=
a_1\begin{pmatrix}1 \\ 0 \\ \vdots \\ 0\end{pmatrix}
+ \dots +
a_n\begin{pmatrix}0 \\ \vdots \\ 0 \\ 1\end{pmatrix}
+
b_1\begin{pmatrix}\mathrm i\\ 0 \\ \vdots \\ 0\end{pmatrix}
+ \dots +
b_n\begin{pmatrix}0 \\ \vdots \\ 0 \\ \mathrm i\end{pmatrix}.
$$
Hence, the $2n$ vectors
$$
\begin{pmatrix}1 \\ 0 \\ \vdots \\ 0\end{pmatrix}
,\dots,
\begin{pmatrix}0 \\ \vdots \\ 0 \\ 1\end{pmatrix}
,
\begin{pmatrix}\mathrm i\\ 0 \\ \vdots \\ 0\end{pmatrix}
,\dots,
\begin{pmatrix}0 \\ \vdots \\ 0 \\ \mathrm i\end{pmatrix}
$$
form a basis of $\mathbb C^n$ as an $\mathbb R$-vector space.
