In the book "Introductory Functional Analysis with Applications" by Kreyszig Q7 in section 2.1 states:
Let $\{e_1 ... , e_n\}$ be a basis for a complex vector space $X$. Find a basis for $X$ regarded as a real vector space. What is the dimension of $X$ in either case?
I'm not sure what "regarded as" means here. The answer states $\{e_1 ... , e_n, i e_1 ... , i e_n\}, n, 2n$.
On
"regarded as a real vector space" means that all of the scalars in the rules of vector linearity, and all of the scalars in the definition of "basis" are real (rather than complex) valued. In particular, if $\{\vec{b}_i\}$ is a basis of $S$, then for any vector $\vec{v}$, there exist some $\{a_i\}$ such that $$ \vec{v} = \sum_i a_i\vec{b}_i $$ If %$S$ is a real vector space with basis $\{\vec{b}_i\}$, then for any vector $\vec{v}$, there exists some $\{a_i\}$ with all $a_i$ real such that $ \vec{v} = \sum_i a_i\vec{b}_i. $
It's a change of field, take a look at vector spaces definition. Scalars can be real or complex whereas you are a $\mathbb R$-vector space or a $\mathbb C$ vector space.