Let $V = \{(a_1,a_2,\ldots,a_n): a_i \in \mathbb{R}\ \text{ for } i = 1,2,\ldots,n\};$ So $V$ is a vector space over $\mathbb{R}$. Is $V$ a vector space over the field of complex numbers with the operations of coordinatewise addition and multiplication?
I don't need an actual answer for this, as I just want to confirm that I understand what the question is asking. Paraphrasing: If $V = \mathbb{R}^n$ and $\mathbb{F} = \mathbb{R}$, then we know that $V$ is a vector space. If $\mathbb{F} = \mathbb{C}$, then is $V$ still a vector space (Is the set of real vectors $v \in \mathbb{R}^n$ over the field of complex scalars a vector space)?