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)?
Best Answer
One of the axioms of a vector space is that $\lambda \vec{v} \in V$ for all $\lambda \in \mathbb{F}$ and $\vec{v} \in V$.
Let $V=\mathbb{R}$ and $\mathbb{F}=\mathbb{C}$. If $\vec{v}=1$ and $\lambda=\operatorname{i}$ then $\lambda \vec{v} = \operatorname{i}$ and that does not belong to $\mathbb{R}$.
In such a case, the pair $(\mathbb{R},\mathbb{F})$ fails to be closed under scalar multiplication.