Help me determine if ℂ=<b> is a vector space over ℂ

Question

I need help determining if for b∈ℂ, b≠0, ℂ= is a vector space over the field of complex numbers

Thanks

Answer 1

For any $b\in \mathbb{C}, b\neq 0$, $b$ is indeed a basis of the $\mathbb{C}$-vector space $\mathbb{C}$ since this vector space has dimension one and $\{ b\}$ is linearly independent (i.e., $b\neq 0$). To make sure that $b$ produces all of $\mathbb{C}$ note that $\forall c \in \mathbb{C}, \exists ! \alpha \in \mathbb{C}, c=\alpha b \left( \alpha =\frac{c}{b} \right)$.

Note that this is no longer true if you look at $\mathbb{C}$ as an $\mathbb{R}$-vector space (because it has dimension $2$ so you would need two linearly independent vectors at least).