# Doubt in real and complex vector space

Does saying "V is a vector space over reals" same as real vector space? For eg:

$$V={M}_{2}\left( {\mathbb{C}} \right)\\ M_2 \text{ is a } 2 \times 2 \text{ matrix and } \mathbb{C} \text{ is the field of complex numbers}\\ W= {\{\begin{bmatrix} a \; b \\ c \; d\end{bmatrix} | \; a= d' \}}$$ Here my book states that W is not a subspace of complex vector space $${M}_{2}\left( {\mathbb{C}} \right)$$but it is a subspace of real vector space $${M}_{2}\left( {\mathbb{C}} \right)$$ Does real vector space here means that a,b,c,d are real numbers but the chosen field for scalar multiplication is complex?
I am very confused. (here $$\mathbb{C}$$ is referring to the field of complex numbers)

Take the "real vector space $$\mathbb{C}^1$$", it's just the complex numbers with complex addition as the group law, and scaling understood as multiplication of complex numbers by a real numbers, but complex multiplication is forbidden (cause we're not in an $$\mathbb{R}$$-algebra, only in an $$\mathbb{R}$$-vector space). So it's really isomorphic to $$\mathbb{R}^2$$ in this case.
• As for your example, what's your $a=d'$, precisely ? $d'$ doesn't seem defined Commented Mar 18, 2021 at 15:17