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)
Best Answer
"Real" or "complex" here refers to the field of the scalars for your vector space.
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.