Let $V = \{(a_1, a_2, …, a_n): a_i \in \mathbb{C}$ for $ i = 1, 2, .., n\}$; so $V$ is a vector space over $\mathbb{R}$. Is V a vector space over the field of real numbers with the operations of coordinatewise addition and multiplication?
The solution says,
Yes. All the conditions are preserved when the field is the real
numbers.
I don't understand how $V$ can be a vector space over $\mathbb{R}$? $V$ itself is specified as being over the field of complex numbers ($V = \{(a_1, a_2, …, a_n): a_i \in \mathbb{C}$), so how can the question then claim that $V$ is a vector space over the real numbers?
I would greatly appreciate it if someone could please take the time to clarify my misunderstanding.
Best Answer
Check the definition of a vector space over a field, for example on Wikipedia: https://en.wikipedia.org/wiki/Vector_space#Definition
Being over $\mathbb{R}$ means that vectors can be multiplied by real numbers and you get again a vector in $V$. This is certainly true in your example since complex numbers multiplied by real numbers are again complex numbers.