Let $\displaystyle V = \{ ( a_{1} , a_{2} , . . . , a_{n} ) : a_{i} \in C for i = 1 , 2, . . . , n \} $. Is V a vector space over the field of complex numbers with operations of coordinatewise addition and multiplication?

proof. Let x and y be in V, then x and y are consisted of coordinates of complex numbers. x+y exist and ix exist. So would that be a vector space?