# Thread: Vector space over the field of complex numbers

1. ## Vector space over the field of complex numbers

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?

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?
You need to prove a lot of things. You need to show $\displaystyle V$ is an abelian group. Then you need to show: (i)$\displaystyle 1\bold{v} = \bold{v}$ (ii) $\displaystyle a(b\bold{v}) = (ab)\bold{v}$ (iii) $\displaystyle a(\bold{v}+\bold{u}) = a\bold{v}+b\bold{u}$ (iv) $\displaystyle (a+b)\bold{v} = a\bold{v}+b\bold{v}$.