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Math Help - Vector space over the field of complex numbers

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

    Let  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?
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  2. #2
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    Quote Originally Posted by tttcomrader View Post
    Let  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 V is an abelian group. Then you need to show: (i) 1\bold{v} = \bold{v} (ii) a(b\bold{v}) = (ab)\bold{v} (iii) a(\bold{v}+\bold{u}) = a\bold{v}+b\bold{u} (iv) (a+b)\bold{v} = a\bold{v}+b\bold{v}.
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