Vector Space Help

[lollipop2046]

Let V = {(a1, a2, ......., an): ai in C for i = 1, 2, ... n}; (C=complex numbers) ; so, V is a vector space over C. Is V a vector space over the field of real numbers with the operaions of coordinatewise addition and multiplication?

I thought the answer to this question is No since after we perform the operations on any two of the elements in C, we are getting some other complex numbers which is not an element in R. But the book says Yes to the question. Can anyone please tell me what's wrong with my concept and what is the correct way to brainstorm this question?

Thanks!

 

[pka]

From your description, “Let V = {(a<SUB>1</SUB>, a<SUB>2</SUB>, ..., a<SUB>n</SUB>): a<SUB>i</SUB> in C for i = 1, 2, ... n}; (C=complex numbers) ;
so, V is a vector space over C.”, it is very hard to know exactly what you mean.
However, if you mean that V is the set of n-tuples with complex entries then V is certainly a v-space over the complex field.
You need to understand that the real field is a subfield of C.
Therefore V is also a v-space over ℜ.