"Linear map" isn't the correct word here. You mean "Linear vector space".

A set of vectors is "dependent" over a given field if there exist numbers in the field, , , etc., not all 0, such that . The difference between "over C" and "over R" is whether the numbers are allowed to be complex or real. In particular, C^3 "over C" has dimension 3 while "over R" it has dimension 6. Do you see why?What is the difference in dependency if C^3 is over C or over R? The question is not if they're a linear map so the field, be it C or R, has no meaning, no? :S I don't understand it.

You are aware that , surely! That would be just i. But you are missing a sign. It should be (-\sqrt(3)+ 3i- i+ \sqrt{3})/4= 2i/4= i/2.2) Need to solve the following equation:

z^4 = (-\sqrt(3) + i ) / ( -1 - i * \sqrt(3) )

I'll start by saying that I spend too much time doing it all over and over again. I'm not even sure how is this supposed to look :S

Is it

z^4 = ( \sqrt(3)*i^2 + 4i + \sqrt(3) )/4 ? :S

Now write that in "polar form" (which is very easy for i/2) and use "DeMoivres' formula". Writing , the nth root is given by where is the positive nth root of the positive real number r and you can get all n roots by taking k= 0, 1, 2, 3, ..., n-1.If it is, how do I go from here? What do I do?

Thanks in advance!