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I've just started to learn about complex numbers and I got stuch at this question:

Let's look at the vector space $$V=C^2.$$ Does the set $${(1 -i, 1 + i),(1 + i, 1 - i)}$$ spans V while V is a vector space above R?

First of all, I didnt understand this question very well. $$V=C^2.$$ What does it mean "V is above R"? Second, how do i prove span of a set of complex numbers?

Thank you all in advance!

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The space $\mathbb{C}^2$ is a real vector space; you just use the usual addition of vectors and the usual product of a vector by a real number. With these operations, it becomes a vector space over $\mathbb R$. This vector space has a basis consisting of $4$ vectors: $(1,0)$, $(i,0)$, $(0,1)$ and $(0,i)$. Therefore $\dim V=4$ and so no set with only $2$ vectors can span it.

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  • $\begingroup$ How did you find the basis? $\endgroup$ – Elyasaf755 Mar 2 '18 at 7:32
  • $\begingroup$ @user534957 If $(a+bi,c+di)\in\mathbb{C}^2$, then$$(a+bi,c+di)=a(1,0)+b(i,0)+c(0,1)+d(0,i).$$ $\endgroup$ – José Carlos Santos Mar 2 '18 at 7:38

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