Determining if a set of complex numbers spans

Question

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!

Answer 1

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.