We view $\Bbb C_2 = \{{w \choose z}:w,z\in \Bbb C\}$ as a vector space over $\Bbb C$, $\Bbb R$ and $\Bbb Q$. Let $x_1={i \choose 0}$, $x_2={\sqrt2 \choose \sqrt5}$, $x_3={0 \choose 1}$, $x_4={i\sqrt3 \choose \sqrt3}$, $x_5={1 \choose 3}\in C_2$. Determine $\dim_F(\operatorname{Span}_F\{x_1, x_2, x_3, x_4, x_5\})$ for $F=\Bbb C$, $\Bbb R$ and $\Bbb Q$.
I largely need help on understanding the concept of the dimension space over different fields, and have provided an example to help.
Thanks.
EDIT: For $\Bbb C$, we can use the definition of dim$_F$(F$^n$)=n to deduce that the dimension is 2 (i think). Still struggling with $\Bbb R$ and $\Bbb Q$.