Finding the linear span of complex numbers over the real number field.

Question

I'm trying to find the linear span of the following group:

$L=\{(z_1,z_2,z_3 )∈C^3 | z_2=\bar z_1 \}$

Over the field: $F=\mathbb R$

I'm a bit confused as to what the linear span of a group of complex numbers could be over the real number field.

If anyone could explain how to approach this problem, I'd appreciate it.

Answer 1

The set $L$ is already a real vector space, since the sum of any two element of $L$ is again an element of $L$ and the product of an element of $L$ by a real number is again an element of $L$. Therefore, $\operatorname{span}(L)=L$.