How can I interpret the span of two vectors in $\mathbb C^3$? We can interpret the span of two vectors in $\mathbb R^3$ as a plane,or span of one vector in $\mathbb R^3$ as a line, but if I have vectors like $(i,1+i,2)$ and $(\sqrt 2 i,\pi,1)$, then how to interpret the span of these 2 vectors?
Actually I was doing a question involving 4 vectors $(i,1+i,2)$,$(\sqrt 2 i, \pi,1)$ and the vectors $(0,i,2-i)$ and $(e,i,0)$ namely $a_1,a_2,a_3,a_4$, now I am to find $\mbox{span}(a_1,a_2) \cap \mbox{span}(a_3,a_4)$,how to find it.I mean if it we vectors in $\mathbb R^3$, then I would have found the intersection of the two planes, which could be a line.But here I cannot do similar thing, it is a bit complicated.
 A: There is not a geometric visualization for that but we can proceed by solving the system
$$c_1a_1+c_2a_2=c_3a_3+c_4a_4$$
for $c_1,c_2,c_3,c_4 \in \mathbb C$, separating the real and the imaginary part.
A: If you want to stick to the geometric intuition, one way to go is to represent each complex vector $v \in \mathbb{C}^n$ as a real vector $v' \in \mathbb{R}^{2n}$, where $v' = (\Re v, \Im v)$ "stacks" the real and imaginary parts of $v$. 
The span of one vector $v \in \mathbb{C}^3$ can as such be associated with a line in $\mathbb{R}^{6}$, and the span of two 3-dimensional complex vectors is associated with a 2-dimensional subspace of $\mathbb{R}^{6}$ (except if they are parallel). 
The intersection of the span of two 3-dimensional complex vectors will thus be associated with another (at most) 2-dimensional subspace of $\mathbb{R}^{6}$.
In your particular problem, we can use ad hoc inspection to isolate this subspace:
$$a_1' = (0,1,2,1,1,0)$$
$$a_2' = (0,\pi,1,\sqrt{2},0,0)$$
and the span will not cover $(1,0,0,0,0,0)$ and $(0,0,0,0,0,1)$
$$a_3' = (0,0,2,0,1,-1)$$
$$a_4' = (e,0,0,0,1,0)$$
and the span will not cover $(0,1,0,0,0,0)$ and $(0,0,0,1,0,0)$.
So the intersection of the spans is at most the span of $(0,0,1,0,0,0)$ and $(0,0,0,0,1,0)$. Going back to the 3D complex vectors, this means that the resulting subspace is included in the span of $(0,0,1)$ and $(0,i,0)$.
