How to demonstrate that 3 vectors are total in $\mathbb C^3$ I don't know if total is the right term. 
A set A is "total" if  $\langle f,a_i \rangle=0 \iff f=0$ .
I have those vectors: $c_1=(1,2,0), c_2=(0,1,2), c_3=(0,0,1)$ and I have to verify if they're "total" in $\mathbb C^3$.
How do I do it? 

Following your definition of total, $\{c_1,c_2,c_3\}$ is total iif the matrix $$A=\begin{bmatrix} c_1 \\ c_2 \\ c_3\end{bmatrix}\in M_{3\times 3}(\mathbb{C})\cap M_{3\times 3}(\mathbb{R})$$ is non singular.
Infact if exists $f\in \mathbb{C^3}$ such that $Af=0$ then  $\langle c_i,\overline{f}\rangle = 0$ (by definition of $\langle,\rangle$ on $\mathbb{C^n}\times \mathbb{C}^n$) and viceversa.
So you only have to check that $detA\neq 0$.
