How to demonstrate that 3 vectors are total in $\mathbb C^3$

Question

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?

Answer 1

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 $detA\neq 0$.

Answer 2

For a finite-dimensional space $V$ and $A\subseteq V$ we have:

\begin{align} A \text{ is total} &\iff \big(x \perp A \implies x = 0\big)\\ &\iff A^\perp = \{0\}\\ &\iff \mathrm{span}\, A = A^{\perp\perp} = \{0\}^\perp = V \end{align}

So you just have to chech if $\mathrm{span}\,A = V$, or in this case if $\{c_1, c_2, c_3\}$ is a basis for $\mathbb{C}^3$.