Showing that a subspace is invariant under a representation r

Question

I'm having trouble with a problem from Yvette Kosmann-Schwarzbach's book called Groups and Symmetries.
Let $V=\{(z_1,z_2,z_3)\in \mathbb{C^3}|z_1+z_2+z_3=0 \}$ a vector subspace of $\mathbb{C^3}$ and $r:S^3 \mapsto \mathbb{C^3}$ a representation of $S_3$, defined by $r(g)e_i=e_{g(i)}$ for every $g\in S_3$ and for $(e_1,e_2,e_3)$ the canonical base of $\mathbb{C^3}$.Show that $V$ is invariant under r.
$r$ is the permutation representation of $S_3$.
What I've done so far :
From a previous exercise we know that $(c,t)=((123),(23))$ generates $S_3$ and for a subspace to be invariant under a representation $r$ we need $r(g)V\subseteq V$,thus i found a base of $V$ ,call it $B=(v_1,v_2)=((0,-1,-1)^t,(2,0,0)^t)$.I calculated $r(t)v_1=v_1$ and $r(t)v_2=v_2$ but everything ''breaks'' on $r(c)$ ,since $r(c)v_1=(-1,0,-1)^t$ and it is clear that $(-1,0,-1)$ does not belong in $V$ since it cannot be written in regard of the basis.
Something must have slipped from my understanding but I can't tell what.
Any help or even small tip would be firmly appreciated !