the span of a representation's action on a vector Consider the image of the action of a group representation $\rho: G \to V$ on some vector $v \in V$:
$$
\{ \rho(g) v : g \in G \}
$$
It seems that the span of this set:
$$
W_v \equiv \operatorname{span}(\{ \rho(g) v : g \in G \})
$$
forms an invariant subspace, since for any $w \in W_v$, $g_j \in G$ (assuming $G$ is finite):
$$
\rho(g_j) w = \rho(g_j) \sum_i w_i \rho(g_i)v = \sum_i w_i \rho(g_j) \rho(g_i) v = \sum_i w_i \rho(g_j g_i) v \in W_v
$$
The representation $\rho$ restricted to this subspace $W_v$ forms a subrepresentation. Is this subrepresentation necessarily irreducibile?

The argument seems ok. It is, however, more correct to write just 
$\{ \rho(g) v : g \in G \}$, without the "$\forall$" symbol.
It doesn't need to be irreducible in general. A simple counter-example is the representation of $S_3$, the permutation group, on $\mathbb{R}^3$ that acts by permuting the coordinates. $W_{(1,0,0)}$ is then the whole $\mathbb{R}^3$ but is reducible.
