Group Action of Symmetric Group on Complex Vector Space/Orbit

Theorem
Let $n \in \Z_{>0}$ be a (strictly) positive integer.

Let $S_n$ denote the symmetric group on $n$ letters.

Let $V$ denote a vector space over the complex numbers $\C$.

Let $V$ have a basis:
 * $\mathcal B := \set {v_1, v_2, \ldots, v_n}$

Let $*: S_n \times V \to V$ be a group action of $S_n$ on $V$ defined as:
 * $\forall \tuple {\rho, v} \in S_n \times V: \rho * v := \lambda_1 v_{\map \rho 1} + \lambda_2 v_{\map \rho 2} + \dotsb + \lambda_n v_{\map \rho n}$

where:
 * $v = \lambda_1 v_1 + \lambda_2 v_2 + \dotsb + \lambda_n v_n$

The orbit of an element $v \in V$ is:


 * $\Orb v = \displaystyle \set {w \in V: \exists \rho \in S_n: w = \sum_{k \mathop = 1}^n \lambda_k v_\map \rho k}$

Proof
By definition: