Group Action of Symmetric Group on Complex Vector Space

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

Then $*$ is a group action.

Proof
Let $\rho, \sigma \in S_n$.

Let $v = \displaystyle \sum_{k \mathop = 1}^n \lambda_k v_k$.

We have:

Hence $*$ fulfils.

Let $e$ denote the identity element of $S_n$. Then:

Hence $*$ fulfils.