Exercise 26

For any vector $\vec{v}$ in $\mathbb{R}^{n}$ and any permutation $\sigma$ of the numbers $1$, $2$, …, $n$ (that is, $\sigma$ is a rearrangement of those numbers into a new order), define $\sigma(\vec{v})$ to be the vector whose components are $v_{\sigma(1)}$, $v_{\sigma(2)}$, …, and $v_{\sigma(n)}$ (where $\sigma(1)$ is the first number in the rearrangement, etc.). Now fix $\vec{v}$ and let $V$ be the span of $\{\hskip 1.00006pt\sigma(\vec{v})\mid\text{$\sigma$ permutes $1$, \ldots, $n$}\hskip 1.00006pt\}$. What are the possibilities for the dimension of $V$?