# Solutions of permuted linear equations

1. Jun 3, 2012

### anthony2005

Solutions of "permuted" linear equations

Hi everyone,

one little mathematic puzzle. Say I have $m$ vectors $\overrightarrow{\mu}_{i}$ and one vector $\overrightarrow{\rho}$ all in an $n\leq m$ dimensional vector space, which are known. My question is, if $\sigma$ permutes the $m$ indices, how many of the $m!$ equations

$\alpha_{\sigma\left(1\right)}\overrightarrow{{\mu}_{1}}+\alpha_{\sigma\left(2\right)}\overrightarrow{\mu}_{2}+...\alpha_{\sigma\left(m\right)} \overrightarrow{{\mu}_{m}}=\overrightarrow{\rho}$

will be satisfied if $\alpha_{i}$ is a non negative integer?

Thank you