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Solutions of permuted linear equations

  1. Jun 3, 2012 #1
    Solutions of "permuted" linear equations

    Hi everyone,

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

    [itex]\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}[/itex]

    will be satisfied if [itex]\alpha_{i}[/itex] is a non negative integer?

    Thank you
  2. jcsd
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