I'm just starting to learn about tensors, and have a question.
I'm looking at the statement $\Lambda_{\mu}\,^{\alpha}= \eta_{\mu\nu}\eta^{\alpha\beta}\Lambda^{\nu}\,_{\beta}$
What is the difference between $\Lambda_{\beta}\,^{\nu}$ and $\Lambda^{\nu}\,_{\beta}$ ?
$Λ_β{}^ν$ has first a covariant and then a contravariant index and thus describes a tensor $Λ_β{}^ν({\bf θ}^β\otimes {\bf e}_ν)$ in $V^*\otimes V=Hom(V^*)$, while in $Λ^ν{}_β({\bf e}_ν⊗{\bf θ}^β)$ the positions are exchanged, so it is in $V\otimes V^*=Hom(V)$. These tensor products are trivially isomorphic, but in the strict sense different. As matrices, by construction $Λ_β{}^ν$ is the two-times Riesz dual of $Λ^ν{}_β$, meaning for the canonical Minkowski metric the components are the same, except that the mixed space-time components have a switched sign.