There are many possible choices of standard basis obeying the condition η η ( e μ μ , e ν ν ) = η η μ μ ν ν . Any two such bases are related in some sense by a Lorentz transformation, either by a change-of-basis matrix Λ Λ ν ν μ μ , a real 4 × 4 matrix satisfying Λ Λ ρ ρ μ μ η η μ μ ν ν Λ Λ σ σ ν ν = η η ρ ρ σ σ . or Λ, a linear map on the abstract vector space satisfying, for any pair of vectors u, v, η η ( Λ Λ u , Λ Λ v ) = η η ( u , v ) .