Then if two different bases exist, { e 0, e 1, e 2, e 3 } and { e ′ 0, e ′ 1, e ′ 2, e ′ 3 }, e μ μ ′ = e ν ν Λ Λ μ μ ν ν can be represented as e μ μ ′ = e ν ν Λ Λ μ μ ν ν or e μ μ ′ = Λ Λ e μ μ . While it might be tempting to think of Λ Λ ν ν μ μ and Λ as the same thing, mathematically, they are elements of different spaces, and act on the space of standard bases from different sides.