The invariance of the interval under coordinate transformations between inertial frames follows from the invariance of c 2 t 2 − − x 2 − − y 2 − − z 2 provided the transformations are linear. This quadratic form can be used to define a bilinear form u ⋅ ⋅ v = c 2 t 1 t 2 − − x 1 x 2 − − y 1 y 2 − − z 1 z 2 via the polarization identity. This bilinear form can in turn be written as u ⋅ ⋅ v = u T [ η η ] v , where [ η ] is a 4 × × 4 matrix associated with η. While possibly confusing, it is common practice to denote [ η ] with just η. The matrix is read off from the explicit bilinear form as η η = ( 1 0 0 0 0 − − 1 0 0 0 0 − − 1 0 0 0 0 − − 1 ) , and the bilinear form u ⋅ ⋅ v = η η ( u , v ) , with which this section started by assuming its existence, is now identified.