The obvious extension of bilinear interpolation to three dimensions is called trilinear interpolation. The interpolated surface is smoother than corresponding surfaces obtained by bilinear interpolation or nearest-neighbor interpolation. A bilinear form is symmetric (resp. Since is symmetric as a bilinear mapping, it follows that is a symmetric tensor. It allows the study of bilinear or multilinear operations via linear operations. When and , this correspondence between quadratic forms and symmetric bilinear forms breaks down. Depending on the bilinear form, the vector space may contain nonzero self-orthogonal vectors. The quadratic form can be expressed in terms of the bilinear form&colon; formula 3. Then formula 71 is bilinear and symmetric in formula 64 and formula 70. Spacetime mathematically viewed as endowed with this bilinear form is known as Minkowski space . However, the matrices of a bilinear on different bases are all congruent. A vector space with a bilinear form generalizes the case of an inner product. Bilinear filtering for magnification filtering is common. It preserves fine detail better than the common bilinear algorithm. Since formula 8, boundedness and ellipticity of the bilinear form apply to formula 20.