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In mathematics, a bilinear form is a bilinear map V × V → K on a vector space V (the elements of which are called vectors) over a field K (the elements of which are called scalars). In other words, a bilinear form is a function B : V × V → K that is linear in each argument separately:
Trilinear interpolation is the extension of linear interpolation, which operates in spaces with dimension =, and bilinear interpolation, which operates with dimension =, to dimension =. These interpolation schemes all use polynomials of order 1, giving an accuracy of order 2, and it requires 2 D = 8 {\displaystyle 2^{D}=8} adjacent pre-defined ...
1.2 Bilinear interpolation. 1.3 Bicubic interpolation. 1.4 Fourier-based interpolation. ... It will also remove small details if in-between larger ones which connect ...
Trilinear filtering solves this by doing a texture lookup and bilinear filtering on the two closest mipmap levels (one higher and one lower quality), and then linearly interpolating the results. [9] This results in a smooth degradation of texture quality as distance from the viewer increases, rather than a series of sudden drops.
Bilinear and trilinear interpolation, using multivariate polynomials with two or three variables; Zhegalkin polynomial, a multilinear polynomial over ; Multilinear form and multilinear map, multilinear functions that are strictly linear (not affine) in each variable; Linear form, a multivariate linear function
In general, for a vector space V over a field F, a bilinear form on V is the same as a bilinear map V × V → F. If V is a vector space with dual space V ∗, then the canonical evaluation map, b(f, v) = f(v) is a bilinear map from V ∗ × V to the base field. Let V and W be vector spaces over the same base field F.
Any bilinear map is a multilinear map. For example, any inner product on a -vector space is a multilinear map, as is the cross product of vectors in .; The determinant of a matrix is an alternating multilinear function of the columns (or rows) of a square matrix.
A duality between two vector spaces over a field F is a non-degenerate bilinear form V 1 × V 2 → F , {\displaystyle V_{1}\times V_{2}\to F,} i.e., for each non-zero vector v in one of the two vector spaces, the pairing with v is a non-zero linear functional on the other.