Then every linear operator T in V can be written as the sum of a diagonalizable operator D and a nilpotent operator N which commute.

Then is a derivation and is linear, i. e., and, and a Lie algebra homomorphism, i. e.,, but it is not always an algebra homomorphism, i. e. the identity does not hold in general.

Then the first argument becomes conjugate linear, rather than the second.

Let X be a normed topological vector space over F, compatible with the absolute value in F. Then in X *, the topological dual space X of continuous F-valued linear functionals on X, all norm-closed balls are compact in the weak -* topology.

Then any vector in R < sup > 3 </ sup > is a linear combination of e < sub > 1 </ sub >, e < sub > 2 </ sub > and e < sub > 3 </ sub >.

Then express the seed ( the initial conditions of the LRS ) as a linear combination of the eigenbasis vectors:

Then standard methods can be used to solve the linear difference equation in.

Abstractly, we can say that D is a linear transformation from some vector space V to another one, W. We know that D ( c ) = 0 for any constant function c. We can by general theory ( mean value theorem ) identify the subspace C of V, consisting of all constant functions as the whole kernel of D. Then by linear algebra we can establish that D < sup >− 1 </ sup > is a well-defined linear transformation that is bijective on Im D and takes values in V / C.

Then the resultant perturbation can be written as a linear sum of the individual types of perturbations,

Then it transfers to the anilox roll ( or meter roll ) whose texture holds a specific amount of ink since it is covered with thousands of small wells or cups that enable it to meter ink to the printing plate in a uniform thickness evenly and quickly ( the number of cells per linear inch can vary according to the type of print job and the quality required ).

Then they became entirely geometric or linear with raised dots.

Using the natural isomorphism between and the space of linear transformations from V to V ,< ref name =" natural iso "> Let L ( V, V ) be the space of linear transformations from V to V. Then the natural map

Then the zero vector of this space can be expressed as a linear combination of no elements, which again is an empty sum.

Then any linear functional can be written in these coordinates as a sum of the form:

Then every Hodge class on X is a linear combination with rational coefficients of the cohomology classes of complex subvarieties of X.

Then every Hodge class on X is a linear combination with rational coefficients of the cohomology classes of complex subvarieties of X.

Then every Hodge class on X is a linear combination with rational coefficients of Chern classes of vector bundles on X.

Then every Hodge class on X is a linear combination with rational coefficients of Chern classes of coherent sheaves on X.

Then, L < sub > 0 </ sub > is a Lie algebra, L < sub > 1 </ sub > is a linear representation of L < sub > 0 </ sub >, and there exists a symmetric L < sub > 0 </ sub >- equivariant linear map such that for all x, y and z in L < sub > 1 </ sub >,

Let e < sub > i </ sub > denote the trivial path at vertex i. Then we can associate to the vertex i, the projective KΓ module KΓe < sub > i </ sub > consisting of linear combinations of paths which have starting vertex i. This corresponds to the representation of Γ obtained by putting a copy of K at each vertex which lies on a path starting at i and 0 on each other vertex.

Then the map T admits one and only one fixed-point x < sup >*</ sup > in X ( this means T ( x < sup >*</ sup >) = x < sup >*</ sup >).

Then a fuzzy subset s: S of a set S is recursively enumerable if a recursive map h: S × N Ü exists such that, for every x in S, the function h ( x, n ) is increasing with respect to n and s ( x ) = lim h ( x, n ).

Then the map

Then he estimated the boundaries of each of the outcrops of rock, filled them in with colour and ended up with a crude geological map.

Then he drew up a map depicting streets, public buildings, and parks in what he called the " City of Superior ", boasting that he would make the area known throughout the country.

Then during the direct illumination calculation, instead of sending out a ray from the surface to the light that tests collisions with objects, the photon map is queried for shadow photons.

Then the map zoomed into Harrogate's location in Yorkshire, followed by an introduction video spotlighting the town.

Then there exists a lift of f ( that is, a continuous map for which and ) if and only if the induced homomorphisms and at the level of fundamental groups satisfy

Then there exists an open neighbourhood V of F ( 0 ) in Y and a continuously differentiable map G: V → X such that F ( G ( y )) = y for all y in V. Moreover, G ( y ) is the only sufficiently small solution x of the equation F ( x ) = y.

If we take E to be the sum of the even exterior powers of the cotangent bundle, and F to be the sum of the odd powers, define D = d + d *, considered as a map from E to F. Then the topological index of D is the Euler characteristic of the Hodge cohomology of M, and the analytical index is the Euler class of the manifold.

Then is the group of bordism classes of pairs with a closed-dimensional manifold ( with G-structure ) and a map.

Let be the set of smooth functions with compact support on the real line Then, the map

Then it transforms each element of the heightmap into a column of pixels, determines which are visible ( that is, have not been covered up by pixels that have been drawn in front ), and draws them with the corresponding color from the texture map.

Then no matter what the choice of system of parameters or lifting, the last map from R = K < sub > d </ sub > → F < sub > d </ sub > is not 0.

Then the map Tor < sub > i </ sub >< sup > A </ sup >( W, R ) → Tor < sub > i </ sub >< sup > A </ sup >( W, S ) is zero for all i ≥ 1.

Let φ: M → N be a smooth map between ( smooth ) manifolds M and N, and suppose f: N → R is a smooth function on N. Then the pullback of f by φ is the smooth function φ < sup >*</ sup > f on M defined by

Then we can define a linear map R by

Then the Hurewicz map

Then f is a proper morphism defined above if and only if is a proper map in the sense of Bourbaki and is separated.

Then an affine connection on M is a bilinear map

# Let be a scheme of finite type over C. Then there is a topological space X < sup > an </ sup > which as a set consists of the closed points of X with a continuous inclusion map λ < sub > X </ sub >: X < sup > an </ sup > → X.