A common problem in computer graphics is to generate a non-zero vector in R3 that is orthogonal to a given non-zero one. There is no single continuous function that can do this for all non-zero vector inputs. This is a corollary of the hairy ball theorem. To see this, consider the given vector as the radius of a sphere and note that finding a non-zero vector orthogonal to the given one is equivalent to finding a non-zero vector that is tangent to the surface of that sphere where it touches the radius.
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Alternatively, each number, when written in binary, can be identified with a non-zero vector of length three over the binary field. Three vectors that generate a subspace form a line; in this case, that is equivalent to their vector sum being the zero vector.
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Define an eigenvector v associated with the eigenvalue λ to be any vector that, given λ, satisfies Equation (). Given the eigenvalue, the zero vector is among the vectors that satisfy Equation (), so the zero vector is included among the eigenvectors by this alternate definition.
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For bit values 0 = FALSE or 1 = TRUE, this is equivalent to the XOR operation. As MLS are periodic and shift registers cycle through every possible binary value (with the exception of the zero vector), registers can be initialized to any state, with the exception of the zero vector.
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Equation () can be stated equivalently as where is the by identity matrix and 0 is the zero vector.
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While the definition of an eigenvector used in this article excludes the zero vector, it is possible to define eigenvalues and eigenvectors such that the zero vector is an eigenvector. Consider again the eigenvalue equation, Equation (). Define an eigenvalue to be any scalar λ ∈ K such that there exists a nonzero vector v ∈ V satisfying Equation (). It is important that this version of the definition of an eigenvalue specify that the vector be nonzero, otherwise by this definition the zero vector would allow any scalar in K to be an eigenvalue.
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Every non- zero vector of II25,1 can be written uniquely as a positive integer multiple of a primitive vector, so to classify all vectors it is sufficient to classify the primitive vectors.
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That is, it is not possible to obtain a non-zero vector in the same direction as the original. Yet another example of group without identity element involves the additive semigroup of positive natural numbers.
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A duality between two vector spaces over a field is a non-degenerate bilinear form : V_1\times V_2\to F, i.e., for each non-zero vector in one of the two vector spaces, the pairing with is a non- zero linear functional on the other. Similarly, a triality between three vector spaces over a field is a non-degenerate trilinear form : V_1\times V_2\times V_3\to F, i.e., each non-zero vector in one of the three vector spaces induces a duality between the other two.
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In the general case, the proportion of r yielding the zero vector may be less than 1/2, and a larger number of trials (such as 20) would be used, rendering the probability of error very small.
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The simplest example of a vector space is the trivial one: {0}, which contains only the zero vector (see the third axiom in the Vector space article). Both vector addition and scalar multiplication are trivial. A basis for this vector space is the empty set, so that {0} is the 0-dimensional vector space over F. Every vector space over F contains a subspace isomorphic to this one. The zero vector space is different from the null space of a linear operator L, which is the kernel of L.
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If n be a factor of (q^m - 1) for some m. The only vector in GF(q)^n of weight d - 1 or less that has d - 1 consecutive components of its spectrum equal to zero is all-zero vector.
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This is an illustration of the shortest vector problem (basis vectors in blue, shortest vector in red). In the SVP, a basis of a vector space V and a norm N (often L2) are given for a lattice L and one must find the shortest non-zero vector in V, as measured by N, in L. In other words, the algorithm should output a non-zero vector v such that N(v)=\lambda(L). In the γ-approximation version SVPγ, one must find a non-zero lattice vector of length at most \gamma \cdot \lambda(L) for given \gamma \geq 1.
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This is because in the true distribution, the zero vector occurs half the time, and those occurrences are randomly mixed in with the nonzero vectors. Even a small sample will see both zero and nonzero vectors. But Gibbs sampling will alternate between returning only the zero vector for long periods (about 2^{99} in a row), then only nonzero vectors for long periods (about 2^{99} in a row). Thus convergence to the true distribution is extremely slow, requiring much more than 2^{99} steps; taking this many steps is not computationally feasible in a reasonable time period.
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Cases of do not offer anything new: is the real line, whereas (the space containing the empty column vector) is a singleton, understood as a zero vector space. However, it is useful to include these as trivial cases of theories that describe different .
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In general, the zero element of a ring is unique, and typically denoted as 0 without any subscript to indicate the parent ring. Hence the examples above represent zero matrices over any ring. The zero matrix also represents the linear transformation which sends all vectors to the zero vector.
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This lattice has no vectors of type 1. The groups Co2 (of order ) and Co3 (of order ) consist of the automorphisms of Λ fixing a lattice vector of type 2 and a vector of type 3 respectively. As the scalar −1 fixes no non-zero vector, these two groups are isomorphic to subgroups of Co1.
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In mathematics, the zero tensor is a tensor, of any order, all of whose components are zero. The zero tensor of order 1 is sometimes known as the zero vector. Taking a tensor product of any tensor with any zero tensor results in another zero tensor. Adding the zero tensor is equivalent to the identity operation.
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One distinguishes three separate cases: #T − λ is not injective. That is, there exist two distinct elements x,y in X such that (T − λ)(x) = (T − λ)(y). Then z = x − y is a non-zero vector such that T(z) = λz. In other words, λ is an eigenvalue of T in the sense of linear algebra.
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A given is integrable iff everywhere. There is a global foliation theory, because topological constraints exist. For example, in the surface case, an everywhere non-zero vector field can exist on an orientable compact surface only for the torus. This is a consequence of the Poincaré–Hopf index theorem, which shows the Euler characteristic will have to be 0.
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An essential question in linear algebra is testing whether a linear map is an isomorphism or not, and, if it is not an isomorphism, finding its range (or image) and the set of elements that are mapped to the zero vector, called the kernel of the map. All these questions can be solved by using Gaussian elimination or some variant of this algorithm.
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If n be a factor of (q^m - 1) for some m, and b an integer that is coprime with n. The only vector v in GF(q)^n of weight d - 1 or less whose spectral components V_j equal zero for j = \ell_1 + \ell_2b (\mod n), where \ell_1 = 0,...., d - s - 1 and \ell_2 = 0,...., s - 1, is the all zero vector.
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The kernel of a matrix A over a field K is a linear subspace of Kn. That is, the kernel of A, the set Null(A), has the following three properties: # Null(A) always contains the zero vector, since . # If and , then . This follows from the distributivity of matrix multiplication over addition. # If and c is a scalar , then , since .
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This situation is impossible in finite dimensions. The tangent cone to the cube at the zero vector is the whole space. Every subset of the Hilbert cube inherits from the Hilbert cube the properties of being both metrizable (and therefore T4) and second countable. It is more interesting that the converse also holds: Every second countable T4 space is homeomorphic to a subset of the Hilbert cube.
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The augmented matrix has rank 3, so the system is inconsistent. The nullity is 0, which means that the null space contains only the zero vector and thus has no basis. In linear algebra the concepts of row space, column space and null space are important for determining the properties of matrices. The informal discussion of constraints and degrees of freedom above relates directly to these more formal concepts.
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In functional analysis, a total set (also called a complete set) in a vector space is a set of linear functionals T such that if t(s) = 0 for all t in T, then s = 0 is the zero vector. In a more general setting, a subset T of a topological vector space V is a total set or fundamental set if the linear span of T is dense in V.
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If n be a factor of q^m - 1 for some m and GCD(n,b) = 1. The only vector in GF(q)^n of weight d - 1 or less whose spectral components V_j equal to zero for j = l_1 + l_2b (\mod n), where l_1 = 0,..., d - s - 2 and l_2 takes at least s + 1 values in the range 0,...., d - 2, is the all-zero vector.
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Given a vector space over a field of the real numbers or complex numbers , a norm on V is a nonnegative-valued function with the following properties: For all and all , # (being subadditive or satisfying the triangle inequality). # (being absolutely homogeneous or absolutely scalable). # If then is the zero vector (being positive definite or being point-separating). A seminorm on V is a function with the properties 1 and 2 above.
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For Minkowski addition, the zero set containing only the zero vector has special importance: For every non-empty subset S of a vector space :; in algebraic terminology, is the identity element of Minkowski addition (on the collection of non-empty sets).The empty set is important in Minkowski addition, because the empty set annihilates every other subset: For every subset of a vector space, its sum with the empty set is empty: .
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In functional analysis and related areas of mathematics, a barrelled space (also written barreled space) is a topological vector spaces (TVS) for which every barrelled set in the space is a neighbourhood for the zero vector. A barrelled set or a barrel in a topological vector space is a set that is convex, balanced, absorbing, and closed. Barrelled spaces are studied because a form of the Banach–Steinhaus theorem still holds for them.
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In other words, there is a ring homomorphism from the field into the endomorphism ring of the group of vectors. Then scalar multiplication is defined as .. Bourbaki calls the group homomorphisms homotheties. There are a number of direct consequences of the vector space axioms. Some of them derive from elementary group theory, applied to the additive group of vectors: for example, the zero vector of and the additive inverse of any vector are unique.
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A maximum length sequence (MLS) is a type of pseudorandom binary sequence. They are bit sequences generated using maximal linear feedback shift registers and are so called because they are periodic and reproduce every binary sequence (except the zero vector) that can be represented by the shift registers (i.e., for length-m registers they produce a sequence of length 2m − 1). An MLS is also sometimes called an n-sequence or an m-sequence.
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At each image point, the gradient vector points in the direction of largest possible intensity increase, and the length of the gradient vector corresponds to the rate of change in that direction. This implies that the result of the Prewitt operator at an image point which is in a region of constant image intensity is a zero vector and at a point on an edge is a vector which points across the edge, from darker to brighter values.
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In geometry, a glide plane operation is a type of isometry of the Euclidean space: the combination of a reflection in a plane and a translation in that plane. Reversing the order of combining gives the same result. Depending on context, we may consider a reflection a special case, where the translation vector is the zero vector. The combination of a reflection in a plane and a translation in a perpendicular direction is a reflection in a parallel plane.
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For , if Q has diagonalization diag(a), that is there is a non-zero vector x such that , then is algebra-isomorphic to a K-algebra generated by an element x satisfying , the quadratic algebra . In particular, if (that is, Q is the zero quadratic form) then is algebra-isomorphic to the dual numbers algebra over K. If a is a non-zero square in K, then . Otherwise, is isomorphic to the quadratic field extension K() of K.
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The result of this differentiating process is mathematically equivalent to a global motion compensation capable of panning. Further down the encoding pipeline, an entropy coder will take advantage of the resulting statistical distribution of the motion vectors around the zero vector to reduce the output size. It is possible to shift a block by a non-integer number of pixels, which is called sub-pixel precision. The in-between pixels are generated by interpolating neighboring pixels.
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The discreteness condition means that there is some positive real number ε, such that for every translation Tv in the group, the vector v has length at least ε (except of course in the case that v is the zero vector, but the independent translations condition prevents this, since any set that contains the zero vector is linearly dependent by definition and thus disallowed). The purpose of this condition is to ensure that the group has a compact fundamental domain, or in other words, a "cell" of nonzero, finite area, which is repeated through the plane. Without this condition, we might have for example a group containing the translation Tx for every rational number x, which would not correspond to any reasonable wallpaper pattern. One important and nontrivial consequence of the discreteness condition in combination with the independent translations condition is that the group can only contain rotations of order 2, 3, 4, or 6; that is, every rotation in the group must be a rotation by 180°, 120°, 90°, or 60°.
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Let Fm×n denote the set of m×n matrices with entries in F. Then Fm×n is a vector space over F. Vector addition is just matrix addition and scalar multiplication is defined in the obvious way (by multiplying each entry by the same scalar). The zero vector is just the zero matrix. The dimension of Fm×n is mn. One possible choice of basis is the matrices with a single entry equal to 1 and all other entries 0.
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This set can be the set of equivalence classes under the equivalence relation between vectors defined by "one vector is the product of the other by a nonzero scalar". In other words, this amounts to defining a projective space as the set of vector lines in which the zero vector has been removed. A third equivalent definition is to define a projective space of dimension as the set of pairs of antipodal points in a sphere of dimension (in a space of dimension ).
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Due to the Helmholtz decomposition theorem, Gauss's law for magnetism is equivalent to the following statement: The vector field is called the magnetic vector potential. Note that there is more than one possible which satisfies this equation for a given field. In fact, there are infinitely many: any field of the form can be added onto to get an alternative choice for , by the identity (see Vector calculus identities): since the curl of a gradient is the zero vector field: This arbitrariness in is called gauge freedom.
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A first immediate consequence of the definition is that whenever or . This may be seen by writing the zero vector 0V as (and similarly for 0W) and moving the scalar 0 "outside", in front of B, by linearity. The set of all bilinear maps is a linear subspace of the space (viz. vector space, module) of all maps from into X. associates of this are taken to the other three possibilities using duality and the musical isomorphism If V, W, X are finite-dimensional, then so is .
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To summarize, the basic quadratic sieve algorithm has these main steps: # Choose a smoothness bound B. The number π(B), denoting the number of prime numbers less than B, will control both the length of the vectors and the number of vectors needed. # Use sieving to locate π(B) + 1 numbers ai such that bi=(ai2 mod n) is B-smooth. #Factor the bi and generate exponent vectors mod 2 for each one. # Use linear algebra to find a subset of these vectors which add to the zero vector.
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In mathematics, the discussion of vector fields on spheres was a classical problem of differential topology, beginning with the hairy ball theorem, and early work on the classification of division algebras. Specifically, the question is how many linearly independent smooth nowhere-zero vector fields can be constructed on a sphere in N-dimensional Euclidean space. A definitive answer was provided in 1962 by Frank Adams. It was already known, by direct construction using Clifford algebras, that there were at least ρ(N)-1 such fields (see definition below).
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In functional analysis and related areas of mathematics, a set in a topological vector space is called bounded or von Neumann bounded, if every neighborhood of the zero vector can be inflated to include the set. A set that is not bounded is called unbounded. Bounded sets are a natural way to define a locally convex polar topologies on the vector spaces in a dual pair, as the polar of a bounded set is an absolutely convex and absorbing set. The concept was first introduced by John von Neumann and Andrey Kolmogorov in 1935.
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Equivalently, a set of vectors is linearly independent if the only way to express the zero vector as a linear combination of elements of is to take zero for every coefficient a_i. A set of vectors that spans a vector space is called a spanning set or generating set. If a spanning set is linearly dependent (that is not linearly independent), then some element of is in the span of the other elements of , and the span would remain the same if one remove from . One may continue to remove elements of until getting a linearly independent spanning set.
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In this case, a linear subspace contains the zero vector, while an affine subspace does not necessarily contain it. Subspaces of V are vector spaces (over the same field) in their own right. The intersection of all subspaces containing a given set S of vectors is called its span, and it is the smallest subspace of V containing the set S. Expressed in terms of elements, the span is the subspace consisting of all the linear combinations of elements of S. A linear subspace of dimension 1 is a vector line. A linear subspace of dimension 2 is a vector plane.
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The vector length is limited by the available on-chip storage divided by the number of bytes of storage needed for each entry. (Added hardware limits may also exist, which in turn may permit SIMD-style implementations.) Outside of vector loops, the application can request zero- vector registers, saving the operating system the work of preserving them on context switches. The vector length is not only architecturally variable, but designed to vary at run time also. To achieve this flexibility, the instruction set is likely to use variable-width data paths and variable-type operations using polymorphic overloading.
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If any one of these is changed (such as rotating axes instead of vectors, a passive transformation), then the inverse of the example matrix should be used, which coincides with its transpose. Since matrix multiplication has no effect on the zero vector (the coordinates of the origin), rotation matrices describe rotations about the origin. Rotation matrices provide an algebraic description of such rotations, and are used extensively for computations in geometry, physics, and computer graphics. In some literature, the term rotation is generalized to include improper rotations, characterized by orthogonal matrices with determinant −1 (instead of +1).
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For example, in calculus if f is a differentiable function defined on some interval, then it is sufficient to show that the derivative is always positive or always negative on that interval. In linear algebra, if f is a linear transformation it is sufficient to show that the kernel of f contains only the zero vector. If f is a function with finite domain it is sufficient to look through the list of images of each domain element and check that no image occurs twice on the list. A graphical approach for a real-valued function f of a real variable x is the horizontal line test.
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If V is a vector space over a field K and if W is a subset of V, then W is a subspace of V if under the operations of V, W is a vector space over K. Equivalently, a nonempty subset W is a subspace of V if, whenever w_1, w_2 are elements of W and \alpha, \beta are elements of K, it follows that \alpha w_1 + \beta w_2 is in W. As a corollary, all vector spaces are equipped with at least two subspaces: the singleton set with the zero vector and the vector space itself. These are called the trivial subspaces of the vector space.
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In functional analysis and related areas of mathematics, locally convex topological vector spaces (LCTVS) or locally convex spaces are examples of topological vector spaces (TVS) that generalize normed spaces. They can be defined as topological vector spaces whose topology is generated by translations of balanced, absorbent, convex sets. Alternatively they can be defined as a vector space with a family of seminorms, and a topology can be defined in terms of that family. Although in general such spaces are not necessarily normable, the existence of a convex local base for the zero vector is strong enough for the Hahn–Banach theorem to hold, yielding a sufficiently rich theory of continuous linear functionals.
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In the mathematical field of representation theory, a trivial representation is a representation of a group G on which all elements of G act as the identity mapping of V. A trivial representation of an associative or Lie algebra is a (Lie) algebra representation for which all elements of the algebra act as the zero linear map (endomorphism) which sends every element of V to the zero vector. For any group or Lie algebra, an irreducible trivial representation always exists over any field, and is one-dimensional, hence unique up to isomorphism. The same is true for associative algebras unless one restricts attention to unital algebras and unital representations. Although the trivial representation is constructed in such a way as to make its properties seem tautologous, it is a fundamental object of the theory.
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An empty matrix is a matrix in which the number of rows or columns (or both) is zero."Empty Matrix: A matrix is empty if either its row or column dimension is zero", Glossary , O-Matrix v6 User Guide"A matrix having at least one dimension equal to zero is called an empty matrix", MATLAB Data Structures Empty matrices help dealing with maps involving the zero vector space. For example, if A is a 3-by-0 matrix and B is a 0-by-3 matrix, then AB is the 3-by-3 zero matrix corresponding to the null map from a 3-dimensional space V to itself, while BA is a 0-by-0 matrix. There is no common notation for empty matrices, but most computer algebra systems allow creating and computing with them.
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Building on his results on matrix games and on his model of an expanding economy, von Neumann invented the theory of duality in linear programming when George Dantzig described his work in a few minutes, and an impatient von Neumann asked him to get to the point. Dantzig then listened dumbfounded while von Neumann provided an hourlong lecture on convex sets, fixed-point theory, and duality, conjecturing the equivalence between matrix games and linear programming. Later, von Neumann suggested a new method of linear programming, using the homogeneous linear system of Paul Gordan (1873), which was later popularized by Karmarkar's algorithm. Von Neumann's method used a pivoting algorithm between simplices, with the pivoting decision determined by a nonnegative least squares subproblem with a convexity constraint (projecting the zero-vector onto the convex hull of the active simplex).
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The radiometric description of the electromagnetic radiative field at a point in space and time is completely represented by the spectral radiance (or specific intensity) at that point. In a region in which the material is uniform and the radiative field is isotropic and homogeneous, let the spectral radiance (or specific intensity) be denoted by , a scalar-valued function of its arguments , , , and , where denotes a unit vector with the direction and sense of the geometrical vector from the source point to the detection point , where denotes the coordinates of , at time and wave frequency . Then, in the region, takes a constant scalar value, which we here denote by . In this case, the value of the vector flux density at is the zero vector, while the scalar or hemispheric flux density at in every direction in both senses takes the constant scalar value .
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In the case of Mal'cev algebras, this construction can be simplified. Every Mal'cev algebra has a special neutral element (the zero vector in the case of vector spaces, the identity element in the case of commutative groups, and the zero element in the case of rings or modules). The characteristic feature of a Mal'cev algebra is that we can recover the entire equivalence relation ker f from the equivalence class of the neutral element. To be specific, let A and B be Mal'cev algebraic structures of a given type and let f be a homomorphism of that type from A to B. If eB is the neutral element of B, then the kernel of f is the preimage of the singleton set {eB}; that is, the subset of A consisting of all those elements of A that are mapped by f to the element eB.
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For any d-dimensional polytope, one can specify its collection of facet directions and measures by a finite set of d-dimensional nonzero vectors, one per facet, pointing perpendicularly outward from the facet, with length equal to the (d-1)-dimensional measure of its facet. As Hermann Minkowski proved, a finite set of nonzero vectors describes a polytope in this way if and only if it spans the whole d-dimensional space, no two are collinear with the same sign, and the sum of the set is the zero vector. The polytope described by this set has a unique shape, in the sense that any two polytopes described by the same set of vectors are translates of each other. The Blaschke sum X\\# Y of two polytopes X and Y is defined by combining the vectors describing their facet directions and measures, in the obvious way: form the union of the two sets of vectors, except that when both sets contain vectors that are parallel and have the same sign, replace each such pair of parallel vectors by its sum.
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In that case, we often speak of a linear combination of the vectors v1,...,vn, with the coefficients unspecified (except that they must belong to K). Or, if S is a subset of V, we may speak of a linear combination of vectors in S, where both the coefficients and the vectors are unspecified, except that the vectors must belong to the set S (and the coefficients must belong to K). Finally, we may speak simply of a linear combination, where nothing is specified (except that the vectors must belong to V and the coefficients must belong to K); in this case one is probably referring to the expression, since every vector in V is certainly the value of some linear combination. Note that by definition, a linear combination involves only finitely many vectors (except as described in Generalizations below). However, the set S that the vectors are taken from (if one is mentioned) can still be infinite; each individual linear combination will only involve finitely many vectors. Also, there is no reason that n cannot be zero; in that case, we declare by convention that the result of the linear combination is the zero vector in V.
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