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The tensor product of two vector spaces is a vector space that is defined up to an isomorphism.There are several equivalent ways to define it. Most consist of defining explicitly a vector space that is called a tensor product, and, generally, the equivalence proof results almost immediately from the basic properties of the vector spaces that are so defined.
In linear algebra, the outer product of two coordinate vectors is the matrix whose entries are all products of an element in the first vector with an element in the second vector. If the two coordinate vectors have dimensions n and m , then their outer product is an n × m matrix.
Then, for the minimal product measure the measure of a set is the sum of the measures of its horizontal sections, while for the maximal product measure a set has measure infinity unless it is contained in the union of a countable number of sets of the form A×B, where either A has Lebesgue measure 0 or B is a single point. (In this case the ...
The tensor product of commutative algebras is of frequent use in algebraic geometry. For affine schemes X , Y , Z with morphisms from X and Z to Y , so X = Spec( A ), Y = Spec( R ), and Z = Spec( B ) for some commutative rings A , R , B , the fiber product scheme is the affine scheme corresponding to the tensor product of algebras:
In linear algebra, the Schmidt decomposition (named after its originator Erhard Schmidt) refers to a particular way of expressing a vector in the tensor product of two inner product spaces. It has numerous applications in quantum information theory, for example in entanglement characterization and in state purification, and plasticity.
the vertex set of G × H is the Cartesian product V(G) × V(H); and; vertices (g,h) and (g',h' ) are adjacent in G × H if and only if. g is adjacent to g' in G, and; h is adjacent to h' in H. The tensor product is also called the direct product, Kronecker product, categorical product, cardinal product, relational product, weak direct product ...
In mathematics, the tensor algebra of a vector space V, denoted T(V) or T • (V), is the algebra of tensors on V (of any rank) with multiplication being the tensor product.It is the free algebra on V, in the sense of being left adjoint to the forgetful functor from algebras to vector spaces: it is the "most general" algebra containing V, in the sense of the corresponding universal property ...
In the special case of the category of groups, a product always exists: the underlying set of is the Cartesian product of the underlying sets of the , the group operation is componentwise multiplication, and the (homo)morphism : is the projection sending each tuple to its th coordinate.