However, the bilinear transformation can also be interpreted as a single linear transformation from the tensor product to Z.

There are various norms that can be placed on the tensor product of the underlying vector spaces, amongst others the projective cross norm and injective cross norm.

In general, the tensor product of complete spaces is not complete again.

Two Hilbert spaces V and W may form a third space by a tensor product.

If a system is composed of two subsystems described in V and W respectively, then the Hilbert space of the entire system is the tensor product of the two spaces.

The difference is that the Cartesian product can be interpreted simply as a pair of items ( or a list ), whereas the tensor product, used to define a monoidal category, is suitable for describing entangled quantum states.

The Hilbert space of the electron pair is, the tensor product of the two electrons ' Hilbert spaces.

Tensor products: If C denotes the category of vector spaces over a fixed field, with linear maps as morphisms, then the tensor product defines a functor C × C → C which is covariant in both arguments.

Functors are often defined by universal properties ; examples are the tensor product, the direct sum and direct product of groups or vector spaces, construction of free groups and modules, direct and inverse limits.

which is simply the canonical way of constructing a basis for a tensor product space of the combined system from the individual spaces.

When it acts on a tensor product of two state vectors, it exchanges the values of the state vectors:

* The Hilbert space of a composite system is the Hilbert space tensor product of the state spaces associated with the component systems ( for instance, J. M.

The composite system is then described by the tensor product, and its state is

The Hilbert space of the composite system is the tensor product

Here a rank 1 tensor ( matrix product of a column vector and a row vector ) is the same thing as a rank 1 matrix of the given size.

Thus the ( real < ref > The complex spinors are obtained as the representations of the tensor product H ⊗< sub > R </ sub > C = Mat < sub > 2 </ sub >( C ).

In mathematics, the tensor product, denoted by ⊗, may be applied in different contexts to vectors, matrices, tensors, vector spaces, algebras, topological vector spaces, and modules, among many other structures or objects.

The term " tensor product " is also used in relation to monoidal categories.

The tensor product V ⊗< sub > K </ sub > W of two vector spaces V and W over a field K can be defined by the method of generators and relations.

( The tensor product is often denoted V ⊗ W when the underlying field K is understood.

The tensor product arises by defining the following four equivalence relations in F ( V × W ):

where v, v < sub > 1 </ sub > and v < sub > 2 </ sub > are vectors from V, while w, w < sub > 1 </ sub >, and w < sub > 2 </ sub > are vectors from W, and c is from the underlying field K. Denoting by R the space generated by these four equivalence relations, the tensor product of the two vector spaces V and W is then the quotient space

A metric tensor g on M assigns to each point p of M a metric g < sub > p </ sub > in the tangent space at p in a way that varies smoothly with p. More precisely, given any open subset U of manifold M and any ( smooth ) vector fields X and Y on U, the real function

The soft palate is tensed by tensor palatini ( Vc ), and then elevated by levator palatini ( pharyngeal plexus — IX, X ) to close the nasopharynx.

The actions of the levator palatini ( pharyngeal plexus — IX, X ), tensor palatini ( Vc ) and salpingopharyngeus ( pharyngeal plexus — IX, X ) in the closure of the nasopharynx and elevation of the pharynx opens the auditory tube, which equalises the pressure between the nasopharynx and the middle ear.

The free algebra on a set X is for practical purposes the same as the tensor algebra on the vector space with X as basis.

Just like SmProj ( k ), the category Corr ( k ) has direct sums () and tensor products ( X ⊗ Y := X × Y ).

If R is a ring and T is a right R-module, we can define a functor H < sub > T </ sub > from the abelian category of all left R-modules to Ab by using the tensor product over R: H < sub > T </ sub >( X ) = T ⊗ X.

The components of this tensor are calculated as the scalar product of tangent vectors X < sub > 1 </ sub > and X < sub > 2 </ sub >:

The individual expressions on the right depend on the choice of the smooth vector fields X and Y, but the left side actually depends only on the pointwise values of X and Y, which is why N < sub > A </ sub > is a tensor.

In mathematics, an invertible sheaf is a coherent sheaf S on a ringed space X, for which there is an inverse T with respect to tensor product of O < sub > X </ sub >- modules.

An invertible sheaf is a coherent sheaf S on a ringed space X, for which there is an inverse T with respect to tensor product of O < sub > X </ sub >- modules, that is, we have

isomorphic to O < sub > X </ sub >, which acts as identity element for the tensor product.

That is, the condition of a tensor inverse then implies, locally on X, that S is the sheaf form of a free rank 1 module over a commutative ring.

Quite generally, the isomorphism classes of invertible sheaves on X themselves form an abelian group under tensor product.

The tensor product of two elements v and w is the equivalence class ( e < sub >( v, w )</ sub > + R ) of e < sub >( v, w )</ sub > in V ⊗ W, denoted v ⊗ w. This notation can somewhat obscure the fact that tensors are always cosets: manipulations performed via the representatives ( v, w ) must always be checked that they do not depend on the particular choice of representative.

The space R is mapped to zero in V ⊗ W, so that the above three equivalence relations become equalities in the tensor product space:

Given a vector x ∈ V and y * ∈ W *, then the tensor product y * ⊗ x corresponds to the map A: W → V given by

* An abelian group A is torsion-free if and only if it is flat as a Z-module, which means that whenever C is a subgroup of some abelian group B, then the natural map from the tensor product C ⊗ A to B ⊗ A is injective.

( Here ⊗ refers to the tensor product over K and id is the identity function.

For a compact topological group, G, there exists a C *- algebra homomorphism Δ: C ( G ) → C ( G ) ⊗ C ( G ) ( where C ( G ) ⊗ C ( G ) is the C *- algebra tensor product-the completion of the algebraic tensor product of C ( G ) and C ( G )), such that Δ ( f )( x, y )

:* There exists a C *- algebra homomorphism Δ: C → C ⊗ C ( where C ⊗ C is the C *- algebra tensor product-the completion of the algebraic tensor product of C and C ) such that for all i, j ( Δ is called the comultiplication );

An E-valued differential form of degree r is a section of the tensor product bundle E ⊗ Λ < sup > r </ sup > T * M. The space of such forms is denoted by

Given central simple algebras A and B, one can look at the their tensor product A ⊗ B as a K-algebra ( see tensor product of R-algebras ).

A module homomorphism M → K is called pure injective if the induced homomorphism between the tensor products C ⊗ M → C ⊗ K is injective for every right R-module C. The module M is pure-injective if any pure injective homomorphism j: M → K splits ( i. e. there exists f: K → M with fj

Note that under the tensor product, symmetric tensors are not a subalgebra: given vectors v and w, they are trivially symmetric 1-tensors, but v ⊗ w is not a symmetric 2-tensor.