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The vector spaces of a tensor product need not be the same, and sometimes the elements of such a more general tensor product are called "tensors". For example, an element of the tensor product space V ⊗ W is a second-order "tensor" in this more general sense, [29] and an order-d tensor may likewise be defined as an element of a tensor product ...
In machine learning, the term tensor informally refers to two different concepts (i) a way of organizing data and (ii) a multilinear (tensor) transformation. Data may be organized in a multidimensional array (M-way array), informally referred to as a "data tensor"; however, in the strict mathematical sense, a tensor is a multilinear mapping over a set of domain vector spaces to a range vector ...
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.
For example, in the Pascal programming language, the declaration type MyTable = array [1.. 4, 1.. 2] of integer, defines a new array data type called MyTable. The declaration var A: MyTable then defines a variable A of that type, which is an aggregate of eight elements, each being an integer variable identified by two indices.
A simple tensor (also called a tensor of rank one, elementary tensor or decomposable tensor [1]) is a tensor that can be written as a product of tensors of the form = where a, b, ..., d are nonzero and in V or V ∗ – that is, if the tensor is nonzero and completely factorizable. Every tensor can be expressed as a sum of simple tensors.
As a tensor is a generalization of a scalar (a pure number representing a value, for example speed) and a vector (a magnitude and a direction, like velocity), a tensor field is a generalization of a scalar field and a vector field that assigns, respectively, a scalar or vector to each point of space. If a tensor A is defined on a vector fields ...
A (1,0) tensor is a vector. A (0,1) tensor is a covector. A (0,2) tensor is a bilinear form. An example is the metric tensor . A (1,1) tensor is a linear map. An example is the delta, , which is the identity map, or a Lorentz transformation .
Monoidal categories can be seen as a generalization of these and other examples. Every monoidal category may also be viewed as a "categorification" of an underlying monoid, namely the monoid whose elements are the isomorphism classes of the category's objects and whose binary operation is given by the category's tensor product.