In mathematics, the modern component-free approach to the theory of a tensor views a tensor as an abstract object, expressing some definite type of multilinear concept.
Their properties can be derived from their definitions, as linear maps or more generally; and the rules for manipulations of tensors arise as an extension of linear algebra to multilinear algebra.
In differential geometry, an intrinsic[definition needed] geometric statement may be described by a tensor field on a manifold, and then doesn't need to make reference to coordinates at all.
The same is true in general relativity, of tensor fields describing a physical property.
The component-free approach is also used extensively in abstract algebra and homological algebra, where tensors arise naturally.
Given a finite set {V1, ..., Vn} of vector spaces over a common field F, one may form their tensor product V1 ⊗ ... ⊗ Vn, an element of which is termed a tensor.
The space of all tensors of type (m, n) is denoted
is isomorphic in a natural way to the space of linear transformations from V to V. Example 2.
A bilinear form on a real vector space V,
corresponds in a natural way to a type (0, 2) tensor in
A nonzero order 0 or 1 tensor always has rank 1.
The rank of a non-zero order 2 or higher tensor is less than or equal to the product of the dimensions of all but the highest-dimensioned vectors in (a sum of products of) which the tensor can be expressed, which is dn−1 when each product is of n vectors from a finite-dimensional vector space of dimension d. The term rank of a tensor extends the notion of the rank of a matrix in linear algebra, although the term is also often used to mean the order (or degree) of a tensor.
The rank of a matrix is the minimum number of column vectors needed to span the range of the matrix.
A matrix thus has rank one if it can be written as an outer product of two nonzero vectors:
The rank of a matrix A is the smallest number of such outer products that can be summed to produce it:
The rank of a tensor of order 2 agrees with the rank when the tensor is regarded as a matrix,[3] and can be determined from Gaussian elimination for instance.
The rank of an order 3 or higher tensor is however often very difficult to determine, and low rank decompositions of tensors are sometimes of great practical interest.
[4] In fact, the problem of finding the rank of an order 3 tensor over any finite field is NP-Complete, and over the rationals, is NP-Hard.
[5] Computational tasks such as the efficient multiplication of matrices and the efficient evaluation of polynomials can be recast as the problem of simultaneously evaluating a set of bilinear forms
If a low-rank decomposition of the tensor T is known, then an efficient evaluation strategy is known.
can be characterized by a universal property in terms of multilinear mappings.
Amongst the advantages of this approach are that it gives a way to show that many linear mappings are "natural" or "geometric" (in other words are independent of any choice of basis).
Explicit computational information can then be written down using bases, and this order of priorities can be more convenient than proving a formula gives rise to a natural mapping.
Another aspect is that tensor products are not used only for free modules, and the "universal" approach carries over more easily to more general situations.
A scalar-valued function on a Cartesian product (or direct sum) of vector spaces
The universal characterization of the tensor product implies that, for each multilinear function
(where W can represent the field of scalars, a vector space, or a tensor space) there exists a unique linear function
Using the universal property, it follows, when V is finite dimensional, that the space of (m, n)-tensors admits a natural isomorphism
Each V in the definition of the tensor corresponds to a V∗ inside the argument of the linear maps, and vice versa.
Differential geometry, physics and engineering must often deal with tensor fields on smooth manifolds.