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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.

**Note:**This article assumes an understanding of the tensor product of vector spaces without chosen bases. An overview of the subject can be found in the main tensor article.

Given a finite set { *V*_{1}, ..., *V*_{n} } of vector spaces over a common field *F*, one may form their tensor product *V*_{1} ⊗ ... ⊗ *V*_{n}, an element of which is termed a **tensor**.

A **tensor on the vector space** *V* is then defined to be an element of (i.e., a vector in) a vector space of the form:

where *V*^{∗} is the dual space of *V*.

If there are *m* copies of *V* and *n* copies of *V*^{∗} in our product, the tensor is said to be of **type ( m, n)** and contravariant of order

**Example 1.** The space of type (1, 1) tensors, 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 An example of such a bilinear form may be defined, termed the associated *metric tensor*, and is usually denoted *g*.

A **simple tensor** (also called a tensor of rank one, elementary tensor or decomposable tensor (Hackbusch 2012, pp. 4)) 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. The **rank of a tensor** *T* is the minimum number of simple tensors that sum to *T* (Bourbaki 1989, II, §7, no. 8).

The zero tensor has rank zero. 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 *d*^{n−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:

In indices, a tensor of rank 1 is a tensor of the form

The rank of a tensor of order 2 agrees with the rank when the tensor is regarded as a matrix (Halmos 1974, §51), and can be determined from Gaussian elimination for instance. The rank of an order 3 or higher tensor is however often *very hard* to determine, and low rank decompositions of tensors are sometimes of great practical interest (de Groote 1987). 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

for given inputs *x _{i}* and

The space 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

is multilinear if it is linear in each argument. The space of all multilinear mappings from *V*_{1} × ... × *V _{N}* to

The universal characterization of the tensor product implies that, for each multilinear function

(where can represent the field of scalars, a vector space, or a tensor space) there exists a unique linear function

such that

for all and

Using the universal property, it follows 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. (Note that in the former case, there are *m* copies of *V* and *n* copies of *V*^{*}, and in the latter case vice versa). In particular, one has

Differential geometry, physics and engineering must often deal with tensor fields on smooth manifolds. The term *tensor* is sometimes used as a shorthand for *tensor field*. A tensor field expresses the concept of a tensor that varies from point to point on the manifold.

- Abraham, Ralph; Marsden, Jerrold E. (1985),
*Foundations of Mechanics*(2 ed.), Reading, Mass.: Addison-Wesley, ISBN 0-201-40840-6. - Bourbaki, Nicolas (1989),
*Elements of Mathematics, Algebra I*, Springer-Verlag, ISBN 3-540-64243-9. - de Groote, H. F. (1987),
*Lectures on the Complexity of Bilinear Problems*, Lecture Notes in Computer Science, vol. 245, Springer, ISBN 3-540-17205-X. - Halmos, Paul (1974),
*Finite-dimensional Vector Spaces*, Springer, ISBN 0-387-90093-4. - Jeevanjee, Nadir (2011), "An Introduction to Tensors and Group Theory for Physicists",
*Physics Today*,**65**(4): 64, Bibcode:2012PhT....65d..64P, doi:10.1063/PT.3.1523, ISBN 978-0-8176-4714-8 - Knuth, Donald E. (1998) [1969],
*The Art of Computer Programming vol. 2*(3rd ed.), pp. 145–146, ISBN 978-0-201-89684-8. - Hackbusch, Wolfgang (2012),
*Tensor Spaces and Numerical Tensor Calculus*, Springer, p. 4, ISBN 978-3-642-28027-6.