Editing Tensor (intrinsic definition) (section)

{{Short description|Coordinate-free definition of a tensor}}
{{hatnote|This article assumes an understanding of the [[tensor product]] of [[vector space]]s without chosen [[Basis (linear algebra)|bases]]. An introduction to the nature and significance of tensors in a broad context can be found in the main [[Tensor]] article.}}
{{More footnotes needed|date=November 2024}}

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_map|multilinear]] concept. Their  properties can be derived from their definitions, as [[linear map]]s 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|date=May 2020}} 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.