**Tensor (intrinsic Definition)**

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 multi-linear concept. Their well-known 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 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 heavily 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.

Read more about Tensor (intrinsic Definition): Definition Via Tensor Products of Vector Spaces, Tensor Rank, Universal Property, Tensor Fields, Basis