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 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.
Definition via tensor products of vector spaces 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. 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:
V ⊗ ⋯ ⊗ V ⊗ V ∗ ⊗ ⋯ ⊗ V ∗ {\displaystyle V\otimes \cdots \otimes V\otimes V^{*}\otimes \cdots \otimes V^{*}}
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 m and covariant of order n and of total order m + n. The tensors of order zero are just the scalars (elements of the field F), those of contravariant order 1 are the vectors in V, and those of covariant order 1 are the one-forms in V∗ (for this reason, the elements of the last two spaces are often called the contravariant and covariant vectors). The space of all tensors of type (m, n) is denoted
T n m ( V ) = V ⊗ ⋯ ⊗ V ⏟ m ⊗ V ∗ ⊗ ⋯ ⊗ V ∗ ⏟ n . {\displaystyle T_{n}^{m}(V)=\underbrace {V\otimes \dots \otimes V} _{m}\otimes \underbrace {V^{*}\otimes \dots \otimes V^{*}} _{n}.}
Example 1. When V {\displaystyle V} is finite-dimensional, the space of type (1, 1) tensors, T 1 1 ( V ) = V ⊗ V ∗ , {\displaystyle T_{1}^{1}(V)=V\otimes V^{*},} is isomorphic in a natural way to the space of linear transformations from V to V. Example 2. When V {\displaystyle V} is finite-dimensional, a bilinear form on a real vector space V, V × V → F , {\displaystyle V\times V\to F,} corresponds in a natural way to a type (0, 2) tensor in T 2 0 ( V ) = V ∗ ⊗ V ∗ . {\displaystyle T_{2}^{0}(V)=V^{*}\otimes V^{*}.} An example of such a bilinear form may be defined, termed the associated metric tensor, and is usually denoted g.
Tensor rank
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:
A = v w T . {\displaystyle A=vw^{\mathrm {T} }.}
The rank of a matrix A {\displaystyle A} is the minimum number of rank one matrices that sum to A {\displaystyle A} :
A = v 1 w 1 T + ⋯ + v k w k T . {\displaystyle A=v_{1}w_{1}^{\mathrm {T} }+\cdots +v_{k}w_{k}^{\mathrm {T} }.}
A simple tensor (also called a tensor of rank one, elementary tensor or decomposable tensor) is a tensor that can be written as a product of tensors of the form
T = a ⊗ b ⊗ ⋯ ⊗ d {\displaystyle T=a\otimes b\otimes \cdots \otimes d}
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