# General Definition Consider a point $P \in M$ where $M$ is some [[manifold]]. A *tensor of type $(N, N')$* at $P$ is defined to be a multilinear [[map|function]] which takes as an argument $N$ [[one-form|one-forms]] and $N'$ [[Vector|vectors]], whose value is a real number. If we want to speak of e.g. a (2,2) tensor $\overline{F}$ without specifying its arguments, we may write $\overline{F}(., .; . , .)$. Note that [[Vector|vectors]], as a special case of tensors, are $(1, 0)$ tensors, while [[one-form|one-forms]] are $(0, 1)$ tensors. By convention, a scalar function on the [[manifold]] is taken to be a type $(0,0)$ tensor. In older literature, tensors were defined by their [[Basis Transformation|transformation rules]] under a change of basis. # Examples of Tensors A common example of a $(1, 1)$ tensor is a matrix in linear algebra. If column vectors are vectors and row vectors are one-forms, then a matrix is a $(1,1)$ tensor, since multiplying it by a vector gives a vector, and letting it operate on both in the usual way gives a number. An orthogonal transformation (e.g. in $O(3)$) takes a matrix into another matrix, i.e. a $(1,1)$ tensor into another $(1,1)$ tensor. Thus, it is a $(2,2)$ tensor.