^^ [[Schutz - 2 - Differentiable Manifolds and Tensors|2. Differentiable Manifolds and Tensors]] << [[Schutz - 2.22 - Tensors and Tensor Fields|2.22 Tensors and Tensor Fields]] | [[Schutz - 2.24 - Components of Tensors and the Outer Product|2.24 Components of Tensors and the Outer Product]] >> # 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.