Tensor (intrinsic Definition) - Basis

Basis

For any given coordinate system we have a basis {e_i} for the tangent space V (this may vary from point to point if the manifold is not linear), and a corresponding dual basis {ei} for the cotangent space V* (see dual space). The difference between the raised and lowered indices is there to remind us of the way the components transform.

For example purposes, then, take a tensor A in the space

The components relative to our coordinate system can be written

.

Here we used the Einstein notation, a convention useful when dealing with coordinate equations: when an index variable appears both raised and lowered on the same side of an equation, we are summing over all its possible values. In physics we often use the expression

to represent the tensor, just as vectors are usually treated in terms of their components. This can be visualized as an n × n × n array of numbers. In a different coordinate system, say given to us as a basis {e_i′}, the components will be different. If (xi′_i) is our transformation matrix (note it is not a tensor, since it represents a change of basis rather than a geometrical entity) and if (yi_i′) is its inverse, then our components vary per

In older texts this transformation rule often serves as the definition of a tensor. Formally, this means that tensors were introduced as specific representations of the group of all changes of coordinate systems.