Tensors are defined independent of any basis, although they are often referred to by their components in a basis related to a particular coordinate system; those components form an array, which can be thought of as a high-dimensional matrix.
Comment on number box o number box
Eq0@literature.cafe 3 days agoWhere does this definition come from?
All the geometric definitions of tensors I have met always assumed a base, such that a change of coordinate or of parametrization would change the values of the tensor. Unless you define the tensor by its action instead of its values?
Dadifer@lemmy.world 2 days ago
CompassRed@discuss.tchncs.de 2 days ago
That’s exactly correct. It’s similar to how a vector in R^2 is just an arrow with a magnitude and a direction. When you represent that arrow in different bases, the arrow itself isn’t changing, just the list of numbers you use to represent them. Likewise, tensors do not change when you change bases, but their representations as n dimensional grids of numbers do change.