On Sunday, November 9, 2025 at 3:06:45 PM UTC-7 Russell Standish wrote:
On Sun, Nov 09, 2025 at 01:11:47AM -0800, Alan Grayson wrote: > > > On Saturday, November 8, 2025 at 6:25:17 AM UTC-7 Alan Grayson wrote: > > In some treatments of tensors, they're described as linear maps. So, in GR, > if we have a linear map described as a 4x4 matrix of real numbers, which > operates on a 4-vector described as a column matrix with entries (ct, x, y, > z), which transforms to another 4-vector, what must be added in this > description to claim that the linear transformation satisfies the > definition of a tensor? TY, AG > > > Let's call the linear transformation T, then the answer to my question might be > that T is a tensor iff it has a continuous inverse. I'm not sure if this is > correct, but I seem to recall this claim in a video about tensors I viewed in > another life. But even if it's true, it seems to conflict with the claim that > an ordinary vector in Euclidean space is a tensor because it's invariant under > linear (?) transformations. In this formulation, it is the argument of T, which > we can call V, which is invariant, not the map T. I'd appreciate it if someone > here could clarity my confusion. TY, AG It's got nothing to do with being invertible (which is the conjunction of being 1:1 and onto). Rather a tensor is a multilinear map - *If it's a map, how can an ordinary vector in Euclidean space be a tensor?* *Such vectors are NOT maps! See my problem? AG* is a map with multiple arguments, and linear in each. Obviously a standard linear map R^n -> R^n is a rank 2 tensor. We recognise them generally as matrices. Vectors correspond to linear maps by means of transposing them and forming the inner product, ie a linear map from R^n->R, and are rank 1 tensors as a result. > Dr Russell Standish Phone 0425 253119 (mobile) Principal, High Performance Coders [email protected] http://www.hpcoders.com.au ---------------------------------------------------------------------------- -- You received this message because you are subscribed to the Google Groups "Everything List" group. To unsubscribe from this group and stop receiving emails from it, send an email to [email protected]. To view this discussion visit https://groups.google.com/d/msgid/everything-list/3216b479-b1ee-4a36-9b17-3011529404b4n%40googlegroups.com.
