On Thursday, November 13, 2025 at 11:14:13 PM UTC-7 Russell Standish wrote:
On Sat, Nov 08, 2025 at 05:25:16AM -0800, 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 The operation of matrix multiplication is linear over the vectors on the right hand side. Is that what you're missing? *Suppose T is a tensor which operates on vector v in some vector space. Then T(v)* *maps to a real number and is linear in v. If we change coordinates, v's coordinates will* *change but v remains the same, so T is invariant wrt coordinate transformations.* *Now let's look at the situation using matrix representation and assume we're dealing* *with a 4 dimensional spacetime manifold. Then T can be represented by a 4x4 matrix,* *and v can be represented as a column matrix with entries ct, x, y, z. When T operates * *on v, we get another column vector, and not a real constant! What am I doing wrong? * *Doesn't every tensor map to a real number? Also, if T is a function of two vectors, **T(u,v),* *then using matrix notation, how is T evaluated to get a real number as the result? **Do * *we model u as a column matrix and v as a row matrix? Thanks for your time. AG* ---------------------------------------------------------------------------- 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/c749d018-e829-4aab-83b1-c3e1523c9ccfn%40googlegroups.com.
