On Friday, November 21, 2025 at 3:27:17 PM UTC-7 Russell Standish wrote:
On Thu, Nov 20, 2025 at 07:02:53PM -0800, Alan Grayson wrote: > > > I studied linear algebra, but my questions involve tensors. If a tensor > T is defined as a linear function whose domain is a vector space, and > maps to a real number, how does one get a real number from T(u), if we > do the calculation using matrices? Here there is no v, just u. AG A matrix corresponds to a rank 2 tensor, ie T(u,v)∈R. T(u)∈R corresponds to a rank 1 tensor. In matrix notation, a rank 1 tensor is a transposed vector, ie vᵀ for some vector v∈Rⁿ. vᵀu in matrix notation corresponds to v.u (ie dot or inner product of two vectors). I'm seeking an unambiguous definition of a TENSOR. You wrote earlier that a tensor is a MAP whose arguments are VECTORS in a vector space, which MAP to real numbers, and is INVARIANT under changes in coordinate systems. Your definition seems OK, but upon more analysis I find it kind-of vacuous. Firstly, any function which depends on elements in a vector space which are invariant under changes in coordinate systems, will necessarily be invariant under changes in coordinate systems, and it doesn't matter if that function is linear or not in its arguments. So, is a tensor just limited by the condition of linearity of its arguments? The invariance under coordinate transformations is a direct result of what its arguments are, and since vectors are invariant, so the tensor T must also have this property. That is, the invariance property of T is totally dependent on the invariance property of its domain, the invariant vectors in some vector space. TY, AG Moreover, you claim an invariant vector is in fact a tensor of rank 1. What is its MAP? Why do you need to introduce v to evaluate T(u), which is just a function of u? Using matrices, there's no way to get a constant as the result of T(u) (which I assume is a row matrix). I suppose a constant is a tensor of rank 0. What is its MAP, your entity that DEFINES a tensor? TY, 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/5d526512-286f-4d53-b598-c790f7b6c347n%40googlegroups.com.
