On 11/9/2025 3:49 PM, Alan Grayson wrote:
On Sunday, November 9, 2025 at 2:07:30 PM UTC-7 Brent Meeker wrote: On 11/9/2025 1:11 AM, 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*A tensor is a geometric object (possibly in an abstract space). It transforms covariantly; which means that changes in coordinates (even non-linear changes in coordinates) leave it the same. * **Round and round we go, but what a tensor is remains elusive! Please define the property that allows it to transform covariantly. Is it a map represented by a matix, and if so, what property must it have that allows it to transform covariantly? AG*
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