On 11/9/2025 5:24 PM, Alan Grayson wrote:
On Sunday, November 9, 2025 at 6:16:15 PM UTC-7 Alan Grayson wrote:
On Sunday, November 9, 2025 at 5:12:54 PM UTC-7 Russell Standish
wrote:
On Sun, Nov 09, 2025 at 03:56:16PM -0800, Alan Grayson wrote:
>
> 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
I did explain that in my post if you read it. In an inner product
space, every vector is isomorphic to a linear map from the
space to
its field. Eg R^n->R in the case of the space R^n. That linear
map is
the rank 1 tensor. In mathematics, something walks and quacks
like a
duck is a duck.
Even the inner product operation is an example of a bilinear map,
hence a rank 2 tensor. In Minkowski spacetime, the inner
product is
known as the Levi-Civita tensor.
So a tensor is nothing more than a multi linear map to the reals?
But if
we represent a tensor by a matrix, will it be automatically invariant
under coordinate transformations? Do we need an inner product space
to define a tensor? TY, AG
If the tensor, represented by a matrix, is "unchanged" under a coordinate
transformation, does this mean its determinant is unchanged? AG
No, in general it transforms like a density. So it's only unchanged if
the determinant of the transformation matrix is 1
Brent
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