Tensor product is the first general construction to understand in the algebra. This Demonstrates ho the Algebra is multiplexed from Vectors, just as Matrices are multiplexed or bundled from vectors
Tensor Algebra then follows.
The fact that Tensors are constructed is a way of describing the structure of a tensor as picked out by perceptual processes. The perception of a tensor is still forming, and i use the term to describe the collection of attributes foud in a measuring tool for space,
Tensors like Matrices have a wider use than geometry, but that need only concern us as it helps to define a tensor's geometrical application.
So a tensor product is a fearfu thing! and rightly so as it is as abstract as hell and a good deal colder!
But before we get to that the reliance on multiplexing to quantify a vector means that i have to reflect on reflection.
Addressing the issue of the general directed magnitude, which is a linear combination of directed magnitudes of trigonometric ratios: for example in the plane acos0+bsin(π/2) where general algebraic rule applies and different kinds cannot be condensed into one;
I have already included the notioin of rotation and scalar magnitude and orientation. Reflection arises as a kind of rotation through π, but also as a "opposite" through a point. Thus a mirror reflection has to be constructed from these rotational opposites through points or point reflections.
Now prior to this i had put a block on reflection at the vector level, but i withdraw that block on reflection!, as i can see that directed magnitudes at all levels exhibit this attribute.
The interesting thing is that as i expand the definition of a vector to include rotation and reflection through a point, i also explain the weird attribute f mirror reflection as the summetria arrived at through reflection in a point to an opposite rotational centre. Because it is a reflection in a point no other point is atributable and the "sum" or aggregate result or summetria of all such reflections is precisely a mirror reflection. This means that although rotation around each individual point is sensible the whole is not defined by rotation about a single point, but by rotations around multiple centres of rotation.
Thus rotationally it is as if the object in the mirror was formed by every point rotatinf in a point in the mirror irrespective of all the others, maintaining only there relative positions. Such a transformation is precisely a mirror reflection, and may be exhibited by disconnected regions which nevertheless maintain their spatial distance relationships even though they have a different rotational relationship.
In general, with directed magnitudes, the Bombelli operator has to be rigorously described in terms of rotational references as well: thus a ++ or — is within the same radial orientation, but a +- or-+ are in radial orientations exactly π radians from each other.
The product of these orientations has to be measured in terms of reflection through a point and the -*- is first a reflection in the initial centre and then a reflection in the rotated by π/2 centre. This produces a + but rotates by π around the centre. Thus physically we should expect rotations by π when we multiply 2 directed magnitudes which are π reflections through a point and π/2 out of phase rotationally.
Given then reflection in a point, i would expect to find it in regionally distributed spaces with freedom of relative movement between the regions. I would not expect it in a rigid body or rigid "structure", where the relative relationships are fixed. It therefore constitutes a dynamic property attributable to systems in dynamic flux or dynamic equilibrium, amongst which i would include cell mitosis and plasma dynamical situations in outer space, or fluid motions in hydrodynamics etc.
Back to Tensors.
If we construct measuring tools and recording structures, like matrices from the trigonometric dynamic magnitudes we should expect to preserve the fundamental attributes of rotation, rotational extension/expansion and reflection. Thus to derive a relativistic measure of general utility we need to multiplex two basis together and tht is precisely what a tensor product describes how to do, to preserve the trigonometric ratio values and attributes. We do not need to worry about bilinear projections etc etc at this stage. All i want to be able to do is to use trigonometic ratios to measure magnitudes that are dynamic and to record them in an appropriated form.