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Reality is this phenomenon, which is constantly manifested through our sensations. Unlike feelings, the latter play a pioneering role in understanding and determining it.
Knowing the finer boundary between unreality (virtuality, film, social networks, dreams, etc.) and reality; between the two worlds, the first is well deduced from the second.

But what can we say more concretely of this reality, that our minds swim like a fish in its environment?

Considering this questioning, it is probably possible to give an infinity of answers.
This reality, to which it is almost impossible to renounce those transmitted to us by our various senses, which seems to be made only of mathematics.
In fact, reality is nothing but various mathematical objects, in this sense characterized by spaces; sizes; measurements; surfaces; lengths; distances; of the circle; cubes; spheres; polytopes; matrices; systems; assemblies ... etc.

Life itself is mathematical, for only our life expectations are intervals of time, a period, a mathematical measure. Our senses communicate mathematically to us, a spider is often afraid, because its shape through our eyes, we feel a danger, similar for hearing, touch, taste and smell in various contexts ...etc.

In this case, how are we going to unravel the mysteries of mathematics?

We live in mathematical spaces and always use mathematical objects.
To live without mathematics is to compare to a lifeless body, since nothingness is also a mathematical object (empty space).

Whatever the position of the object, it seems to belong to a space. We know the Euclidean space; hermitian or prehilbertian, which are also algebraic objects.
We have always learned that a linear mapping or a linear transformation of elements belonging to a vector space (ring, abelian group, body, etc.) is delimited by a starting set (which gives the values of the antecedents) (the images or result of the transformations of the variable elements).
In the 0, 1, 2 and 3 dimensions, objects change shapes and are easily perceived by the eye.

In the limit, any object in dimension greater than 3 is hardly observable.
Geometry is the more simplified representation of the numerical or matrix complex of mathematics.
A mathematical object is formalized by this transformation on elements identifiable by sets and characterized by stops.

Having a base or several vertices, displacement or mobility in search of an ideal environment in a space is equivalent to changing from one position to another (vector space). To do this, we will need locomotion, is nothing more than a base made up of the family of elements (eigenvectors associated with the values or solutions of the system) free and generating made possible by the diagonalization (or triangularization and through the matrix of passage ), which consists in bringing a set (endomorphism) into a diagonal (trigonal) matrix.

The latter is translated by the homothety (movement on the same direction) or the rotation (from one direction to another) in the whole of the vector space.
Indeed, geometrically, to move is to go from one point to another represents a distance.
Physically, it amounts to calculating the determinant (or inversibility) of a matrix with the unit matrix of the same dimension with an unknown.
The new position is known by the values of these unknowns fixed in the set and which will allow the installation of the elements (vectors) of the base.
One manages to move in a space, while changing base by the financial means of diagonalization ...

Abdi-Basid ADAN,02, 2017