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## Summary:

- Vectors and Tensors types of transformations

## Main Question or Discussion Point

Hello,

I was pondering on the following: a vector is a specific entity whose existence is independent of the coordinate system used to describe it.

To start, I guess I need to state that we are describing the vector from the same reference frame using different coordinate systems (Cartesian, Cartesian but rotated, cylindrical, spherical, elliptical, etc.). These coordinate system examples are all "applied" to the flat 3-dimensional Euclidean space (we are not dealing with curved geometries).

Question: in 2D, using the simple Cartesian coordinate system ##C_1##, an arbitrary vector ##A## has a description ##A= [A_1, A_2]## where the coordinates are the scaling for the basis vectors. The coordinates ##A_1## and ##A_2## varies if we change coordinate system, for example, using a coordinate system ##C_2## with the same origin ##O_1=O_2## but a rotated version of ##C_1##. So far so good.

There are coordinate transformations (2x2 matrices), like rotations, that just change the descriptions of the vector without changing its length or actual direction. What is the category of these types of transformations called? The group of rotations is an example. Any other examples? A translated coordinate system ##C_3## has a different origin than ##C_1## but parallel axes.

There are also transformations that change the length and/or direction of a vector converting it into a totally new vector. How do we call those transformations?

Thank you!!

I was pondering on the following: a vector is a specific entity whose existence is independent of the coordinate system used to describe it.

To start, I guess I need to state that we are describing the vector from the same reference frame using different coordinate systems (Cartesian, Cartesian but rotated, cylindrical, spherical, elliptical, etc.). These coordinate system examples are all "applied" to the flat 3-dimensional Euclidean space (we are not dealing with curved geometries).

Question: in 2D, using the simple Cartesian coordinate system ##C_1##, an arbitrary vector ##A## has a description ##A= [A_1, A_2]## where the coordinates are the scaling for the basis vectors. The coordinates ##A_1## and ##A_2## varies if we change coordinate system, for example, using a coordinate system ##C_2## with the same origin ##O_1=O_2## but a rotated version of ##C_1##. So far so good.

There are coordinate transformations (2x2 matrices), like rotations, that just change the descriptions of the vector without changing its length or actual direction. What is the category of these types of transformations called? The group of rotations is an example. Any other examples? A translated coordinate system ##C_3## has a different origin than ##C_1## but parallel axes.

There are also transformations that change the length and/or direction of a vector converting it into a totally new vector. How do we call those transformations?

Thank you!!