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Since I think I am missing some basic understanding about coordinate systems and their basis, I would really appreciate your help answering my questions. Also if something isn't written mathematically correct, a hint would be nice :)

Let's say we have a set of points in a 2D Cartesian system $F_{qp}=u_q\cdot \vec{e_u}+v_p \cdot \vec{e_v}$.

  1. These points can also be written as $F_{qp}(u_q,v_p)$, right?

Now let's say we want to shift these points by $t_x$ in the $x$-direction and by $t_y$ in the $y$-direction. This leads to the following equation:

$\begin{bmatrix} x_k\\y_j \end{bmatrix}=\begin{bmatrix} u_q\\v_p \end{bmatrix}+\begin{bmatrix} t_x\\t_y \end{bmatrix} \ \ \ \ $ (1)

  1. The points with components $x_k$ and $y_k$ do also have the basis $\vec{e_u}, \vec{e_v}$, right?

  2. Now, if I am saying that I have some input points $F_{qp}=u_q\cdot \vec{e_u}+v_p \cdot \vec{e_v}$ which will be transformed to another set of points $G_{kj}=x_k\cdot \vec{e_x}+y_j \cdot \vec{e_y}$ (note the different unit vectors) is the equation (1) still correct? I am asking this because I declared the unit vectors different.

  3. If equation (1) is still correct, does that mean that $\vec{e_u}=\vec{e_x}$ and $\vec{e_v}=\vec{e_y}$?

Thanks in advance.

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