# Questions about the basics of coordinate systems and their basis

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 $$\vec{e_v}=\vec{e_y}$$?