Let's consider a vector space V with $\dim(V)=3$. By taking the basis S = ($\vec{i}, \vec{j}, \vec{k}$) with $\vec{i}=\begin{bmatrix}1\\0\\0\end{bmatrix}$ , $\vec{j} = \begin{bmatrix}0\\1\\0\end{bmatrix}$ and $\vec{k} = \begin{bmatrix}0\\0\\1\end{bmatrix}$, we can express the vector $\vec{v} = 2\vec{i}+2\vec{j}+2\vec{k}$ in the $S$ basis as: $\vec{v}_S= \begin{bmatrix}2\\2\\2\end{bmatrix}$
However, if we set the basis $S'$ = ($2\vec{i}, 2\vec{j}, 2\vec{k}$), the coordinate vector of $\vec{v}$ relative to the basis $S'$ will be: $\vec{v}_{S'} = \begin{bmatrix}1\\1\\1\end{bmatrix}$.
I am not sure to understand, tell me if I am wrong: Coordinate vectors can be different even if they are coordinate vectors of a same vector? And on the other way, can coordinate vectors be the same even if they're coordinate vectors of different two vectors? Do they just depend on basis, not the vector's physical value?