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Coordinates in a basis
A basis gives a coordinate system inside a vector space. If
$$B=(\mathbf b_1,\ldots,\mathbf b_n)$$
is an ordered basis, every vector $\mathbf v$ can be written uniquely as
$$\mathbf v=c_1\mathbf b_1+\cdots+c_n\mathbf b_n.$$
The coefficient list is the coordinate vector of $\mathbf v$ relative to $B$:
$$[\mathbf v]_B= \begin{pmatrix}c_1\ \vdots\ c_n\end{pmatrix}.$$
The vector and its coordinates are different objects
In $\mathbb R^2$, let
$$B=((1,1),(1,-1)).$$
The vector
$$\mathbf v=(4,2)$$
satisfies
$$\mathbf v=3(1,1)+1(1,-1),$$
so
$$[\mathbf v]_B=\begin{pmatrix}3\1\end{pmatrix}.$$
The vector is still $(4,2)$ in the plane; $(3,1)$ is only its description in the basis $B$.
Why order matters
If the basis vectors are reordered, the coordinate entries must be reordered as well. Coordinates therefore belong to an ordered basis.
Choosing a basis converts abstract vectors into coordinate columns and allows vector-space problems to be handled with matrices.