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Change of basis
A vector does not change when we choose a different basis; only its coordinates change. A change-of-basis matrix converts one coordinate description into another.
Suppose $B=(\mathbf b_1,\ldots,\mathbf b_n)$ is a basis of $\mathbb R^n$. Form the matrix
$$P_B=\begin{pmatrix}\mathbf b_1&\cdots&\mathbf b_n\end{pmatrix},$$
whose columns are the basis vectors written in standard coordinates.
Then
$$\mathbf v=P_B[\mathbf v]_B.$$
From standard coordinates to basis coordinates
Because $P_B$ is invertible,
$$[\mathbf v]_B=P_B^{-1}\mathbf v.$$
For example, with
$$B=((1,1),(1,-1)),$$
we have
$$P_B=\begin{pmatrix}1&1\1&-1\end{pmatrix}.$$
The vector $(4,2)$ has basis coordinates $(3,1)$ because
$$\begin{pmatrix}4\2\end{pmatrix}
P_B\begin{pmatrix}3\1\end{pmatrix}.$$
Between two arbitrary bases
If $B$ and $C$ are two bases, coordinates can be converted by first interpreting the $B$ coordinates as the underlying vector and then expressing that vector in $C$ coordinates:
$$[\mathbf v]_C=P_C^{-1}P_B[\mathbf v]_B.$$
The matrix $P_C^{-1}P_B$ is therefore the change-of-coordinates matrix from basis $B$ to basis $C$.
Changing basis is a change of description, not a transformation of the underlying vector. This distinction becomes especially important when the same linear transformation is represented by different matrices.