Answer
The desired matrix is
$\mathbf {A}=\mathbf {B^{-1}}$
Therefore $\mathbf {B^{-1}}$ is the matrix that implements the change of basis and is the coordinate mapping.
Work Step by Step
If we write $\mathbf { x }$ in terms of basis $b_{i}$ then we can find $c_{i}$
$\mathbf { x } = c _ { 1 } \mathbf { b } _ { 1 } + \cdots + c _ { n } \mathbf { b } _ { n }$
This can be written as:
$x =\mathbf { B c}$
Since the columns of $\mathbf {B}$ are linearly independent, then to perform the transformation you can take the inverse of $\mathbf {B}$ on both sides.
$c = \mathbf {B^{-1}x}$
The desired matrix is
$\mathbf {A}=\mathbf {B^{-1}}$
Therefore $\mathbf {B^{-1}}$ is the matrix that implements the change of basis and is the coordinate mapping.
