Given
v = \begin{pmatrix}X \\
Y \end{pmatrix}
and a basis
\mathcal B = \left\{
\begin{pmatrix} A \\ C\end{pmatrix},
\begin{pmatrix} B \\ D \end{pmatrix}
\right\}.
Calculate the coordinate vector [v]_{\mathcal B} =
\begin{pmatrix} {\color{red}X} \\
{\color{blue}Y} \end{pmatrix} of v
with respect to the basis
\mathcal B .
\color{red} X
=
(D*X-B*Y)/det
\color{blue} Y
=
(A*Y-C*X)/det
We looking for {\color{red}X}
and {\color{blue}Y} to form the unique solution of the system of linear equations
\begin{pmatrix}X \\
Y \end{pmatrix} =
{\color{red}X} \begin{pmatrix} A \\ C\end{pmatrix} +
{\color{blue}Y} \begin{pmatrix} B \\ D\end{pmatrix}.
This can be done either using Gaussian elimination or with the matrix
A=
\begin{pmatrix} A & B \\
C & D \end{pmatrix}
.
This matrix is invertible with \det(A) = det and
A^{-1}=
\dfrac 1{det}\begin{pmatrix} D & -B \\
-C & A \end{pmatrix}.
Write the system of linear equations
\begin{pmatrix}X \\
Y \end{pmatrix} =
{\color{red}X} \begin{pmatrix} A \\ C\end{pmatrix} +
{\color{blue}Y} \begin{pmatrix} B \\ D\end{pmatrix}
as \begin{pmatrix}X \\ Y \end{pmatrix} =
\begin{pmatrix} A & B \\ C & D\end{pmatrix}
\begin{pmatrix} {\color{red}X} \\ {\color{blue}Y} \end{pmatrix}
and multiply by A^{-1} from the left.
This yields \begin{pmatrix} {\color{red}X} \\ {\color{blue}Y} \end{pmatrix} =
\dfrac 1{det}\begin{pmatrix} D & -B \\
-C & A \end{pmatrix}
\begin{pmatrix}X \\ Y \end{pmatrix} =
\begin{pmatrix} {\color{red}SX} \\
{\color{blue}SY} \end{pmatrix}
.