Coordinates of a matrix So on the textbook, it gives an example: 
If the basis of B matrix is{$\begin{bmatrix}1&0\\ 0&0\end{bmatrix}, \begin{bmatrix}0&1\\ 0&0\end{bmatrix},\begin{bmatrix}0&0\\ 1&1\end{bmatrix}, \begin{bmatrix}1&0\\ 1&0\end{bmatrix}$} 
Then the B-coordinates of a matrix $\begin{bmatrix}a&b\\ c&d\end{bmatrix}$ are $\begin{bmatrix}a-c+d\\ b\\d\\c-d\end{bmatrix}$
But can I write the B-coordinates as $\begin{bmatrix}a-d\\ b\\c-d\\d\end{bmatrix}$? I found it much easier to write the column matrix in this way, but is it correct?

$$(a-c+d)\begin{bmatrix}1&0\\ 0&0\end{bmatrix}+b \begin{bmatrix}0&1\\ 0&0\end{bmatrix}+d\begin{bmatrix}0&0\\ 1&1\end{bmatrix}+(c-d)\begin{bmatrix}1&0\\ 1&0\end{bmatrix}=\begin{bmatrix}a&b\\ c&d\end{bmatrix},$$
while
$$(a-d)\begin{bmatrix}1&0\\ 0&0\end{bmatrix}+b \begin{bmatrix}0&1\\ 0&0\end{bmatrix}+(c-d)\begin{bmatrix}0&0\\ 1&1\end{bmatrix}+d\begin{bmatrix}1&0\\ 1&0\end{bmatrix}=\begin{bmatrix}a&b\\ c&{\color{red}{c-d}}\end{bmatrix}$$
