Coordinates of a matrix

Question

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?

Answer 1

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

Answer 2

The coordinates of a vector with respect to an ordered basis are unique so there is only one correct answer. In your case,

$$ \begin{pmatrix} a & b \\ c & d \end{pmatrix} = (a + d - c) \begin{pmatrix} 1 & 0 \\ 0 & 0 \end{pmatrix} + b \begin{pmatrix} 0 & 1 \\ 0 & 0 \end{pmatrix} + d \begin{pmatrix} 0 & 0 \\ 1 & 1 \end{pmatrix} + (c - d) \begin{pmatrix} 1 & 0 \\ 1 & 0 \end{pmatrix} $$

so

$$ \begin{pmatrix} a & b \\ c & d \end{pmatrix}_{B} = \left[ \begin{matrix} a + d - c \\ b \\ d \\ c - d \end{matrix} \right]. $$