I know how to find the coordinate vector respect to a vector matrix. However, in my textbook, I see the following problem, which asks to find the coordinate vector of M respect to a matrix basis:
$M=\begin{bmatrix}1 & 1\\0& 1\end{bmatrix}\in Mat(2,2,\mathbb{Q})$
$\mathcal{B} _ { 1 } = \left\{ A = \left[ \begin{array} { c c } { 1 } & { 0 } \\ { 0 } & { 1 } \end{array} \right] , B = \left[ \begin{array} { c c } { 0 } & { 1 } \\ { 0 } & { 0 } \end{array} \right] , C = \left[ \begin{array} { c c } { 0 } & { 0 } \\ { 1 } & { 0 } \end{array} \right] , D = \left[ \begin{array} { c c } { 1 } & { 0 } \\ { 0 } & { – 1 } \end{array} \right] \right\}$
How can I proceed? Can I explicit the matrix components as column vectors?
Best Answer
Start by setting up the equation you want to solve:
$$\begin{bmatrix} 1 & 1 \\ 0 & 1 \end{bmatrix} = a\begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix} + b\begin{bmatrix} 0 & 1 \\ 0 & 0 \end{bmatrix} + c\begin{bmatrix} 0 & 0 \\ 1 & 0 \end{bmatrix} + d \begin{bmatrix} 1 & 0 \\ 0 & -1 \end{bmatrix}.$$
Next, simplify:
$$\begin{bmatrix} 1 & 1 \\ 0 & 1 \end{bmatrix} = \begin{bmatrix} a + d & b \\ c & a - d \end{bmatrix}.$$
Then turn into a system of equations:
\begin{align*} 1 &= a + d \\ 1 &= b \\ 0 &= c \\ 1 &= a - d. \end{align*}
Finally, solve!