Find a change of coordinates matrix

Question

Find the change of coordinates matrix that changes coordinates in the basis $1$, $1+t$ in $P_1$ to the coordinates in $1-t$, $2t$.

Answer 1

Write $1$ as $a \cdot (1-t)+b \cdot 2t$, then first column of matrix is $\begin{vmatrix} a \\ b \end{vmatrix}$. Next write $1+t$ as $c \cdot (1-t)+d \cdot 2t$, second column of matrix is $\begin{vmatrix} c \\ d \end{vmatrix}$ .