Answer
The coordinates of $p$ relative to $B=(5,-4,3)$
Work Step by Step
Given;
Basis,$B=\left\{ 1,1 - t , 2 - 4 t + t ^ { 2 } \right\}$
$p ( t ) = 7 - 8 t + 3 t ^ { 2 }$
We are required to find the coordinates of $p$ relative to $B$
Taking $a_{1},a_{2},a_{3}$ as constants.
We need to satisfy,
$a _ { 1 } ( 1 ) + a_ { 2 } ( 1 - t ) + a _ { 3 } \left( 2 - 4 t + t ^ { 2 } \right) = p ( t ) = 7 - 8 t + 3 t ^ { 2 }$
By equating the coefficients we have.
$\begin{matrix} { a _ { 1 } + a_ { 2 } + 2 a _ { 3 } = 7 } \\ { - a _ { 2 } - 4 a _ { 3 } = - 8 } \\ { a _ { 3 } = 3 } \end{matrix}$
We form the augmented matrix as;
Let, $\mathbf{A}=\begin{bmatrix}1&1&2&7\\0&-1&-4&-8\\0&0&1&3\end{bmatrix}$
Row reducing the augmented matrix $A$:
$\mathbf{A}=\begin{bmatrix}1&1&2&7\\0&-1&-4&-8\\0&0&1&3\end{bmatrix}\sim\begin{bmatrix}1&0&0&5\\0&1&0&-4\\0&0&1&3\end{bmatrix}$
The coordinates of $p$ relative to $B=(5,-4,3)$
