All bases considered in these are assumed to be ordered bases. In Exercise, compute coordinate vector v with respect to the giving basis S for V.

We are given the following ordered basis S for the vector space$V=P_1$ as well as the following vector v in V:$S=t+1,t-2,v=t+4$We have to compute the coordinate vector $[v{]}_S,$ of v with respect to the basis S.We have$v=t+4$
$=a(t+1)+b(t-2)$
$=(a+b)t+(a-2b).$Equating coefficients yields the following linear system:$a+b=1$
$a-2b=4$The associated augmented matrix for this system is$A^{\prime }=\left[\begin{array}{ccc}1& 1& 1\\ 1& -2& 4\end{array}\right]$Subtracting the first row from the second yields$\left[\begin{array}{ccc}1& 1& 1\\ 0& -3& 3\end{array}\right]$Dividing the second row by -3 yields$\left[\begin{array}{ccc}1& 1& 1\\ 0& 1& -1\end{array}\right]$Finally, subtracting the second row from the first yields the following reduced row echelon form of$A^{\prime }:$
$A_R^{\prime }=\left[\begin{array}{ccc}1& 0& 2\\ 0& 1& -1\end{array}\right]$Thus, we have the solution$a=2$
$b=-1$Therefore, the coordinate vector $[v{]}_S$ of v respect to the basis S is $[v{]}_S=\left[\begin{array}{c}2\\ -1\end{array}\right]$