# 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.

All bases considered in these are assumed to be ordered bases. In Exercise, compute the coordinate vector of v with respect to the giving basis S for V. $Vis{P}_{1},S=t+1,t-2,v=t+4$
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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 $\left[v{\right]}_{S},$ of v with respect to the basis S.We have$v=t+4$
$=a\left(t+1\right)+b\left(t-2\right)$
$=\left(a+b\right)t+\left(a-2b\right).$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 $\left[v{\right]}_{S}$ of v respect to the basis S is $\left[v{\right]}_{S}=\left[\begin{array}{c}2\\ -1\end{array}\right]$