 # All bases considered in these are assumed to be ordered bases. Daniaal Sanchez 2021-02-21 Answered

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. V is ${R}^{2},S=\left\{\left[\begin{array}{c}1\\ 0\end{array}\right]\left[\begin{array}{c}0\\ 1\end{array}\right]\right\},v=\left[\begin{array}{c}3\\ -2\end{array}\right]$

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We are given the following ordered basis S for the vector space $V={R}^{2}$as well as the following vector v in V: $S=\left\{\left[\begin{array}{c}1\\ 0\end{array}\right]\left[\begin{array}{c}0\\ 1\end{array}\right]\right\};v=\left[\begin{array}{c}3\\ -2\end{array}\right]$ We have to compute the coordinate vector [v]_s of v with respect to the basis S. We have $\left[v{\right]}_{s}=\left[\begin{array}{c}a\\ b\end{array}\right]$ Where $v=\left[\begin{array}{c}3\\ -2\end{array}\right]$
$=a\left[\begin{array}{c}1\\ 0\end{array}\right]+b\left[\begin{array}{c}0\\ 1\end{array}\right]$
$=\left[\begin{array}{c}a\\ b\end{array}\right]$ Equating components yields the following linear system: $a=3\phantom{\rule{1em}{0ex}}\text{and}\phantom{\rule{1em}{0ex}}b=-2$ Threfore, the coordinate vector ${\left[v\right]}_{S}$ of v with respect to the basis S is $\left[v{\right]}_{S}=\left[\begin{array}{c}3\\ -2\end{array}\right]$