1. ## plz help

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2. ## Re: plz help

Originally Posted by saqifriends
Let $B=\{b_1, b_2\}$ and $C=\{c_1, c_2\}$ be the bases for $\mathbb{R}^2$. Find the change-of-coordinates matrix from $B$ to $C$, and change-of-coordinates matrix from $C$ to $B$ where;

$b_1=\begin{bmatrix}-1\\8\end{bmatrix}\!,\, b_2= \begin{bmatrix}1\\-5\end{bmatrix}\!,\quad c_1=\begin{bmatrix}1\\4\end{bmatrix}\!,\,c_2 = \begin{bmatrix}1 \\ 1 \end{bmatrix}\!.$
${\begin{gathered}C=BT~~\Rightarrow~~ T={B^{-1}}C={\left[\!\!\begin{array}{*{20}{r}}{-1}&1\\8&{-5}\end{array}\!\!\right]^{-1}} \left[\!\!\begin{array}{*{20}{r}}1&1\\4&1\end{array}\!\! \right]= \frac{1}{3}\left[\!\!\begin{array}{*{20}{r}}5&1\\8&1\end{array}\!\! \right] \left[\!\!\begin{array}{*{20}{r}}1&1\\4&1\end{array}\!\! \right]=\ldots= \left[\!\!\begin{array}{*{20}{r}}3&2\\4&3\end{array}\!\! \right]\!.\hfill\\[9pt] B=CT~~\Rightarrow~~ T={C^{-1}}B={\left[\!\!\begin{array}{*{20}{r}}1&1\\4&1\end{array}\!\! \right]^{-1}} \left[\!\!\begin{array}{*{20}{r}}{-1}&1\\8&{-5}\end{array}\!\!\right]= \frac{1}{3}\left[\!\!\begin{array}{*{20}{r}}{-1}&1\\4&{-1}\end{array}\!\!\right] \left[\!\!\begin{array}{*{20}{r}}5&1\\8&1\end{array}\!\! \right]=\ldots= \left[\!\!\begin{array}{*{20}{r}}3&{-2}\\{-4}&3\end{array}\!\!\right]\!.\hfill\\ \end{gathered}}$