We consider here a vector basis $B_R$ of $\mathbb{R}^2$ that is derived from the standard basis
$$ E=\left\{ \left( \begin{array}{c} 1\\ 0\\ \end{array} \right) ,\;\left( \begin{array}{c} 0\\ 1\\ \end{array} \right) \right\} $$
$B_R$ being rotated by 90° in counter clock wise direction gives the new vector basis:
$$ B_R=\left\{ \left( \begin{array}{c} 0\\ 1\\ \end{array} \right) ,\;\left( \begin{array}{c} -1\\ 0\\ \end{array} \right) \right\} $$
So, from my understanding, this new vector basis $B_R$ should transform every vector in standard basis $E$ to new basis $B_R$.
From this matrix we can easily derive its inverse by transposing it: $$ R=\left( \begin{matrix} 0& -1\\ 1& 0\\ \end{matrix} \right) ,R^{-1}=\left( \begin{matrix} 0& 1\\ -1& 0\\ \end{matrix} \right) $$
Question: In terms of vector basis change the matrix $R$ transforms a vectors coordinates from basis $B_R$ to basis $E$, which is opposite to my understanding as I see that it should transforms a vectors coordinates from basis $E$ to basis $B_R$. Can you please show me what is wrong with my reasoning?
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