Rotating a point vs rotating coordinate system Let's suppose we rotate point $P$ by angle $\alpha$ to obtain point $P'$. How to prove that coordinates of point $P$ are $P'$ in the new coordinate system realized by rotating our coordinate axes by $-\alpha$? Is it something that needs to be proven? Or can we just accept it because it's obvious?
While the formula for rotating a point around origin is simple, and the proof does take all  four quadrants of coordinate system into account, the proof presented here of the effect rotating the axes has on point coordinates works only for the first quadrant (and only if we don't cross point $P$ while move the $X$ axis - meaning this proof is valid only if the angle between $|OP|$ and $X$ is greater than $\alpha$ on the figure).
 A: Active rotation of the vector $x = \vec{OP}$ to a new vector $x'$:
$$
x' = A x
$$
where $A$ is the matrix of the rotation.
We want to get the same new coordinates $x_i'$ by a change of point of view, the vector stays the same, by a linear transformation of the coordinate system, where the base vectors $e_i$
get transformed into new base vectors $e_i'$ according to
$$
e_i' = B e_i
$$
for $i \in \{ 1, \ldots, n \}$.
\begin{align}
\underbrace{A^{-1} x'}_{(4)} = x
&= \underbrace{\sum_{i=1}^n x_i e_i}_{(1)}
\\
&=^! \underbrace{\sum_{i=1}^n x_i' e_i'}_{(2)}
= \sum_{i=1}^n x_i' B e_i
= \sum_{i=1}^n B x_i' e_i
= B \sum_{i=1}^n x_i' e_i
= \underbrace{B x'}_{(3)}
\end{align}
which leads to $B = A^{-1}$.
$(1)$ vector $x$ as linear combination of original base vectors, having coordinates $x_i$
$(2)$ the same vector $x$ expressed as linear combination of new base vectors, having new coordinates $x_i'$
$(3)$ the effect of $B$ (interpreted as active transformation within the old coordinate system) is $B x' = x$
$(4)$ this should be consistent with the active transformation $A x = x'$
