Exercise 6.11.18

Let $\beta =\left\{x,x^{\prime }\right\}$ be an orthonormal basis of $V$. Since $\parallel y\parallel =1$, we may write ${\varphi }_{\beta }(y)=(\cos <!--\; FUNCTION\; APPLICATION\; -->\varphi ,\sin <!--\; FUNCTION\; APPLICATION\; -->\varphi )$ for some angle $\varphi$. Let

and $T$ be the transformation with ${[T]}_{\beta }=A$. We have that $T(x)=y$ and $T$ is a rotation.

On the other hand, by the definition of a rotation, we must have

and

Thus we must have $\cos <!--\; FUNCTION\; APPLICATION\; -->\varphi =\cos <!--\; FUNCTION\; APPLICATION\; -->𝜃$ and $\sin <!--\; FUNCTION\; APPLICATION\; -->\varphi =\sin <!--\; FUNCTION\; APPLICATION\; -->𝜃$. If $0\le \varphi ,𝜃<2\pi$, we must have $\varphi =𝜃$. So the rotation is unique.