Past exam of the mathematics course of the University of Oxford / 2015 / Prelims / Mathematics I / 3 / a

Choose the rotation taking $e_1$ to $e_2$, $e_2$ to $-e_1$ and $e_3$ to itself. The columns of the matrix of a linear map are these images of the standard basis, soThis fixes the orientation used throughout this question.

Using the convention ${\chi }_T(x)=\det (xI-A)$, the characteristic polynomial isSince $x^2+1>0$ for every $x\in \mathbb{R}$, the only eigenvalue over the real numbers is $1$. Solving $(A-I)v=0$ givesThus the only eigenspace over the real numbers is the axis of the rotation.