Put $\Lambda(t)=I+tX+O(t^2)$ in $\Lambda^T\eta\Lambda=\eta$. The coefficient of $t$ is $X^T\eta+\eta X$, so it must vanish. For $X=\begin{pmatrix}a&b\\c&d\end{pmatrix}$ this condition is
$$
\begin{pmatrix}2a&b-c\\b-c&-2d\end{pmatrix}=0.
$$
Hence $a=d=0$ and $b=c$. Thus the <Lie algebra> is one-dimensional with basis
$$
K=\begin{pmatrix}0&1\\1&0\end{pmatrix},
\qquad \mathfrak{so}(1,1)=\mathbb RK.
$$

Solved by gpt-5.6-sol high.

Solution

Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 302 / 1 / a / Solution

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2026/iii/paper-302/1/a/solution