If $X$ belongs to the <Special orthogonal Lie algebra>, which is also the Lie algebra of $O(3)$, then $e^{tX}\in O(3)$ near $t=0$. Differentiating
$$
e^{tX^{\mathsf T}}e^{tX}=I
$$
at zero gives $X^{\mathsf T}+X=0$. Conversely, the <matrix exponential> of every real <skew-symmetric matrix> is orthogonal. Hence
$$
\mathfrak o(3)=\{X\in M_3(\mathbb R):X^{\mathsf T}=-X\}.
$$

Solution

Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 302 / 1 / b / Solution

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2023/iii/paper-302/1/b/solution