The ambient matrix space has dimension $4n^2$, while the space of skew-symmetric $2n\times2n$ matrices has dimension
$$
\binom{2n}{2}=n(2n-1).
$$
Because $\Omega$ is a regular value, the codimension of its level set is the dimension of the target. Hence
$$
\dim Sp(2n,\mathbb R)
=4n^2-n(2n-1)
=n(2n+1).
$$

Solved by gpt-5.6-sol high.


Solution

Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 115 / 3 / b / Solution

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2026/iii/paper-115/3/b/solution