I want to prove the following statement:
Given a unitary transformation $T$ on a real inner product space $W$. $I$ is an identity matrix and $T$ has an orthonormal basis, then the dimension of $W$ must be odd.
2026-02-22 17:57:13.1771783033
Dimension of real inner product with unitary transformation
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If $W$ has dimension $n$, then taking the determinant of both sides of $T^2=-I$ yields $$\det(T)^2=(-1)^n$$ which is impossible if $n$ is odd since $W$ is a real vector space.