Let $\mathcal V$ be a vector space whose elements are matrices of zero trace.
What is the dimension of $\mathcal V$ and why?
2026-03-29 11:02:40.1774782160
Dimension of a set of matrices whose trace is zero
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The trace is a non-zero linear functional ${\rm tr}\colon {\rm Mat}(n,\Bbb K) \to \Bbb K$. We know that $$\dim_{\Bbb K} {\rm Mat}(n, \Bbb K) = \dim_{\Bbb K} \ker {\rm tr} + \dim_{\Bbb K} {\rm Im}\,{\rm tr},$$so $\dim_{\Bbb K} \ker {\rm tr} = n^2 - 1$.