Ivan Shishkin, Rye (1878)

Problems/Linear algebraUnreviewed

Intersection d’un hyperplan de Mn(C)M_n(\C) avec le groupe des inversibles 22

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Let n≥1n \ge 1 be a natural number and K\mathbb{K} be a field. Let E:=Mn(K)E := \mathcal{M}_n(\mathbb{K}).

  1. Let ϕ\phi be a linear form on EE. Show that there exists a unique matrix A∈EA \in E such that:
    ∀M∈E,ϕ(M)=tr(AM)\forall M \in E, \quad \phi(M) = \text{tr}(AM)
  2. Deduce that for any hyperplane HH of EE, the intersection H∩GLn(K)H \cap \text{GL}_n(\mathbb{K}) is non-empty.
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