Ivan Shishkin, Rye (1878)

Problems/Linear algebraReviewed

Morphismes multiplicatifs sur Mn(K)\mathcal{M}_n(\mathbb{K})

by Uettechat·
45
Difficulty scaleÉchelle de difficulté

This score reflects both the level of the required concepts and the difficulty of the solution.Ce score tient compte à la fois du niveau des notions nécessaires et de la difficulté de la résolution.

  1. 1–10First steps / middle schoolPremiers pas / collège
  2. 11–25Beginner / high schoolDébutant / lycée
  3. 26–50Intermediate / undergraduateIntermédiaire / licence
  4. 51–70Advanced / graduateAvancé / master
  5. 71–90Expert / specializedExpert / spécialisé
  6. 91–100Research levelNiveau recherche
These levels are approximate guides.Ces niveaux sont des repères approximatifs.
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Soit φ:Mn(K)→K\varphi : \mathcal{M}_n(\mathbb{K}) \to \mathbb{K} non constante et telle que ∀A,B∈Mn(K),ϕ(AB)=ϕ(A)ϕ(B)\forall A,B \in \mathcal{M}_n(\mathbb{K}), \phi(AB)=\phi(A)\phi(B)

  1. Montrer que ϕ(A)≠0⟺A∈GLn(K)\phi(A) \neq 0 \Longleftrightarrow A \in \mathrm{GL}_n(K)
  2. Déterminer les A∈Mn(K)A \in \mathcal{M}_n(\mathbb{K}) tels que ∀X∈Mn(K),ϕ(A+X)=ϕ(A)+ϕ(X)\forall X \in \mathcal{M}_n(\mathbb{K}), \phi(A+X)=\phi(A)+\phi(X)
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References

  1. oral X-MP 2025 (1ère question)
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