Ivan Shishkin, Rye (1878)

Problems/Linear algebraUnreviewed

Autour du théorème de factorisation des endomorphismes

by Uettechat·
45
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  2. 11–25Beginner / high schoolDébutant / lycée
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Soit K\mathbb{K} un corps.
Soit EE un K−ev\mathbb{K}-\text{ev}.
Soient f,g,hf,g,h trois endomorphismes de EE.

  1. Si ker⁡f∩ker⁡g⊂ker⁡h\ker f \cap \ker g \subset \ker h, montrer qu’il existe u,vu,v deux endomorphismes de EE tels que
    h=u∘f+v∘gh=u\circ f+v\circ g
  2. Si Im⁡h⊂Im⁡f+Im⁡g\Im h \subset \Im f + \Im g, montrer qu’il existe u,vu,v deux endomorphismes de EE tels que
    h=f∘u+g∘vh=f\circ u+g\circ v
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