Exercices de mathématiques oraux X-ens Algebre 1

Endomorphisms u such that Ker u = Im u.

Let E be a vector space (any dimension, but in Coq we reason in finite dimension).

Question 1.

Let u be an endomorphism of E, such that Ker u = Im u and S be a complement of Im u (supplémentaire in french), so that E is the direct sum of S and Im u.

Question 1.a.

Show that for all x in E, there is a unique pair (y, z) in S² such that x = y + u (z) and pose v and z so that y = v(x) and z = w(x).

Note that we used locking in order to protect w and v from expanding unexpectedly during proofs.

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Question 1.a.i.

Prove the following lemmas.

Question 1.a.ii.

Reuse and adapt and the proof in the course.

• (hint: use mulmxKpV)

Question 1.a.iii.

From the proof

deduce

Question 1.a.iii.

Show some simplification lemmas

• the two first are direct
• the two last use Su_dec_uniq.

Question 1.b.

• Show that v is linear.
• Show that u o v + v o u = 1.

Question 1.c.

• Show that w is linear.
• Show that u o w + w o u = u.

Endomorphisms u such that Ker u = Im u.

• What are the morphisms such that u o v = 0 and v + u is invertible ?

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