From 5ef9f3a28f44dcf4346d82ea97dc068b4c0a5ef8 Mon Sep 17 00:00:00 2001 From: vcncolin Date: Mon, 1 Jul 2024 13:03:20 +0200 Subject: [PATCH] posts --- _posts/2024-07-01-welcome-to-jekyll.markdown | 67 +++++++------------- 1 file changed, 23 insertions(+), 44 deletions(-) diff --git a/_posts/2024-07-01-welcome-to-jekyll.markdown b/_posts/2024-07-01-welcome-to-jekyll.markdown index 43c56d0..29f0711 100644 --- a/_posts/2024-07-01-welcome-to-jekyll.markdown +++ b/_posts/2024-07-01-welcome-to-jekyll.markdown @@ -5,57 +5,36 @@ title: "Welcome to Jekyll!" date: 2024-07-01 12:19:36 +0200 categories: jekyll update --- -You’ll find this post in your `_posts` directory. Go ahead and edit it and re-build the site to see your changes. You can rebuild the site in many different ways, but the most common way is to run `jekyll serve`, which launches a web server and auto-regenerates your site when a file is updated. -Jekyll requires blog post files to be named according to the following format: +# Chapitre 3 : Compléments d'algèbre linéaire -`YEAR-MONTH-DAY-title.MARKUP` +## Sous-espaces vectoriels et familles de vecteurs -Where `YEAR` is a four-digit number, `MONTH` and `DAY` are both two-digit numbers, and `MARKUP` is the file extension representing the format used in the file. After that, include the necessary front matter. Take a look at the source for this post to get an idea about how it works. +## Applications linéaires -Jekyll also offers powerful support for code snippets: +### Exercice 11 -{% highlight ruby %} -def print_hi(name) - puts "Hi, #{name}" -end -print_hi('Tom') -#=> prints 'Hi, Tom' to STDOUT. -{% endhighlight %} +Soient $n \in \mathbb{N}$ et $D$ l'application : +$$ +\begin{array}{rlll} +D: \mathbb{R}_n[X] & \longrightarrow & \mathbb{R}_n[X] \\ +P & \longrightarrow & P^{\prime} +\end{array} +$$ -Check out the [Jekyll docs][jekyll-docs] for more info on how to get the most out of Jekyll. File all bugs/feature requests at [Jekyll’s GitHub repo][jekyll-gh]. If you have questions, you can ask them on [Jekyll Talk][jekyll-talk]. +1. Vérifier que $D$ est un automorphisme de $\mathbb{R}_n[X]$ +2. On pose $\Gamma = id_{\mathbb{R}_n[X]} + D + D^2 + ... + D^n$. Montrer que $\Gamma$ est un automorphisme. +3. Rappeler une factorisation usuelle de $1 - X^{n+1}$ +4. En déduire $\Gamma \circ (id - D)$ +5. En déduire l'application réciproque de $\Gamma$ -[jekyll-docs]: https://jekyllrb.com/docs/home -[jekyll-gh]: https://github.com/jekyll/jekyll -[jekyll-talk]: https://talk.jekyllrb.com/ -======= ---- -layout: post -title: "Welcome to Jekyll!" -date: 2024-07-01 12:19:36 +0200 -categories: jekyll update ---- -You’ll find this post in your `_posts` directory. Go ahead and edit it and re-build the site to see your changes. You can rebuild the site in many different ways, but the most common way is to run `jekyll serve`, which launches a web server and auto-regenerates your site when a file is updated. +#### Corrigé : -Jekyll requires blog post files to be named according to the following format: +1. $D$ est un endomorphisme car $D$ est linéaire, et $P' \in \mathbb{R}_n[X]$ +2. $\Gamma$ transforme la base canonique de $\mathbb{R}_n[X]$ en famille libre de $\mathbb{R}_n[X]$, de dimension $n+1$. On peut donc dire que $\Gamma$ est une application linéaire (car somme d'applications linéaires), bijective, de $\mathbb{R}_n[X]$ vers $\mathbb{R}_n[X]$, donc $\Gamma$ est un automorphisme. +3. $(1-X^{n+1}) = (1-X) . \sum_{k=0}^n X^k$ TMTC +4. En réécrivant l'égalité précédente en plissant un peu les yeux, on obtient : +$\Gamma \circ (id - D) = (id - D^{n+1})$ +Il est alors important de remarquer que dans $\mathbb{R}_n[X]$, $D^{n+1} = 0$. On en déduit alors que dans $\mathbb{R}_n[X]$, $\Gamma \circ (id - D) = id$ -`YEAR-MONTH-DAY-title.MARKUP` -Where `YEAR` is a four-digit number, `MONTH` and `DAY` are both two-digit numbers, and `MARKUP` is the file extension representing the format used in the file. After that, include the necessary front matter. Take a look at the source for this post to get an idea about how it works. - -Jekyll also offers powerful support for code snippets: - -{% highlight ruby %} -def print_hi(name) - puts "Hi, #{name}" -end -print_hi('Tom') -#=> prints 'Hi, Tom' to STDOUT. -{% endhighlight %} - -Check out the [Jekyll docs][jekyll-docs] for more info on how to get the most out of Jekyll. File all bugs/feature requests at [Jekyll’s GitHub repo][jekyll-gh]. If you have questions, you can ask them on [Jekyll Talk][jekyll-talk]. - -[jekyll-docs]: https://jekyllrb.com/docs/home -[jekyll-gh]: https://github.com/jekyll/jekyll -[jekyll-talk]: https://talk.jekyllrb.com/ ->>>>>>> f477aae2909248464a80bc7dc06c2cf0c6f5a6ab