From a1a5447b8b040b100bad89766066ae4ba8d6d920 Mon Sep 17 00:00:00 2001 From: Anhgelus Morhtuuzh Date: Thu, 13 Feb 2025 17:42:52 +0100 Subject: =?UTF-8?q?Ajout=20de=20la=20semaine=20des=20cours=20du=207=20au?= =?UTF-8?q?=2013=20f=C3=A9vrier?= MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit --- semestre 2/maths/td/02-13.tex | 215 ++++++++++++++++++++++++++++++++++++++++++ 1 file changed, 215 insertions(+) create mode 100644 semestre 2/maths/td/02-13.tex (limited to 'semestre 2/maths/td/02-13.tex') diff --git a/semestre 2/maths/td/02-13.tex b/semestre 2/maths/td/02-13.tex new file mode 100644 index 0000000..f90b2da --- /dev/null +++ b/semestre 2/maths/td/02-13.tex @@ -0,0 +1,215 @@ +%%===================================================================================== +%% +%% Filename: cours.tex +%% +%% Description: +%% +%% Version: 1.0 +%% Created: 03/06/2024 +%% Revision: none +%% +%% Author: YOUR NAME (), +%% Organization: +%% Copyright: Copyright (c) 2024, YOUR NAME +%% +%% Notes: +%% +%%===================================================================================== +\documentclass[a4paper, titlepage]{article} + +\usepackage[utf8]{inputenc} +\usepackage[T1]{fontenc} +\usepackage{textcomp} +\usepackage[french]{babel} +\usepackage{amsmath, amssymb} +\usepackage{amsthm} +\usepackage[svgnames]{xcolor} +\usepackage{thmtools} +\usepackage{lipsum} +\usepackage{framed} +\usepackage{parskip} +\usepackage{titlesec} + +\renewcommand{\familydefault}{\sfdefault} + +% figure support +\usepackage{import} +\usepackage{xifthen} +\pdfminorversion=7 +\usepackage{pdfpages} +\usepackage{transparent} +\newcommand{\incfig}[1]{% + \def\svgwidth{\columnwidth} + \import{./figures/}{#1.pdf_tex} +} + +\pdfsuppresswarningpagegroup=1 + +\colorlet{defn-color}{DarkBlue} +\colorlet{props-color}{Blue} +\colorlet{warn-color}{Red} +\colorlet{exemple-color}{Green} +\colorlet{corol-color}{Orange} +\newenvironment{defn-leftbar}{% + \def\FrameCommand{{\color{defn-color}\vrule width 3pt} \hspace{10pt}}% + \MakeFramed {\advance\hsize-\width \FrameRestore}}% + {\endMakeFramed} +\newenvironment{warn-leftbar}{% + \def\FrameCommand{{\color{warn-color}\vrule width 3pt} \hspace{10pt}}% + \MakeFramed {\advance\hsize-\width \FrameRestore}}% + {\endMakeFramed} +\newenvironment{exemple-leftbar}{% + \def\FrameCommand{{\color{exemple-color}\vrule width 3pt} \hspace{10pt}}% + \MakeFramed {\advance\hsize-\width \FrameRestore}}% + {\endMakeFramed} +\newenvironment{props-leftbar}{% + \def\FrameCommand{{\color{props-color}\vrule width 3pt} \hspace{10pt}}% + \MakeFramed {\advance\hsize-\width \FrameRestore}}% + {\endMakeFramed} +\newenvironment{corol-leftbar}{% + \def\FrameCommand{{\color{corol-color}\vrule width 3pt} \hspace{10pt}}% + \MakeFramed {\advance\hsize-\width \FrameRestore}}% + {\endMakeFramed} + +\def \freespace {1em} +\declaretheoremstyle[headfont=\sffamily\bfseries,% + notefont=\sffamily\bfseries,% + notebraces={}{},% + headpunct=,% + bodyfont=\sffamily,% + headformat=\color{defn-color}Définition~\NUMBER\hfill\NOTE\smallskip\linebreak,% + preheadhook=\vspace{\freespace}\begin{defn-leftbar},% + postfoothook=\end{defn-leftbar},% +]{better-defn} +\declaretheoremstyle[headfont=\sffamily\bfseries,% + notefont=\sffamily\bfseries,% + notebraces={}{},% + headpunct=,% + bodyfont=\sffamily,% + headformat=\color{warn-color}Attention\hfill\NOTE\smallskip\linebreak,% + preheadhook=\vspace{\freespace}\begin{warn-leftbar},% + postfoothook=\end{warn-leftbar},% +]{better-warn} +\declaretheoremstyle[headfont=\sffamily\bfseries,% + notefont=\sffamily\bfseries,% +notebraces={}{},% +headpunct=,% + bodyfont=\sffamily,% + headformat=\color{exemple-color}Exemple~\NUMBER\hfill\NOTE\smallskip\linebreak,% + preheadhook=\vspace{\freespace}\begin{exemple-leftbar},% + postfoothook=\end{exemple-leftbar},% +]{better-exemple} +\declaretheoremstyle[headfont=\sffamily\bfseries,% + notefont=\sffamily\bfseries,% + notebraces={}{},% + headpunct=,% + bodyfont=\sffamily,% + headformat=\color{props-color}Proposition~\NUMBER\hfill\NOTE\smallskip\linebreak,% + preheadhook=\vspace{\freespace}\begin{props-leftbar},% + postfoothook=\end{props-leftbar},% +]{better-props} +\declaretheoremstyle[headfont=\sffamily\bfseries,% + notefont=\sffamily\bfseries,% + notebraces={}{},% + headpunct=,% + bodyfont=\sffamily,% + headformat=\color{props-color}Théorème~\NUMBER\hfill\NOTE\smallskip\linebreak,% + preheadhook=\vspace{\freespace}\begin{props-leftbar},% + postfoothook=\end{props-leftbar},% +]{better-thm} +\declaretheoremstyle[headfont=\sffamily\bfseries,% + notefont=\sffamily\bfseries,% + notebraces={}{},% + headpunct=,% + bodyfont=\sffamily,% + headformat=\color{corol-color}Corollaire~\NUMBER\hfill\NOTE\smallskip\linebreak,% + preheadhook=\vspace{\freespace}\begin{corol-leftbar},% + postfoothook=\end{corol-leftbar},% +]{better-corol} + +\declaretheorem[style=better-defn]{defn} +\declaretheorem[style=better-warn]{warn} +\declaretheorem[style=better-exemple]{exemple} +\declaretheorem[style=better-corol]{corol} +\declaretheorem[style=better-props, numberwithin=defn]{props} +\declaretheorem[style=better-thm, sibling=props]{thm} +\newtheorem*{lemme}{Lemme}%[subsection] +%\newtheorem{props}{Propriétés}[defn] + +\newenvironment{system}% +{\left\lbrace\begin{align}}% +{\end{align}\right.} + +\newenvironment{AQT}{{\fontfamily{qbk}\selectfont AQT}} + +\usepackage{LobsterTwo} +\titleformat{\section}{\newpage\LobsterTwo \huge\bfseries}{\thesection.}{1em}{} +\titleformat{\subsection}{\vspace{2em}\LobsterTwo \Large\bfseries}{\thesubsection.}{1em}{} +\titleformat{\subsubsection}{\vspace{1em}\LobsterTwo \large\bfseries}{\thesubsubsection.}{1em}{} + +\newenvironment{lititle}% +{\vspace{7mm}\LobsterTwo \large}% +{\\} + +\renewenvironment{proof}{$\square$ \footnotesize\textit{Démonstration.}}{\begin{flushright}$\blacksquare$\end{flushright}} + +\title{TD du 13 février} +\author{William Hergès\thanks{Sorbonne Université - Faculté des Sciences, Faculté des Lettres}} + +\begin{document} + \maketitle + \newpage + \section*{Exercice 1} + Trivial + \section*{Exercice 2} + Trivial + \section*{Exercice 3} + Trivial + \section*{Exercice 4} + On a : + \begin{align*} + 2 &= a-b+c \\ + 1 &= a+b-c \\ + 3 &= -a+b+c + \end{align*} + où $(a,b,c)$ sont les coordonnées du vecteur dans la base. + + D'où : + $$\begin{pmatrix} 1&-1&1&|&2\\1&1&-1&|&1\\-1&1&1&|&3 \end{pmatrix}\iff\begin{pmatrix} 1&-1&1&|&2\\0&2&-2&|&-1\\0&0&2&|&5\end{pmatrix}$$ + Ainsi, + \begin{align*} + c &= 5/2 \\ + b &= 2 \\ + a &= 3/2 + \end{align*} + \section*{Exercice 5} + Si $x=0$, alors $v = u + w$, donc ce n'est pas libre.\\ + Si $x\neq 0$, alors $(uvx)$ est échelonné, donc elle est libre. + + Il s'agit d'une base si et seulement si $x\neq 0$. + \section*{Exercice 6} + \begin{lititle} + Matrice de changement de base. + \end{lititle} + On a $v$ un vecteur dans la base canonique. $\lambda$ est dans la base $A$ et $\mu$ dans la base $B$. + + Ainsi, on a : + \begin{align*} + \mathrm{I}_n v &= A\lambda \\ + \mathrm{I}_n v &= B\mu + \end{align*} + d'où : + $$ A\lambda = B\mu \iff \mu = B^{-1}A\lambda $$ + Ainsi, $B^{-1}A$ est la matrice de passage de $A$ à $B$. + \section*{Exercice 7} + $\lambda v_1+v_2 = (\lambda x_1+x_2,\lambda y_1 + y_2)$\\ + $\lambda y_1+y_2=\lambda x_1+x_2+\lambda+1 \neq \lambda x_1+x_2+1$\\ + donc $A$ n'est pas un sev. + + $\lambda v_1+v_2 = (\lambda x_1 + x_2, \lambda x_1 + x_2)$\\ + donc $A_1$ est un sev. + + $\lambda v_1+v_2 = (\lambda x_1+x_2,\lambda y_1 + y_2)$\\ + $\lambda y_1 = \lambda^2 x^2 \neq x^2$\\ + donc $A_2$ n'est pas un sev. +\end{document} -- cgit v1.2.3