{ "cells": [ { "cell_type": "code", "execution_count": 2, "metadata": {}, "outputs": [ { "data": { "application/javascript": "IPython.notebook.set_autosave_interval(300000)" }, "metadata": {}, "output_type": "display_data" }, { "name": "stdout", "output_type": "stream", "text": [ "Autosaving every 300 seconds\n" ] } ], "source": [ "from IPython.display import display, Latex\n", "from IPython.core.display import HTML\n", "%reset -f\n", "%matplotlib inline\n", "%autosave 300\n", "from matplotlib.pylab import *\n", "import sympy as sy\n", "sy.init_printing() " ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Exercice Géométrie\n", "- Exercice 1\n", "On se place dans le plan euclidien $\\mathbb R^2$ muni de la norme Euclidienne. \n", "On se donne une famille $H$ de points définie implicitement par la relation\n", "$$\n", "H = \\{(x,y) \\in \\mathbb R^2 \\, | \\, x^2y^2 = 1\\}.\n", "$$\n", "On se donne $E$ une famille de points définie implicitement par la relation\n", "$$\n", "E = \\{(x,y) \\in \\mathbb R^2 \\, | \\, 2 x^2 + y^2 = 1\\}.\n", "$$\n", "Trouver les points communs à $H$ et $E$.\n" ] }, { "cell_type": "code", "execution_count": null, "metadata": {}, "outputs": [], "source": [] }, { "cell_type": "markdown", "metadata": {}, "source": [ "- Exercice 2\n", " - Tracer la courbe paramétrique définie par les équations suivantes $$\\begin{cases}\n", "x(t) = \\cos(t)\\\\\n", "y(t) = \\sin(2t)\\end{cases}$$ pour $t \\in [0, 2\\pi]$.\n", " - Calculer la longueur de cette courbe.\n", " - Trouver les points d'intersection de cette courbe avec le cercle de centre $(0,0)$ et de rayon $\\frac 1 2$.\n" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Algèbre linéaire\n", "Pour diagonaliser une matrice $A$ avec sympy, on utilise la méthode `diagonalize` de la bibliothèque `sympy`, comme sur l'exemple suivant:\n" ] }, { "cell_type": "code", "execution_count": 4, "metadata": {}, "outputs": [ { "data": { "image/png": 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"text/latex": [ "$\\displaystyle \\left[\\begin{matrix}1 & 2 & 3\\\\4 & 5 & 6\\\\7 & 8 & 9\\end{matrix}\\right]$" ], "text/plain": [ "⎡1 2 3⎤\n", "⎢ ⎥\n", "⎢4 5 6⎥\n", "⎢ ⎥\n", "⎣7 8 9⎦" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "image/png": 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"text/latex": [ "$\\displaystyle \\left[\\begin{matrix}1 & - \\frac{3 \\sqrt{33}}{22} - \\frac{1}{2} & - \\frac{1}{2} + \\frac{3 \\sqrt{33}}{22}\\\\-2 & \\frac{1}{4} - \\frac{3 \\sqrt{33}}{44} & \\frac{1}{4} + \\frac{3 \\sqrt{33}}{44}\\\\1 & 1 & 1\\end{matrix}\\right]$" ], "text/plain": [ "⎡ 3⋅√33 1 1 3⋅√33⎤\n", "⎢1 - ───── - ─ - ─ + ─────⎥\n", "⎢ 22 2 2 22 ⎥\n", "⎢ ⎥\n", "⎢ 1 3⋅√33 1 3⋅√33 ⎥\n", "⎢-2 ─ - ───── ─ + ───── ⎥\n", "⎢ 4 44 4 44 ⎥\n", "⎢ ⎥\n", "⎣1 1 1 ⎦" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "image/png": 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"text/latex": [ "$\\displaystyle \\left[\\begin{matrix}0 & 0 & 0\\\\0 & \\frac{15}{2} - \\frac{3 \\sqrt{33}}{2} & 0\\\\0 & 0 & \\frac{15}{2} + \\frac{3 \\sqrt{33}}{2}\\end{matrix}\\right]$" ], "text/plain": [ "⎡0 0 0 ⎤\n", "⎢ ⎥\n", "⎢ 15 3⋅√33 ⎥\n", "⎢0 ── - ───── 0 ⎥\n", "⎢ 2 2 ⎥\n", "⎢ ⎥\n", "⎢ 15 3⋅√33⎥\n", "⎢0 0 ── + ─────⎥\n", "⎣ 2 2 ⎦" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "image/png": 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"text/latex": [ "$\\displaystyle \\left[\\begin{matrix}1 & 2 & 3\\\\4 & 5 & 6\\\\7 & 8 & 9\\end{matrix}\\right]$" ], "text/plain": [ "⎡1 2 3⎤\n", "⎢ ⎥\n", "⎢4 5 6⎥\n", "⎢ ⎥\n", "⎣7 8 9⎦" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "A=sy.Matrix([[1,2,3],[4,5,6],[7,8,9]])\n", "display(A)\n", "P,D=A.diagonalize()\n", "display(P)\n", "display(D)\n", "display(sy.simplify(P*D*P.inv()))" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "- Exercice 3\n", "On définit $A$ comme\n", "$$A=\\left[\\begin{matrix}2& 1 & 0\\\\1 & 2 & 1\\\\1 & 0 & 2\\end{matrix}\\right].$$\n", "- Trouver les valeurs propres de $A$ et les vecteurs propres associés.\n", "- Diagonaliser $A$, préciser la matrice de passage $P$ et la matrice diagonale $D$ et vérifier $A=PDP^{-1}$.\n", "\n", "- Exercice 4\n", "On définit $S$ comme\n", "$$S=\\left[\\begin{matrix}1 & 0 & 0\\\\0 & \\frac{\\sqrt{3}}{2} & \\frac{1} 2\\\\0 & -\\frac{1}{2}& \\frac{\\sqrt{3}}{2}\\end{matrix}\\right].$$\n", "Trouver le plus petit $p>0$ entier tel que $S^p$ est l'identité. Pouvait-on déviner le résultat?\n", "\n", "Vérifier que $S^{-1}=S^t$. Comment appelle-ton la propriété d'une telle matrice? Pourquoi ce résultat?\n", "\n", "Indication: On va calculer $S^p$, avec $p$ entier croissant, tant que ce résultat n'est pas l'identité. On connaîtra $p$ à la sortie de la boucle.\n" ] }, { "cell_type": "code", "execution_count": null, "metadata": {}, "outputs": [], "source": [] }, { "cell_type": "markdown", "metadata": {}, "source": [ "- Exercice 5\n", "Soit la matrice,\n", "$$\n", "A = \\begin{pmatrix}\n", "0 & 2 & 2\\\\\n", "-2 & 0 & -1\\\\\n", "-2 & 1 & 0\n", "\\end{pmatrix}\n", "$$\n", "On considère la suite de $\\mathbb R^3$ définie par\n", "$$\n", "(Id-A)\\left(\\begin{matrix}x^{n+1}\\\\y^{n+1}\\\\z^{n+1}\\end{matrix}\\right)=(Id+A)\\left(\\begin{matrix}x^{n}\\\\y^{n}\\\\z^{n}\\end{matrix}\\right),\n", "$$\n", "et de premièr itéré donné quelconque, $A$ la matrice introduite à la question précédente et $Id$ la matrice identité.\n", "\n", "Calculer les 50 premiers itérés ainsi que la norme Euclidienne de chaque itéré. Qu'observe-t-on quant à la norme euclidienne de chaque itéré? Démontrer mathématiquement ce que vous observez.\n", "\n", "La suite est-elle convergente? " ] }, { "cell_type": "code", "execution_count": null, "metadata": {}, "outputs": [], "source": [] }, { "cell_type": "markdown", "metadata": {}, "source": [] }, { "cell_type": "code", "execution_count": null, "metadata": {}, "outputs": [], "source": [] }, { "cell_type": "code", "execution_count": null, "metadata": {}, "outputs": [], "source": [] }, { "cell_type": "code", "execution_count": null, "metadata": {}, "outputs": [], "source": [] } ], "metadata": { "kernelspec": { "display_name": "base", "language": "python", "name": "python3" }, "language_info": { "codemirror_mode": { "name": "ipython", "version": 3 }, "file_extension": ".py", "mimetype": "text/x-python", "name": "python", "nbconvert_exporter": "python", "pygments_lexer": "ipython3", "version": "3.8.5" } }, "nbformat": 4, "nbformat_minor": 2 }