diff --git "a/Chapitre 9 - Produits scalaires et espaces euclidens/9.8-9.9 La projection orthogonale sur un sous-espace vectoriel - La projection orthogonale examples et remarques suppl\303\251mentaires.ipynb" "b/Chapitre 9 - Produits scalaires et espaces euclidens/9.8-9.9 La projection orthogonale sur un sous-espace vectoriel - La projection orthogonale examples et remarques suppl\303\251mentaires.ipynb" index c2195f1..34c9e6f 100644 --- "a/Chapitre 9 - Produits scalaires et espaces euclidens/9.8-9.9 La projection orthogonale sur un sous-espace vectoriel - La projection orthogonale examples et remarques suppl\303\251mentaires.ipynb" +++ "b/Chapitre 9 - Produits scalaires et espaces euclidens/9.8-9.9 La projection orthogonale sur un sous-espace vectoriel - La projection orthogonale examples et remarques suppl\303\251mentaires.ipynb" @@ -1,338 +1,32067 @@ { "cells": [ { "cell_type": "markdown", "metadata": {}, "source": [ "# **Concept(s)-clé(s) et théorie**\n", "\n", "## Définition 1\n", "Soient $V$ un $\\mathbb{R}$-espace vectoriel muni d'un produit scalaire $\\langle \\cdot, \\cdot \\rangle$ et $W \\subset V$ un sous-espace vectoriel de $V$. L'**orthogonal à $W$ dans $V$** est le sous-ensemble de $V$ défini par $$W^\\perp = \\left\\{v \\in V: \\langle v,w \\rangle = 0 \\quad \\forall \\ w \\in W\\right\\}$$\n", "\n", "## Proposition 1\n", "Soient $V$ un $\\mathbb{R}$-espace vectoriel muni d'un produit scalaire $\\langle \\cdot, \\cdot \\rangle$ et $W \\subset V$ un sous-espace vectoriel de $V$. Alors le sous-ensamble $W^\\perp$ de $V$ est un sous-espace vectoriel de $V$.\n", "\n", "## Proposition 2\n", "Soient $V$ un $\\mathbb{R}$-espace vectoriel et $W \\subset V$ un sous-espace vectoriel de $V$. Alors pour tout $v \\in V$, il existe $w \\in W$ et $x \\in W^\\perp$ tels que $v = w + x$. De plus, $w$ et $x$ sont uniquement déterminés par $v$. Donc, cela revient à dire que $V = W \\oplus W^{\\perp}$.\n", "\n", "## Définition 2\n", "Soient $V$ un $\\mathbb{R}$-espace vectoriel et $W \\subset V$ un sous-espace vectoriel de $V$. Soient également $v \\in V$ et $w \\in W, x \\in W^\\perp$ tels que $v = w+x$, come ci-dessus. On appelle $w$ la **projection orthogonale de $v$ sur $W$** et on écrit $w = proj_Wv$.\n", "\n", "## Proposition 3\n", "Soient $V$ un $\\mathbb{R}$-espace euclidien de dimension $n$ et $W \\subset V$ un sous-espace vectoriel de $V$ de dimension $k$. Soit également $\\{v_1, \\dots, v_k, v_{k+1}, \\dots, v_n\\}$ un base orthonormée de $V$ tel que $\\{v_1, \\dots, v_k\\}$ est une base orthonormée de $W$ et $\\{v_{k+1}, \\dots, v_n\\}$ est une base orthonormée de $W^\\perp$. Alors nous pouvons calculer les projections orthogonales de $v \\in V$ sur $W$ et sur $W^\\perp$ respectivement comme suit:\n", "\\begin{align*}\n", "proj_Wv &= \\sum\\limits_{i=1}^k \\dfrac{\\langle v, v_i \\rangle}{\\langle v_i,v_i \\rangle} v_i\\\\\n", "proj_{W^\\perp}v &= \\sum\\limits_{j=k+1}^n \\dfrac{\\langle v, v_j \\rangle}{\\langle v_j,v_j \\rangle} v_j\n", "\\end{align*}\n", "\n", "## Corollaire 1 (de Gram-Schmidt)\n", "Soient $V$ un espace euclidien et $W \\subset V$ un sous-espace vectoriel de $V$. Alors $$dim W^\\perp = dim V - dim W$$\n", "\n", "## Corollaire 2\n", "Soient $V$ un espace euclidien et $W \\subset V$ un sous-espace vectoriel de $V$. Alors $$\\left(W^\\perp\\right)^\\perp = W$$" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "# Exercises et Examples" ] }, { "cell_type": "code", - "execution_count": null, + "execution_count": 1, "metadata": {}, - "outputs": [], + "outputs": [ + { + "data": { + "text/html": [ + " \n", + " " + ] + }, + "metadata": {}, + "output_type": "display_data" + } + ], "source": [ "import Librairie.AL_Fct as al\n", "import Corrections.corrections as corrections\n", "import numpy as np\n", "import sympy as sp" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Exercice 1\n", "\n", - "Some theoretical questions, in the form of T/F. " - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "## Exercice 2\n", - "\n", "Soit $V = \\mathbb{R}^n, \\ n=2,3$. Considérez les paires suivantes, faites par un ensemble de vecteurs $\\mathcal{S}$ générant un sous-espace vectoriel $W$ de $V$ et par un élément $v$ de $V$. Calculez les projections orthogonales de $v$ sur $W$ et $W^\\perp$. Tous les produits scalaires sont censés être les produits usuels.\n", "\n", - "1. $V = \\mathbb{R}^2 \\qquad \\mathcal{S} = \\left\\{ \\begin{bmatrix}1 \\\\ 1\\end{bmatrix} \\right\\} \\qquad \\qquad \\quad v = \\begin{bmatrix} -1 \\\\ 0 \\end{bmatrix}$\n", - "2. $V = \\mathbb{R}^2 \\qquad \\mathcal{S} = \\left\\{ \\begin{bmatrix}0 \\\\ 1\\end{bmatrix}, \\begin{bmatrix} 1 \\\\ -1 \\end{bmatrix} \\right\\} \\qquad v = \\begin{bmatrix} -3 \\\\ 5 \\end{bmatrix}$\n", - "3. $V = \\mathbb{R}^3 \\qquad \\mathcal{S} = \\left\\{ \\begin{bmatrix}1 \\\\ 2 \\\\ 1\\end{bmatrix} \\right\\} \\qquad \\qquad \\quad \\ \\ v = \\begin{bmatrix} 1 \\\\ 0 \\\\ 1\\end{bmatrix}$\n", - "4. $V = \\mathbb{R}^3 \\qquad \\mathcal{S} = \\left\\{ \\begin{bmatrix}1 \\\\ -1\\\\ 1\\end{bmatrix} \\begin{bmatrix} 0 \\\\ 1 \\\\ 1 \\end{bmatrix} \\right\\} \\qquad \\quad v = \\begin{bmatrix} -1 \\\\ 0 \\\\ 1 \\end{bmatrix}$" + "1. $V = \\mathbb{R}^2 \\qquad \\mathcal{S} = \\left\\{ \\begin{pmatrix}1 \\\\ 1\\end{pmatrix} \\right\\} \\qquad \\qquad \\quad \\ v = \\begin{pmatrix} -1 \\\\ 0 \\end{pmatrix}$\n", + "2. $V = \\mathbb{R}^2 \\qquad \\mathcal{S} = \\left\\{ \\begin{pmatrix}0 \\\\ 1\\end{pmatrix}, \\begin{pmatrix} 1 \\\\ -1 \\end{pmatrix} \\right\\} \\qquad v = \\begin{pmatrix} -3 \\\\ 5 \\end{pmatrix}$\n", + "3. $V = \\mathbb{R}^3 \\qquad \\mathcal{S} = \\left\\{ \\begin{pmatrix}1 \\\\ 2 \\\\ 1\\end{pmatrix} \\right\\} \\qquad \\qquad \\qquad \\ v = \\begin{pmatrix} 1 \\\\ 0 \\\\ 1\\end{pmatrix}$\n", + "4. $V = \\mathbb{R}^3 \\qquad \\mathcal{S} = \\left\\{ \\begin{pmatrix}1 \\\\ -1\\\\ 1\\end{pmatrix}, \\begin{pmatrix} 0 \\\\ 1 \\\\ 1 \\end{pmatrix}\\right\\} \\qquad \\quad \\ \\ v = \\begin{pmatrix} -1 \\\\ 0 \\\\ 1 \\end{pmatrix}$" ] }, { "cell_type": "code", "execution_count": null, "metadata": {}, "outputs": [], "source": [ "case_number = 4" ] }, { "cell_type": "code", "execution_count": null, "metadata": {}, "outputs": [], "source": [ "if case_number == 1:\n", " S = [[1,1]]\n", " v = [-1,0]\n", " dim=1\n", "elif case_number == 2:\n", " S = [[0,1], [-1,1]]\n", " v = [-3,5]\n", " dim=2\n", "elif case_number == 3:\n", " S = [[1,2,1]]\n", " v = [1,0,1]\n", " dim=1\n", "elif case_number == 4:\n", " S = [[1,-1,1], [0,1,1]]\n", " v = [-1,0,1]\n", " dim=2\n", "else:\n", " print(f\"{case_number} n'est pas un numéro de cas valide!\" \n", " f\"Numéros de cas disponibles: [1,2,3,4]\")\n", "\n", "step = 0\n", "VectorsList = [S]" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### Aide\n", "\n", "Pour calculer la projection orthogonal de $v$ sur $W$, il peut être utile de dériver une base ortogonale (ou orthonormée) pour ce dernier. Vous pouvez utiliser la cellule suivante pour exécuter l'algorithme interactif de Gram-Schmidt.\n", "\n", "#### Instructions\n", "\n", "Pour utiliser la méthode interactive de Gram-Schmidt, procédez comme suit:\n", "\n", "1. Insérez le numéro de dossier souhaité dans la cellule suivante\n", "2. Exécutez la cellule appelée \"SÉLECTION DES PARAMÈTRES\" pour sélectionner le type d'opération et les coefficients nécessaires\n", "3. Exécutez la cellule appelée \"EXÉCUTER L'ÉTAPE DE L'ALGORITHME GRAM-SCHMIDT\" pour exécuter l'étape de l'algorithme de Gram-Schmidt avec les paramètres précédemment sélectionnés\n", "\n", "En outre:\n", "\n", "1. Vous pouvez annuler une opération en sélectionnant le bouton \"Revert\".\n", "\n", "2. Si les coefficients insérés sont incorrects, vous pouvez essayer avec de nouvelles valeurs sans effectuer une opération \"Revert\".\n", "\n", "3. Les coefficients qui ne sont pas liés à l'opération sélectionnée peuvent être définis sur n'importe quelle valeur, car ils ne sont pas utilisés dans le code." ] }, { "cell_type": "code", "execution_count": null, "metadata": {}, "outputs": [], "source": [ "# SÉLECTION DES PARAMÈTRES\n", "norm_coeff, proj_coeffs, operation, step_number = al.manual_GS(dim=dim)" ] }, { "cell_type": "code", "execution_count": null, "metadata": {}, "outputs": [], "source": [ "# EXÉCUTER L'ÉTAPE DE L'ALGORITHME GRAM-SCHMIDT\n", "S = al.interactive_gram_schmidt(norm_coeff, proj_coeffs,\n", " operation, step_number, \n", " S.copy(), VectorsList)" ] }, { "cell_type": "code", "execution_count": null, "metadata": {}, "outputs": [], "source": [ - "# INSERT HERE THE VALUES OF THE TWO PROJECTIONS\n", - "proj1 = [1,1,1] # projection onto W\n", - "proj2 = [0,0,0] # projection onto the orthogonal complement of W in V" + "# IINSÉREZ ICI LES VALEURS DES DEUX PROJECTIONS\n", + "proj1 = [1,1,1] # projection sur W\n", + "proj2 = [0,0,0] # projection sur le complément orthogonal de W en V" ] }, { "cell_type": "code", "execution_count": null, "metadata": { "scrolled": false }, "outputs": [], "source": [ - "corrections.Ex2Chapitre9_8_9(proj1, proj2, case_nb=case_number)" + "corrections.Ex1Chapitre9_8_9(proj1, proj2, case_nb=case_number)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ - "## Exercice 3\n", + "## Exercice 2\n", "\n", "Soit $V = \\mathbb{P}^n(\\mathbb{R}), \\ n=2,3$. Considérez les paires suivantes, faites par un ensemble de fonctions $\\mathcal{S}$ générant un sous-espace vectoriel $W$ de $V$ et par un élément $v$ de $V$. alculez les projections orthogonales de $v$ sur $W$ e $W^\\perp$. Tous les produits scalaires sont censés être les produits usuels, dans l'intervalle spécifié $I$.\n", "\n", "1. $V = \\mathbb{P}^2(\\mathbb{R}) \\qquad \\mathcal{S} = \\left\\{ 1, x \\right\\} \\qquad \\qquad \\ \\ \\ v = x^2 \\qquad \\qquad \\qquad \\\n", "I = [-1,1]$\n", "2. $V = \\mathbb{P}^2(\\mathbb{R}) \\qquad \\mathcal{S} = \\left\\{ 1, x^2 \\right\\} \\qquad \\qquad v = x \\qquad \\qquad \\qquad \\ \\ \\ I = [0,1]$\n", "3. $V = \\mathbb{P}^3(\\mathbb{R}) \\qquad \\mathcal{S} = \\left\\{ 1, 1+x^2, x^3 \\right\\} \\quad \\ v = x^2 + x \\qquad \\qquad \\ \\ \\ I = [-1,1]$\n", "4. $V = \\mathbb{P}^3(\\mathbb{R}) \\qquad \\mathcal{S} = \\left\\{ x^2, x^3 \\right\\} \\qquad \\quad \\ \\ v = 1 + x + x^2 + x^3 \\quad \\ I=[-1,1]$" ] }, { "cell_type": "code", "execution_count": null, "metadata": {}, "outputs": [], "source": [ "case_number = 1" ] }, { "cell_type": "code", "execution_count": null, "metadata": {}, "outputs": [], "source": [ "x = sp.Symbol('x')\n", "if case_number == 1:\n", " S = [1+0*x, x]\n", " v = x**2\n", " int_limits = [-1,1]\n", " dim=2\n", "elif case_number == 2:\n", " S = [1+0*x, x**2]\n", " v = x\n", " int_limits = [0,1]\n", " dim=2\n", "elif case_number == 3:\n", " S = [1+0*x, x**2, x**3]\n", " v = x**2 + x\n", " int_limits = [-1,1]\n", " dim=3\n", "elif case_number == 4:\n", " S = [x**2, x**3]\n", " v = 1 + x + x**2 + x**3\n", " int_limits = [-1,1]\n", " dim=2\n", "else:\n", " print(f\"{case_number} n'est pas un numéro de cas valide!\" \n", " f\"Numéros de cas disponibles: [1,2,3,4]\")\n", "\n", "step = 0\n", "VectorsList = [S]" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### Aide\n", "\n", "Pour calculer la projection de $v$ sur $W$, il peut\n", "aider à dériver une base orthogonale (ou orthonormée) pour ce dernier. Vous pouvez utiliser la cellule suivante pour exécuter l'algorithme interactif de Gram-Schmidt pour fonctions." ] }, { "cell_type": "code", "execution_count": null, "metadata": {}, "outputs": [], "source": [ "# SÉLECTION DES PARAMÈTRES\n", "norm_coeff, proj_coeffs, operation, step_number = al.manual_GS(dim=dim)" ] }, { "cell_type": "code", "execution_count": null, "metadata": {}, "outputs": [], "source": [ "# EXÉCUTER L'ÉTAPE DE L'ALGORITHME GRAM-SCHMIDT\n", "S = al.interactive_gram_schmidt_func(norm_coeff, proj_coeffs,\n", " operation, step_number, \n", " S.copy(), VectorsList,\n", " int_limits=int_limits,\n", " weight_function=None)" ] }, { "cell_type": "code", "execution_count": null, "metadata": {}, "outputs": [], "source": [ - "# INSERT HERE THE VALUES OF THE TWO PROJECTIONS\n", - "proj1 = 1 + 0*x # projection onto W\n", - "proj2 = x # projection onto the orthogonal complement of W in V" + "# INSÉREZ ICI LES VALEURS DES DEUX PROJECTIONS\n", + "proj1 = 1 + 0*x # projection sur W\n", + "proj2 = x # projection sur le complément orthogonal de W en V" ] }, { "cell_type": "code", "execution_count": null, "metadata": {}, "outputs": [], "source": [ - "corrections.Ex3Chapitre9_8_9(proj1, proj2, \n", + "corrections.Ex2Chapitre9_8_9(proj1, proj2, \n", " int_limits=int_limits, \n", " case_nb=case_number)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ - "## Exercice 4\n", + "## Exercice 3\n", + "\n", + "Soit $V = \\mathbb{R}^n, \\ n=2,3$. Étant donné les ensembles de vecteurs suivants $\\mathcal{S}$, générant un sous-espace $W$ de $V$, déterminez une *base orthogonale* pour $W^\\perp$.\n", "\n", - "Give a basis for $W$ and ask to compute a basis for $W^\\perp$." + "1. $ \\qquad V=\\mathbb{R}^2 \\qquad \\mathcal{S} = \\left\\{ \\begin{pmatrix} 1 \\\\ 2 \\end{pmatrix}, \\begin{pmatrix} 1 \\\\ 0 \\end{pmatrix} \\right\\}$\n", + "2. $ \\qquad V=\\mathbb{R}^2 \\qquad \\mathcal{S} = \\left\\{ \\begin{pmatrix} 1 \\\\ -1 \\end{pmatrix}\\right\\}$\n", + "3. $ \\qquad V=\\mathbb{R}^3 \\qquad \\mathcal{S} = \\left\\{ \\begin{pmatrix} 1 \\\\ 2 \\\\1 \\end{pmatrix}, \\begin{pmatrix} 0 \\\\ 1 \\\\ 0 \\end{pmatrix}, \\begin{pmatrix} -2 \\\\ 0 \\\\ -2 \\end{pmatrix} \\right\\}$\n", + "4. $ \\qquad V=\\mathbb{R}^3 \\qquad \\mathcal{S} = \\left\\{ \\begin{pmatrix} -1 \\\\ 1 \\\\ 2 \\end{pmatrix}\\right\\}$\n", + "\n", + "### Instructions \n", + "1. Si $W^\\perp$ ne contient que le vecteur nul (i.e. si $W \\equiv \\mathbb{R}^n$), entrez simplement une liste vide comme suit \"base = $[ \\ ]$\" (et non comme \"base = $[[ \\ ]]$\").\n", + "2. Si $W^\\perp$ a une dimension supérieure à $0$, entrez les éléments de sa base sous forme de liste de listes. Ainsi, même si la base de $W^ \\perp$ contient un élément, entrez-le comme \"base = $[[a, b]]$\" et non comme \"base = $[a, b]$\"." + ] + }, + { + "cell_type": "code", + "execution_count": 6, + "metadata": {}, + "outputs": [], + "source": [ + "case_number = 4\n", + "basis = [[1,1,0], [1,-1,1]]" + ] + }, + { + "cell_type": "code", + "execution_count": 7, + "metadata": {}, + "outputs": [ + { + "data": { + "text/latex": [ + "C'est correct!" + ], + "text/plain": [ + "" + ] + }, + "metadata": {}, + "output_type": "display_data" + }, + { + "data": { + "application/vnd.plotly.v1+json": { + "config": { + "linkText": "Export to plot.ly", + "plotlyServerURL": "https://plot.ly", + "showLink": false + }, + "data": [ + { + "marker": { + "color": [ + "blue" + ], + "line": { + "color": "DarkSlateGrey", + "width": 2 + }, + "size": [ + 0, + 2 + ] + }, + "name": "$w_0$", + "type": "scatter3d", + "x": [ + 0, + -1 + ], + "y": [ + 0, + 1 + ], + "z": [ + 0, + 2 + ] + }, + { + "marker": { + "color": [ + 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