diff --git a/Chapitre 4 - Bases et dimension/Prototype 4.10.ipynb b/Chapitre 4 - Bases et dimension/Prototype 4.10.ipynb
new file mode 100644
index 0000000..c4bd547
--- /dev/null
+++ b/Chapitre 4 - Bases et dimension/Prototype 4.10.ipynb	
@@ -0,0 +1,277 @@
+{
+ "cells": [
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {
+    "jupyter": {
+     "source_hidden": true
+    }
+   },
+   "outputs": [],
+   "source": [
+    "%cd ..\n",
+    "import Librairie.AL_Fct as al\n",
+    "%cd \"Chapitre 4 - Bases et dimension\"\n",
+    "\n",
+    "import numpy as np\n",
+    "from IPython.display import display, Markdown, Latex\n",
+    "import plotly.graph_objs as go\n",
+    "import plotly\n",
+    "import ipywidgets as widgets\n",
+    "from ipywidgets import interact, interactive, fixed, interact_manual\n",
+    "from ipywidgets import Layout, HBox, VBox, Label\n",
+    "#import matplotlib\n",
+    "#import plotly.express as px"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "# Concept(s)-clé(s)\n",
+    "\n",
+    "\n",
+    "\n",
+    "### LEMME 5 :\n",
+    "\n",
+    "Soit $S$ un système de $m$ équations linéaires à $n$ inconnues, $A$ la matrice des coefficients de $S$ et $\\hat{A}$ sa matrice augmentée. Alors $S$ possède une solution si et seulement si le rang colonne de $A$ est égal au rang colonne de $\\hat{A}.$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "### EXERCICE 1.1\n",
+    "\n",
+    "Soit 𝑆 un système de 𝑚 équations linéaires à 𝑛 inconnues:\n",
+    "\n",
+    "$\\begin{equation}\n",
+    "  \\left\\{\n",
+    "      \\begin{aligned}\n",
+    "        x_1 + 4x_2 + 3x_3 + 4x_4 = 1 \\\\ \n",
+    "        2x_1 + 6x_2 + 5x_3 + 8x_4 = 1 \\\\\n",
+    "        x_1  + x_3 + 4x_4 = 1\n",
+    "      \\end{aligned}\n",
+    "    \\right.\n",
+    "\\end{equation}$\n",
+    "\n",
+    "Cette quation peut se repréenseter sous la forme  $AX =b$.\n",
+    "Ecrivez A la matrice et le vecteur b dans la cellule suivante en respectant\n",
+    "\n",
+    "$[[ a, b], [c, d]] = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix}$\n",
+    "\n",
+    "et b en suivant le format \n",
+    "\n",
+    "$[e, f] = \\begin{pmatrix} e \\\\ f \\end{pmatrix}$"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {
+    "jupyter": {
+     "source_hidden": true
+    }
+   },
+   "outputs": [],
+   "source": [
+    "def exercice1_1():\n",
+    "    A_sol = [[1, 4, 3, 4],[2, 6, 5, 8],[1, 0, 1, 4]]\n",
+    "    b_sol = [1, 1, 1]\n",
+    "    if A == [] or b == []:\n",
+    "        print(\"Au moins une des deux entrée est vide\")\n",
+    "    elif not (len(A) == len(b)):\n",
+    "        print(\"Les tailles de la matrice et du vecteur ne correspondent pas\")\n",
+    "    else:\n",
+    "        if A == A_sol:\n",
+    "            if b == b_sol:\n",
+    "                print(\"Correct !\")\n",
+    "            else:\n",
+    "                print(\"Le vecteur b est faux, votre reponse correspond au système suivant:\")\n",
+    "                al.printSyst(A, b)\n",
+    "        elif b == b_sol:\n",
+    "            print(\"La Matrice A est fausse, votre reponse correspond au système suivant:\")\n",
+    "            al.printSyst(A, b)\n",
+    "        else:\n",
+    "            print(\"Faux, votre reponse correspond au système suivant:\")\n",
+    "            al.printSyst(A, b)"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "# n'oubliez pas d'executer la cellule\n",
+    "A = []\n",
+    "b = []\n",
+    "\n",
+    "exercice1_1() # ne pas toucher cette ligne"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "### EXERCICE 1.2\n",
+    "Pour calculer le rang colonne de $A$  nous échelonnons sa matrice transposée. Si et seulement si le rang colonne de $A$  est égal au rang colonne de $\\hat{A}$.\n",
+    "À partir de cette matrice, donnez le rang colonne de $A$  et trouvez une base de l'espace colonne de $A$ ."
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {
+    "jupyter": {
+     "source_hidden": true
+    }
+   },
+   "outputs": [],
+   "source": [
+    "A_sol = np.array([[1, 4, 3, 4],[2, 6, 5, 8],[1, 0, 1, 4]])\n",
+    "A_sol_t = A_sol.transpose()\n",
+    "M = al.echelonMat('E', A_sol_t)\n",
+    "                     "
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {
+    "jupyter": {
+     "source_hidden": true
+    }
+   },
+   "outputs": [],
+   "source": [
+    "def exercice1_2_1():\n",
+    "    \n",
+    "    text_rang = widgets.IntText()\n",
+    "    box = VBox([Label('Rang colonne de A:'), text_rang])\n",
+    "    button = widgets.Button(description='Vérifier')\n",
+    "    out = widgets.Output()\n",
+    "    \n",
+    "    def callback(e):\n",
+    "        out.clear_output()\n",
+    "        with out:\n",
+    "            if(text_rang.value == 2):\n",
+    "                print('Correct !')\n",
+    "            else: \n",
+    "                print('False, try again or check the solution')\n",
+    "                     \n",
+    "    button.on_click(callback)\n",
+    "    display(box)\n",
+    "    display(button)\n",
+    "    display(out)\n",
+    "\n",
+    "exercice1_2_1()"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {
+    "jupyter": {
+     "source_hidden": true
+    }
+   },
+   "outputs": [],
+   "source": [
+    "def exercice1_2_2():\n",
+    "  \n",
+    "    base_sol = [[1,2,1],[0,1,2]]\n",
+    "    \n",
+    "    if base_sol == []:\n",
+    "        print(\"L'entrée est vide\")\n",
+    "    else:\n",
+    "        if base == base_sol:\n",
+    "            print(\"Correct !\")\n",
+    "        else:\n",
+    "            print(\"Faux\")\n",
+    "    "
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "# [vect1,vect2,...] avec vect = [a, b, c] n'oubliez pas d'executer la cellule\n",
+    "base = [[1,2,1],[0,1,2]]\n",
+    "exercice1_2_2()"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "### EXERCICE 1.3 \n",
+    "Le système possède-t-il une solution ? (utilisez le lemme 5)\n"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {
+    "jupyter": {
+     "source_hidden": true
+    }
+   },
+   "outputs": [],
+   "source": [
+    "def exercise_1_3():\n",
+    " \n",
+    "    radio = widgets.RadioButtons(\n",
+    "        options=['Yes', 'No'],\n",
+    "        description='Answer:',\n",
+    "        disabled=False\n",
+    "    )\n",
+    "\n",
+    "    button = widgets.Button(description='Vérifier')\n",
+    "    out = widgets.Output()\n",
+    "\n",
+    "    def callback(e):\n",
+    "        out.clear_output()\n",
+    "        with out:\n",
+    "            if (radio.value == \"Yes\"):\n",
+    "                    print('Mauvaise réponse.')\n",
+    "            else: \n",
+    "                print('Correct !')\n",
+    "                     \n",
+    "    button.on_click(callback)\n",
+    "    display(radio)\n",
+    "    display(button)\n",
+    "    display(out)\n",
+    "    \n",
+    "exercise_1_3()\n",
+    "\n",
+    "\n",
+    "\n"
+   ]
+  }
+ ],
+ "metadata": {
+  "kernelspec": {
+   "display_name": "Python 3",
+   "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": "3.7.4"
+  }
+ },
+ "nbformat": 4,
+ "nbformat_minor": 4
+}
diff --git a/Chapitre 4 - Bases et dimension/Prototype 4.9.ipynb b/Chapitre 4 - Bases et dimension/Prototype 4.9.ipynb
new file mode 100644
index 0000000..22e4385
--- /dev/null
+++ b/Chapitre 4 - Bases et dimension/Prototype 4.9.ipynb	
@@ -0,0 +1,314 @@
+{
+ "cells": [
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {
+    "jupyter": {
+     "source_hidden": true
+    }
+   },
+   "outputs": [],
+   "source": [
+    "%cd ..\n",
+    "import Librairie.AL_Fct as al\n",
+    "%cd \"Chapitre 4 - Bases et dimension\"\n",
+    "\n",
+    "import numpy as np\n",
+    "from IPython.display import display, Markdown, Latex\n",
+    "import plotly.graph_objs as go\n",
+    "import plotly\n",
+    "import ipywidgets as widgets\n",
+    "from ipywidgets import interact, interactive, fixed, interact_manual, Layout, HBox, VBox"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "# Concept(s)-clé(s)\n",
+    "\n",
+    "\n",
+    "\n",
+    "### DÉFINITION 1 :\n",
+    "\n",
+    "Soit $A\\in M_{m\\times n}(\\mathbb{R})$ une matrice de taille $m\\times n$ à coefficients réels.\n",
+    "\n",
+    "\n",
+    "\n",
+    "1. Le *rang ligne* de $A$ est la dimension de l'espace ligne de $A.$\n",
+    "\n",
+    "2. Le *rang colonne* de $A$ est la dimension de l'espace colonne de $A.$\n",
+    "\n",
+    "\n",
+    "\n",
+    "### REMARQUES 2 :\n",
+    "\n",
+    "\n",
+    "\n",
+    "1. Le rang ligne de $A$ est plus petit ou égal à $n,$ car c'est un sous-espace vectoriel de $\\mathbb{R}^n.$\n",
+    "\n",
+    "2. Le rang ligne de $A$ est plus petit ou égal à $m,$ car engendré par $m$ vecteurs.\n",
+    "\n",
+    "3. Le rang colonne de $A$ est plus petit ou égal à $m$ et $n,$ par le même raisonnement.\n",
+    "\n",
+    "4. Le rang colonne de $A$ est égal au rang ligne de $A^T.$\n",
+    "\n",
+    "5. Le rang ligne de $A$ est égal au rang colonne de $A^T.$\n",
+    "\n",
+    "\n",
+    "\n",
+    "### PROPOSITION 3 :\n",
+    "\n",
+    "Soient $A,B\\in M_{m\\times n}(\\mathbb{R})$ des matrices ligne équivalentes. Alors l'espace ligne de $A$ est égal à l'espace ligne de $B.$ Par conséquent, le rang ligne de $A$ est égal au rang ligne de $B.$\n",
+    "\n",
+    "### PROPOSITION 4 :\n",
+    "\n",
+    "Soit $A$ une matrice échelonnée. Alors le rang ligne de $A$ est égal au nombre de pivots. Aussi, une base de l'espace ligne de $A$ est donnée par les lignes contenant un pivots."
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "### EXERCICE 1\n",
+    "\n",
+    "Soit la matrice A :\n",
+    "\n",
+    "$$A = \\begin{pmatrix} 1 & 2 & 3 \\\\ 4 & 5 & 6  \\end{pmatrix}$$\n",
+    "\n",
+    "Sélectionnez les propositions correctes\n",
+    "(maintenez CTRL pour sélectionner plusieurs cellules à la fois)"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {
+    "jupyter": {
+     "source_hidden": true
+    }
+   },
+   "outputs": [],
+   "source": [
+    "def exercise_1():\n",
+    "    r1 = 'Le rang ligne de 𝐴 est plus petit ou égal à 2, car c\\'est un sous-espace vectoriel de ℝ2.'\n",
+    "    r2 = 'Le rang ligne de 𝐴 est plus petit ou égal à 3, car c\\'est un sous-espace vectoriel de ℝ3.'\n",
+    "    r3 = 'Le rang ligne de 𝐴 est plus petit ou égal à 3, car engendré par 3 vecteurs.'\n",
+    "    r4 = 'Le rang ligne de 𝐴 est plus petit ou égal à 2, car engendré par 2 vecteurs.'\n",
+    "    r5 = 'Le rang colonne de A est plus petit ou égal à 2.'\n",
+    "\n",
+    "    select = widgets.SelectMultiple(\n",
+    "        options=[(r1, 1), (r2, 2), (r3, 3), (r4, 4), (r5, 5)],\n",
+    "        description='Sélection :',\n",
+    "        disabled=False,\n",
+    "        layout=Layout(width='auto', height='100px')\n",
+    "    )\n",
+    "\n",
+    "    button = widgets.Button(description='Vérifier')\n",
+    "    out = widgets.Output()\n",
+    "\n",
+    "    def callback(e):\n",
+    "        out.clear_output()\n",
+    "        with out:\n",
+    "            if (1 in select.value or 3 in select.value):\n",
+    "                    print('Mauvaise réponse. \\nAttention à ne pas confondre les espaces des lignes et colonnes')\n",
+    "            elif (2 not in select.value or 4 not in select.value or 5 not in select.value):\n",
+    "                print('Il manque au moins une réponse.')\n",
+    "            elif (2 in select.value and 4 in select.value and 5 in select.value):\n",
+    "                print('Correct !')\n",
+    "                     \n",
+    "    button.on_click(callback)\n",
+    "\n",
+    "    display(select)\n",
+    "    display(button)\n",
+    "    display(out)\n",
+    "    \n",
+    "exercise_1()"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "### EXERCICE 2\n",
+    "#### 2.1\n",
+    "Soit la matrice $A$ :\n",
+    "\n",
+    "$$A = \\begin{pmatrix} 1 & 2 & 3 \\\\ 0 & 1 & 2  \\end{pmatrix}$$\n",
+    "\n",
+    "Donnez le rang ligne de $A$."
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {
+    "jupyter": {
+     "source_hidden": true
+    }
+   },
+   "outputs": [],
+   "source": [
+    "def exercice_2_1():\n",
+    "    text = widgets.IntText(\n",
+    "        description='Réponse :',\n",
+    "        disabled=False\n",
+    "    )\n",
+    "    button = widgets.Button(description='Vérifier')\n",
+    "    out = widgets.Output()\n",
+    "    \n",
+    "    def callback(e):\n",
+    "        out.clear_output()\n",
+    "        r = text.value\n",
+    "        \n",
+    "        with out:\n",
+    "            if r == 2:\n",
+    "                display(Markdown(\"Par la proposition 4 c'est  correct!\"))\n",
+    "            elif r > 2:\n",
+    "                display(Markdown(\"Faux, c'est trop grand\"))\n",
+    "            else:\n",
+    "                display(Markdown(\"Faux, c'est trop petit\"))\n",
+    "        \n",
+    "    button.on_click(callback)\n",
+    "    \n",
+    "    display(text)\n",
+    "    display(button)\n",
+    "    display(out)\n",
+    "exercice_2_1()"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "#### 2.2\n",
+    "Parmi les ensembles suivants, séléctionnez l'ensemble générant l'espace ligne de $A$\n",
+    "\n",
+    "$E_1 = \\begin{Bmatrix}\\begin{pmatrix} 1 \\\\ 2 \\\\ 3 \\end{pmatrix}, \\begin{pmatrix} 0 \\\\ 1 \\\\ 2 \\end{pmatrix}\\end{Bmatrix}$ $E_2 = \\begin{Bmatrix}\\begin{pmatrix} 1 \\\\ 2 \\\\ 3 \\end{pmatrix}, \\begin{pmatrix} 0 \\\\ 1 \\\\ 2 \\end{pmatrix}, \\begin{pmatrix} 0 \\\\ 0 \\\\ 1 \\end{pmatrix}\\end{Bmatrix}$ \n",
+    "$E_3 = \\begin{Bmatrix}\\begin{pmatrix} 1 \\\\ 0\\end{pmatrix}, \\begin{pmatrix} 2 \\\\ 1 \\end{pmatrix}, \\begin{pmatrix} 3 \\\\ 2 \\end{pmatrix}\\end{Bmatrix}$\n",
+    "$E_4 = \\begin{Bmatrix}\\begin{pmatrix} 1 \\\\ 0\\end{pmatrix}, \\begin{pmatrix} 2 \\\\ 1 \\end{pmatrix}\\end{Bmatrix}$"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {
+    "jupyter": {
+     "source_hidden": true
+    }
+   },
+   "outputs": [],
+   "source": [
+    "def exercise_2_2():\n",
+    "    select = widgets.SelectMultiple(\n",
+    "        options=['E1', 'E2', 'E3', 'E4'],\n",
+    "        description='Sélection :',\n",
+    "        disabled=False,\n",
+    "        layout=Layout(width='auto', height='auto')\n",
+    "    )\n",
+    "\n",
+    "    button = widgets.Button(description='Vérifier')\n",
+    "\n",
+    "    out = widgets.Output()\n",
+    "\n",
+    "    def callback(e):\n",
+    "        \n",
+    "        out.clear_output()\n",
+    "        with out:\n",
+    "            if len(select.value) <= 0: # Empty\n",
+    "                pass\n",
+    "            elif len(select.value) > 1:\n",
+    "                display(Markdown(\"Un seul de ces ensembles génère exactement l'espace ligne de A\"))\n",
+    "            elif 'E1' not in select.value:\n",
+    "                display(Markdown(\"Faux\"))\n",
+    "            else:\n",
+    "                display(Markdown(\"Correct !\"))\n",
+    "    button.on_click(callback)\n",
+    "    display(select)\n",
+    "    display(button)\n",
+    "    display(out)\n",
+    "    \n",
+    "exercise_2_2()"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "#### 2.3\n",
+    "Utilizez la remarque 2 ainsi que les propositions 3 et 4 pour trouver le rang colonne de $A$"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {
+    "jupyter": {
+     "source_hidden": true
+    }
+   },
+   "outputs": [],
+   "source": [
+    "\n",
+    "\n",
+    "def exercise_2_3():\n",
+    "    text = widgets.IntText(description='Réponse :', disabled=False)\n",
+    "\n",
+    "    button = widgets.Button(description='Vérifier')\n",
+    "    button2 = widgets.Button(description='Solution', disabled=True)\n",
+    "    box = HBox(children=[button,button2])\n",
+    "    out = widgets.Output()\n",
+    "\n",
+    "    def callback(e):\n",
+    "        out.clear_output()\n",
+    "        button2.disabled = False\n",
+    "        with out:\n",
+    "            if(text.value == 2):\n",
+    "                print('Correct !')\n",
+    "            else: \n",
+    "                print('False, try again or check the solution')\n",
+    "    \n",
+    "    def solution(e):\n",
+    "        out.clear_output()\n",
+    "        with out:\n",
+    "            A = np.array([[1, 2, 3], [0, 1, 2]])\n",
+    "            A_t = A.transpose()\n",
+    "            display(Markdown('Pour trouver le rang colonne de $A$, nous utilisons la remarque 2 et trouvous le rang ligne de la transposée de $A$.'))\n",
+    "            display(Markdown('Par les propositions 3 et 4, nous échelonnons la matrice transposée et observont le nombre de lignes qui contiennent des pivots'))\n",
+    "            M = al.echelonMat('E', A_t)\n",
+    "            display(Markdown('Ainsi le rang colonne de $A$ est 2'))\n",
+    "                      \n",
+    "    button.on_click(callback)\n",
+    "    button2.on_click(solution)\n",
+    "    \n",
+    "    display(text)\n",
+    "    display(box)\n",
+    "    display(out)\n",
+    "    \n",
+    "exercise_2_3()\n"
+   ]
+  }
+ ],
+ "metadata": {
+  "kernelspec": {
+   "display_name": "Python 3",
+   "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": "3.7.4"
+  }
+ },
+ "nbformat": 4,
+ "nbformat_minor": 4
+}