AL_101.000_Espacios_Vectoriales.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Espacios vectoriales"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "\n",
    "Un  **Espacio  Vectorial $V$ de valor real** es:\n",
    "\n",
    "* (I) un __conjunto__ no vacío que tiene:\n",
    "* (II)  una __operación suma__ ($+$) que cumple:\n",
    " *  (A1) la ley clausurativa ${u}+{v} \\in V$\n",
    " *  (A2) la ley conmutativa ${u}+{v}={v}+{u}$\n",
    " *  (A3) la ley asociativa $({u}+{v})+{w}={u}+({v}+{w})$\n",
    "* (III)  un __elemento cero__ ($\\vec{0}$)  que cumple:\n",
    " *  (A4) la ley modulativa ${v}+\\vec{0}={v}$\n",
    "* (IV) Para cada elemento ${v}$ un __elemento opuesto__ $(-{v})$  que cumple:\n",
    " *  (A5) la ley del opuesto ${v}+(-{v})=\\vec{0}$\n",
    "* (V) una __opración multiplicación por escalar__  que cumple:\n",
    " *  (M1) la ley clusurativa $a{v} \\in V$\n",
    " *  (M2) la ley distributiva $(a+b){v}=a{v}+b{v}$\n",
    " *  (M3) la ley distributiva $a({u}+{v})=a{u}+a{v}$\n",
    " *  (M4) la ley `asociativa' $a(b{v})=(ab){v}$\n",
    " *  (M5) la ley de la identidad $1{v}={v}$\n",
    "\n",
    "donde ${u},{v},{w} \\in V$ y $a,b \\in R$\n",
    "\n",
    "\n",
    "Algunos espacios vectoriales son:\n",
    "\n",
    "|Conjunto|Símbolo|\n",
    "|-|-|\n",
    "| números reales | $\\mathbb{R}$ |\n",
    "| vectores de n dimensiones | $\\mathbb{R}^n$ |\n",
    "| funciones $f:\\mathbb{Z} \\rightarrow \\mathbb{R}$ | $\\mathbb{R}^\\mathbb{Z}$ |\n",
    "| funciones $f:\\mathbb{R} \\rightarrow \\mathbb{R}$ | $\\mathbb{R}^\\mathbb{R}$ |\n",
    "| matrices | $M_{m \\times n}$ |\n",
    "| polinomios de grado $n$ o menor | $P_n$ |\n",
    "| El conjunto de los polinomios | $P$ | \n",
    "\n",
    "\n",
    "\n",
    "<img src=\"AL_dim23nZR.png\" /> \n",
    "\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Norma o magnitud de un vector\n",
    "\n",
    "La norma de u"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "\n",
    "\n",
    "\n",
    "\n",
    "\n",
    "En esta página enumeramos algunos conjuntos que forman espacios vectoriales. Además, se ilustra como se definen algunos de sus elementos con python y su librería simbólica sympy."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "from sympy import * #importa la librería simbolica de python\n",
    "from sympy import init_printing ;init_printing() # mejora la presentación"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Números Reales $R$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "x=Rational(1,2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAAsAAAAqBAMAAACXcryGAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAzRAiu5mrdu/dZjKJ\nRFRer8KoAAAAcklEQVQYGWNgYGBUYAACZn8QxeSaD6IYGOqpQYX1T1sAMow88B8EPhCl1zTEG6iO\nKYHh/AUGBjYFBo4GBgaOCQxsv4G++wSmgNKcn8Bm3T8ApqaASVYHMFUGJvkCGIyBjK0MDMXA8Jhz\nJmwCUDvQDRMYAFKIIjdUG6QaAAAAAElFTkSuQmCC\n",
      "text/latex": [
       "$$\\frac{1}{2}$$"
      ],
      "text/plain": [
       "1/2"
      ]
     },
     "execution_count": 3,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "x"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Vectores $R^n$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "v=Matrix([1,2,3])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAABoAAABLCAMAAABZRmeuAAAAPFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAo1xBWAAAAE3RSTlMA\nMquZdlQQQOkwRM0iu+/dZolsKH0JNQAAATZJREFUOBHtldmawyAIRt2SaROXOLz/u1ZAzUdCbudm\n6kVqOeDyC2osUHNmtsIWYyz40NoyiTnwvwNE9rSeveOK9nXAC4oJdLQm7x5QGyp80VTeuwzFHSzw\nRd6hOv5+0bMa0fuURioLoVasiAobxwrkKHNzVlAuaHT9jETUTqmrIh6o9NwWUYQ2qOxyRyUxuZ+y\nH4TQz+vdHVsRncX++xK1vPjmFCN5yrksEhMUFEut1Tva+CXZdr5JNERTzI+ca5qx8+9RbPKmcQcL\nNSKeyQLaDRAAI0BTfsstp1fYUVxNXn1Act4pCe5RtpZBlPOquReYWDwNaHLmV0AgTsBEW7isEKhG\nUt+ziMqU7wWUFD1QogUCzymizObbFdCrXFk8h7SvjJpm7Pw9enyyV3ygQxj7wcWxxXwADY4Zi0gr\naeYAAAAASUVORK5CYII=\n",
      "text/latex": [
       "$$\\left[\\begin{matrix}1\\\\2\\\\3\\end{matrix}\\right]$$"
      ],
      "text/plain": [
       "⎡1⎤\n",
       "⎢ ⎥\n",
       "⎢2⎥\n",
       "⎢ ⎥\n",
       "⎣3⎦"
      ]
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "v"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAFIAAAAZBAMAAABQnWQHAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAu90izRCZq3bvZjKJ\nRFS0c0oyAAAA+UlEQVQ4EdXTsQ7BQBzH8V+0JRJtxGAxtDqwSEgkvEYfgUVXb2AwGMsoBmYPYDKw\nS3Q16TMIiTDUpa3mf3INHd3U633yvV7TomS28H2oZgPGdxaIbkopW1y406/QuXTYe0DQlHROKkMc\nXUI70B6hVMojTuYs5KdEOh4uURMOJ/MGck8iT65yFUv5xkvEu+OjyXKZG2kCck/cZOi8o3JbY7Pw\nffLPyW5XKQQKiySZLfIScyuhOeFgG9A9sdR6YKvx8C3oK7HcAOPYATOg2RJKqXGwDSLX0O7Riexl\nfUVWZN/3qVQHffd9dsISL9N+n4khsvA/TfPH/73+ApdhOk6XQiZRAAAAAElFTkSuQmCC\n",
      "text/latex": [
       "$$\\left[\\begin{matrix}1 & 2 & 3\\end{matrix}\\right]$$"
      ],
      "text/plain": [
       "[1  2  3]"
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "v.T #transpuesta"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "collapsed": true
   },
   "source": [
    "## Matrices $M_{m \\times n}$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "A=Matrix([[1,2],[4,5],[7,8],[10,11]])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAE4AAABkCAMAAAAfQ/Z/AAAAP1BMVEX///8AAAAAAAAAAAAAAAAA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\n8A9o2S9txd0RpQAAAABJRU5ErkJggg==\n",
      "text/latex": [
       "$$\\left[\\begin{matrix}1 & 2\\\\4 & 5\\\\7 & 8\\\\10 & 11\\end{matrix}\\right]$$"
      ],
      "text/plain": [
       "⎡1   2 ⎤\n",
       "⎢      ⎥\n",
       "⎢4   5 ⎥\n",
       "⎢      ⎥\n",
       "⎢7   8 ⎥\n",
       "⎢      ⎥\n",
       "⎣10  11⎦"
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "A"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAH4AAAAyBAMAAABol3KsAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nDw8+mL033p/BYmbv0eMf9ft9xP8PfwH6oMTxd26o+QAAAABJRU5ErkJggg==\n",
      "text/latex": [
       "$$\\left[\\begin{matrix}1 & 4 & 7 & 10\\\\2 & 5 & 8 & 11\\end{matrix}\\right]$$"
      ],
      "text/plain": [
       "⎡1  4  7  10⎤\n",
       "⎢           ⎥\n",
       "⎣2  5  8  11⎦"
      ]
     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "A.T #transpuesta"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Funciones de valor real $F(R)$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "x=symbols('x') #se define la variable de la funcion"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "f=cos(x)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAADkAAAAVBAMAAAAOWFv7AAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAEJmJVCLvMs1Edmar\n3bti/yyrAAABP0lEQVQoFZWRL0gDURzHP+ft3J1uczgwCQ7EYpqoYVNwBovFSxbTRJNlaaAiHtj1\ngsmyjRXRIoIYRLRYTAMRbF5eGjIQTf52vL0NXfEX3vu83/f3770H/7R9FX/bL88KlDdW7CPbyY7T\n60DPXtPcJe1iW+OYJg1OXaOdh9HZOYxczWdv9VwEMw3GxdNZLkRri5J74NIwAuKiJq4gxX3lDYY9\nbI+14gsc7hYwRY3nYYal5ApE6kxIpWgLRjLvDVfUQR+SHAsRbVKuSNCXqEHq5EOp8K1UnTvl8iB5\n7cpEPzFAxpe+5DehfJohJhO1p1oealINp7J2MN1SkaNIgYQr13jEaSWalgQNpGF8eh5rctF3Fi6z\nUlTKGc/V2p2g7cvyy171eV1TF641ynP9sXbz0AxPQe+mf1+H9apsqNMN/AAQ00PuNbV4pAAAAABJ\nRU5ErkJggg==\n",
      "text/latex": [
       "$$\\cos{\\left (x \\right )}$$"
      ],
      "text/plain": [
       "cos(x)"
      ]
     },
     "execution_count": 12,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "f"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": 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zudGm7uc//zkTJ04kMjKSZcuWUVdXd8MxBy9W0tTepdfFIEravXs32dnZJrnU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\nCIKB+P+mXvvotzvbPAAAAABJRU5ErkJggg==\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x5e42150>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/plain": [
       "<sympy.plotting.plot.Plot at 0x5de00d0>"
      ]
     },
     "execution_count": 13,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "plotting.plot(f,(x,-5,5))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "y=f.subs(x,1) #sustituye x por 1"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAADcAAAAVBAMAAAAQkWtIAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAEJmJVCLvMs1Edmar\n3bti/yyrAAABIklEQVQoFWNgIA1UQ5Rvx6aL9QFQtJGBgacAiyyHAAND3EegRAMWycUMDHI6IEkg\nAwOkAkW4QJJiGFIMnBdgkhwODAxCxiYMjFaLJzBUea0AijMrwCSBLNYUhrKA2gCGV4wPGHiB4nwb\nYJLcDQwcDQz+BZcZGOorExmYgeK8DjBJlgsM8goMDOxfGBj4De6/CgCKs02ASbJ/YJi/gIGB5SdQ\n8oFw/yd0SbhO1QCG/UBdCGOB7gbayeCQxMAwf4oBAw/QMSgOYs1mYA4oK2BoZklk4ANaynwAaDY4\nEJgUGBhktEwZWJVtJnBar7YEioMCgenY39MMDBxAS9DBVZhAGIyBRG+Csa1gDCQaZDEIMDaAKVQC\nHNlAIZgiVNloCHcrAwMAdOVEGRxKsBoAAAAASUVORK5CYII=\n",
      "text/latex": [
       "$$\\cos{\\left (1 \\right )}$$"
      ],
      "text/plain": [
       "cos(1)"
      ]
     },
     "execution_count": 15,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "y"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAKAAAAAPBAMAAACRq9klAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nS3MaHzXPDni+W0L+6/UXCHYJ0EbLLOgAAAAASUVORK5CYII=\n",
      "text/latex": [
       "$$0.54030230586814$$"
      ],
      "text/plain": [
       "0.540302305868140"
      ]
     },
     "execution_count": 16,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "y.evalf() # evalua la expreción"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAEoAAAAVBAMAAAATJmVgAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAEM3dMnZUu0TviSKZ\nZquXRLrEAAABSklEQVQoFaWSPUvDYBSFTz6a1ITYUOziIBVEdHIQXc2i4NQidBH8AHHP0kkkgj/A\ndHOSjA62vIOgi+jg6BD8A2bQubhL8b4mNwmazQvJeXI4OfcNBPjn6HFa4Hi/i7a7hXPH+M7A2omY\ngHlGlaFCaxGbBbFT6JRgVkKmTG+93DjPCYsFYvN4xw4Sffm59yHdG7pa+/e79D0Ss1F8aDATqH08\nymWngBI5a1obeKXHbOoDWDATmCEaMXnrgOVanwa9cYTmqpwFsjf64ifVRoO24JK6oPpE6MhbNrMr\nXzCpy89TSKmcsgXezkop2kgtHlyg9LlGhOm4lKLTG+4BapQqnd4YwBS0jjeOgIvDAA/U+CJr06kP\n96718WRmPNkKrsiiNXO91jAmXMoiFaIJNu2Q6a86T+xZXaYKPWGvyVCl+V+Yx6tStjw4jSPwDWX2\nRQ0oD4CmAAAAAElFTkSuQmCC\n",
      "text/latex": [
       "$$- \\sin{\\left (x \\right )}$$"
      ],
      "text/plain": [
       "-sin(x)"
      ]
     },
     "execution_count": 17,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "diff(f,x) #la expreción se puede derivar"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "collapsed": true
   },
   "source": [
    "## Polinomios $P$ "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Los polinomios son un caso particular de las funciones de valor real"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "x=symbols('x') # Se define la variable del polinomio"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "p=3+2*x-5*x**2+7*x**3 #Luego se define el polinomio"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAKoAAAAWBAMAAACiUBq/AAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\ngB6wTFn9G28o9F8aEKe88yuX3AAAAABJRU5ErkJggg==\n",
      "text/latex": [
       "$$7 x^{3} - 5 x^{2} + 2 x + 3$$"
      ],
      "text/plain": [
       "   3      2          \n",
       "7⋅x  - 5⋅x  + 2⋅x + 3"
      ]
     },
     "execution_count": 20,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "p"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": 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GfiiuY7ATUZ/jkL9GoNUJPP15Jg5eLMdTsaMY7ETU5xjuBiaEwF+2ncfOs8X4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\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x9dc6430>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/plain": [
       "<sympy.plotting.plot.Plot at 0x5de0fb0>"
      ]
     },
     "execution_count": 24,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "plotting.plot(p,(x,-1,1))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "y=p.subs(x,-1) #sustituye x por -1"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAACIAAAAPBAMAAABzWwAZAAAAJ1BMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAilU6eAAAADHRSTlMAEM3dMiK7mat272a9noTYAAAAP0lE\nQVQYGWNggANFEAtMQIXCDgIZYAIqIFYBFAETUAEGBk6QGjABE6JQRMgYBFRoYDLMgexdh7sZwARM\nBE4DAPRCGcrmow14AAAAAElFTkSuQmCC\n",
      "text/latex": [
       "$$-11$$"
      ],
      "text/plain": [
       "-11"
      ]
     },
     "execution_count": 23,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "y"
   ]
  },
  {
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