Algebra Lineal.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Algebra Lineal\n",
    "\n",
    "Nos estamos aproximando a estudiar modelos de Machine Learning. Estos modelos tienen como objetivo la extracción de información a partir de los datos, los cuales pueden ser representados como vectores y matrices, por lo que es necesario un recordar algunos conceptos de $\\textbf{Algebra Lineal}$.\n",
    "\n",
    "![Never](images/meme.jpg)\n",
    "\n",
    "\n",
    "El Álgebra Lineal es una rama de las matemáticas que estudia los espacios vectoriales y las transformaciones lineales de estos espacios, tales como rotación, escalamiento, traslación, etc.\n",
    "\n",
    "\n",
    "\n",
    "Créditos: Aurélien Géron (Hands-on Machine Learning with Scikit-learn and Tensorflow)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Vectores\n",
    "\n",
    "$\\textbf{Definición:}$ Un vector es una cantidad definida por una magnitud y una dirección. Por ejemplo la velocidad de un pájaro es un vector tridimensional, cuya magnitud es la rapidez del vuelo y su dirección. Un vector puede ser representado por un arreglo de numeros llamados *escalares*. Cada escalar corresponde a la magnitud del vector a lo largo de caa una de sus dimensiones.\n",
    "\n",
    "Por ejemplo, si un pajaro está volando a una velocidad de 13 km/h hacia el sur, 5 km/h hacia el este y 1 km/h hacia el cielo, podemos representar su velocidad mediante el siguiente vector:\n",
    "\n",
    "**velocidad** $= \\begin{pmatrix}\n",
    "13 \\\\\n",
    "5 \\\\\n",
    "1 \\\\\n",
    "\\end{pmatrix}$\n",
    "\n",
    "\n",
    "Nota: Por convención, los vectores son generalmente presentados en forma de columnas. Además, sus nombres se escriben generalmente en minuscula para distinguirlos de las matrices.\n",
    "\n",
    "Un vector con 1 elemento puede ser representado como una flecha o un punto en sobre un eje, un vector de 2 elementos puede ser representado por un punto o una flecha en un plano y un vector de 3 elementos puede ser representado por un punto o una flecha en un espacio. Así, un vector N dimensional puede ser representado por un punto o una flecha en un espacio N-dimensional.\n",
    "\n",
    "\n",
    "\n",
    "¿Por qué estamos interesados en los vectores?\n",
    "\n",
    "En Machine Learning podemos representar observaciones y predicciones a través de vectores. Por ejemplo supongamos que queremos predecir si un correo corresponde a spam o no, basado en las observaciones que tenemos de él. Podemos representar dichas observaciones como:\n",
    "\n",
    "\n",
    "**correo** $= \\begin{pmatrix}\n",
    "320 \\\\\n",
    "0.8 \\\\\n",
    "3 \\\\\n",
    "\\end{pmatrix}$\n",
    "\n",
    "\n",
    "Este vector podría representar que tenemos un correo que tiene 320 palabras, el 80% de quienes lo leen lo arrojan a la papelera de reciclaje y posee 3 imágenes adjuntas.\n",
    "\n",
    "Basado en este vector, podemos hacer que nuestro algoritmo prediga la probabilidad de que este correo corresponda a spam o no. Suponiendo que nuestro algoritmo arroja que existe un 86% de probabilidad de que corresponda a spam, nuestro vector de predicciones estaría dado por:\n",
    "\n",
    "**class_probabilities** $= \\begin{pmatrix}\n",
    "0.86 \\\\\n",
    "0.14 \\\\\n",
    "\\end{pmatrix}$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Vectores en Python.\n",
    "\n",
    "Es hora de empezar a ocupar nuestra herramienta estrella. Lo primero es saber cómo podemos representar un vector dentro de python y para ello tenemos varias opciones:\n",
    "\n",
    "1. Una lista"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[1, 2, 3, 4]"
      ]
     },
     "execution_count": 1,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "vector = [1,2,3,4]\n",
    "vector"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "2. NumPy: Esta es una de las librerías de Python para realizar cálculos vectoriales de manera fácil y optimizada."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([320. ,   0.8,   3. ])"
      ]
     },
     "execution_count": 2,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "import numpy as np\n",
    "\n",
    "vector = np.array([320, 0.8, 3])\n",
    "vector"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Podemos obtener el tamaño de este vector con el atributo .shape"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(3,)"
      ]
     },
     "execution_count": 3,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "vector.shape"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "el $i^{ésimo}$ elemento (llamado item) de un vector $\\textbf{v}$ es denotado $\\textbf{v}_i$.\n",
    "\n",
    "Observación: En Python, la indexación comienza a partir del elemento 0. Por lo cual, si queremos obtener el tercer elemento del vector, tenemos que:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "320.0"
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "vector[0]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "De manera de hacer este repaso de Álgebra Lineal más amigable y entendible, visualizaremos nuestras operaciones. Esto lo haremos a través de una de las herramientas de visualización más utilizadas de Python: matplotlib."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [],
   "source": [
    "%matplotlib inline\n",
    "import matplotlib.pyplot as plt"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Vectores bidimensionales\n",
    "\n",
    "Empezaremos creando dos vectores 2D para visualizarlos."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [],
   "source": [
    "u = np.array([2, 5])\n",
    "v = np.array([3, 1])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.scatter([1,2], [1,2], color=[\"r\",\"b\"])\n",
    "plt.grid()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "(2, 3) (5, 1)\n"
     ]
    },
    {
     "data": {
      "image/png": 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mrc+OIuLIiDjqwH3gM3R/1K1NZr4IPBcRB3673EXAUzVOmulKGnCapOdZ4JMRcUTv7+FFdF9jqlVEHNf758l0z2/fOd/xg/ztgO/mrfFDFxF3AZ8Gjo2IaeBvM/PWeldxHvAl4IneOWWAv8nMf61x0wnA7b3/ccYSYCIzG3H5XcN8BLi3+3eeZcCdmXlfvZMAuAq4o/ekaSfw5Zr3AL/55nYx8Gd1bwHIzEcjYgOwBdgHbKUZb3+/OyKOAfYCX+n34rJveZekwvjipCQVxnBLUmEMtyQVxnBLUmEMtyQVxnBLUmEMtyQV5v8BzH9TGnB5B4wAAAAASUVORK5CYII=\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "x_coords, y_coords = zip(u, v)\n",
    "print(x_coords, y_coords)\n",
    "plt.scatter(x_coords, y_coords, color=[\"r\",\"b\"])\n",
    "plt.axis([0, 9, 0, 6])\n",
    "plt.grid()\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Como dijimos, los vectores también pueden ser representados como flechas. Matplotlib nos entrea una herramienta para dibujar flechas en un gráfico. Crearemos una función que nos retorne el gráfico de una flecha centrada en el origen."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [],
   "source": [
    "def plot_vector2d(vector2d, origin=[0, 0], **options):\n",
    "    return plt.arrow(origin[0], origin[1], vector2d[0], vector2d[1],\n",
    "              head_width=0.2, head_length=0.3, length_includes_head=True, **options)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plot_vector2d(u, color=\"r\")\n",
    "plot_vector2d(v, color=\"b\")\n",
    "plt.axis([0, 9, 0, 6])\n",
    "plt.grid()\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Vectores tridimensionales\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "metadata": {},
   "outputs": [],
   "source": [
    "a = np.array([1, 2, 8])\n",
    "b = np.array([5, 6, 3])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "from mpl_toolkits.mplot3d import Axes3D\n",
    "\n",
    "subplot3d = plt.subplot(111, projection='3d')\n",
    "x_coords, y_coords, z_coords = zip(a,b)\n",
    "subplot3d.scatter(x_coords, y_coords, z_coords)\n",
    "subplot3d.set_zlim3d([0, 9])\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 27,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "def plot_vectors3d(ax, vectors3d, z0, **options):\n",
    "    for v in vectors3d:\n",
    "        x, y, z = v\n",
    "        ax.plot([x,x], [y,y], [z0, z], color=\"gray\", linestyle='dotted', marker=\".\")\n",
    "    x_coords, y_coords, z_coords = zip(*vectors3d)\n",
    "    ax.scatter(x_coords, y_coords, z_coords, **options)\n",
    "\n",
    "subplot3d = plt.subplot(111, projection='3d')\n",
    "subplot3d.set_zlim([0, 9])\n",
    "plot_vectors3d(subplot3d, [a,b], 0, color=(\"r\",\"b\"))\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Norma\n",
    "\n",
    "La norma de un vector $\\textbf{v}$ denotada como $\\textbf{||v||}$, es la medida de la magnitud de $\\textbf{v}$. El tipo de norma que discutiremos en este repaso será la norma Euclideana, la cual está definida como:\n",
    "\n",
    "$|| \\textbf{u} || = \\sqrt{\\sum_{i}{\\textbf{u}_i}^2}$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 29,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "5.385164807134504 3.1622776601683795\n"
     ]
    }
   ],
   "source": [
    "def euclidean_norm(vector):\n",
    "    squared_vector = [element**2 for element in vector]\n",
    "    return sum(squared_vector)**0.5\n",
    "\n",
    "print(euclidean_norm(u), euclidean_norm(v))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "NumPy tiene un modulo de Álgebra Lineal llamado `linalg`, el cual ofrece un método para calcular la norma llamado `norm`:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 30,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "5.385164807134504"
      ]
     },
     "execution_count": 30,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "import numpy.linalg as LA\n",
    "LA.norm(u)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "¿Es esa la magnitud de u? Comprobémoslo con matplotlib."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 32,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAW4AAAD8CAYAAABXe05zAAAABHNCSVQICAgIfAhkiAAAAAlwSFlzAAALEgAACxIB0t1+/AAAADl0RVh0U29mdHdhcmUAbWF0cGxvdGxpYiB2ZXJzaW9uIDIuMi4zLCBodHRwOi8vbWF0cGxvdGxpYi5vcmcvIxREBQAAHidJREFUeJzt3XuU3WV97/H3d657ZjJJCBkCCRSQGLooyxssbPVoEziKHKwCVQoVLXq64r2CBy8oVLS4pFRbatXTpqB4jIAYCVJaL5ySiG29QFAPkCBEBOZ+y2Tus/fsvb/nj9k7DCHJ7D2z935+v70/r7VmkZnsmd9nsWY+eeb5Pc/zM3dHRETioy50ABERKY6KW0QkZlTcIiIxo+IWEYkZFbeISMyouEVEYqag4jazlWa2zcweN7M9ZvYH5Q4mIiKH1lDg6/4e+L67v8XMmoDWMmYSEZEjsIU24JjZCuCXwItcu3VERIIrZMR9MjAIfM3MXgrsAj7k7pPzX2Rmm4HNAIlE4ozf+Z3fKXXWJctms9TVRXNaP6rZopoLoptNuYoX1WyVzPXEE08MuXtHQS929yO+AWcCaeCVuff/HvirI33Ohg0bPIp27NgROsJhRTVbVHO5RzebchUvqtkqmQt4yBfo4/xbIf+UdAFd7v6z3PvbgFcU+Y+JiIiUyILF7e59QKeZnZr70DnA7rKmEhGRwyp0VckHgW/mVpQ8BbyzfJFERORICipud/8lc3PdIiISWPRu44qIyBGpuEVEYkbFLSISMypuEZGYUXGLiMSMiltEJGZU3CIiMaPiFhGJGRW3iEjMqLhFRGJGxS0iEjMqbhGRmFFxi4jEjIpbRCRmVNwiIjGj4hYRiZlCn4BTFHfnmWeeOfC+mR14q6urO+ybmVFfX/+8NzMrR0QRkdgqS3EDZDKZol6fL2gzyz9dHnc/UPb19fU0NDQ8762xsZGGhgbq6+tLnl9EJKrKVtzFml/WB388k8mQyWRIpVIHPp4v+ny554u8qamJxsbGA2/5kbyISLWITHEXa37Buzuzs7PMzs4yNTX1glJvbGwkkUiQzWZJpVI0NjaqzEUktmJb3EdycKmnUilSqRTpdJru7m7cnYaGBpqbm0kkEjQ3N9PU1ERdne7Vikj0VWVxH0m+1NPpNOl0mqmpqQMfb2hooKWlhZaWFhKJBA0NNfe/R0RioOabaf7oPJ1OMz4+zsTEBO5OfX09iUSC1tbWA0WuKRYRCa3mi/tQ8mWeyWSYnJw8MCoHaGlpoa2tjdbWVq1mEZEgVNwFmD8qn5qaYnp6GnensbHxQIk3NzdrNC4iFaHiXoR8kc/OzrJ//35GR0cBjcZFpDJU3CWQL/L5o/Gmpiba29tpa2vTTU4RKamCGsXMngbGgQyQdvczyxkqzvIlnkql2LdvH8PDwzQ1NbF8+XLa2to0EheRJStmKLjJ3YfKlqQKzS/x4eFhhoaGaG5uPjASV4mLyGLod/gKyZd4MpkklUoxNDREIpFgxYoVtLa2Bk4nInFS6FZBB35oZrvMbHM5A9WCfInPzMyw78EHGb/0UtLp9PPOYhERORw7+FCnQ77IbJ27d5vZMcB9wAfd/YGDXrMZ2AzQ0dFxxtatW8uRd0lmZmZIJBKhYzwnnabpySchnWbsxS8mkUgcONo2KtvvJyYmWLZsWegYhxTVbMpVvKhmq2SuTZs27Sr0/mFBUyXu3p3774CZbQfOAh446DVbgC0AGzZs8PXr1xcVuhL27t1LVHLZ5CTrLryQxqefxpua+L93330gW349eFtbGytWrKC5uTlYzp07d7Jx48Zg1z+SqGZTruJFNVtUcy04rDOzNjNrz/8ZeD3waLmDVbVUimPf+U4aOjuxTIbsQXPc7o67MzExQU9PD11dXUxOTr7gyFsRqU2FjLjXANtzo8AG4DZ3/35ZU1WzbJZjrryS5sceoy43p509wq9i+dMNBwYGMDNWrlzJ8uXLIzOVIiKVt2Bxu/tTwEsrkKUmrPrsZ2n90Y+om5k58LFse/uCn5cfhY+MjDAyMkJ7ezsrV67U5h6RGqSf+gpa/s//zPJvfYu66ennfTy7YkXBXyM/XTI2Nsb4+DgtLS0cddRRQefBRaSyVNwV0nbPPay66abnjbTzMkcdtaiv6e4Httk3NTVx9NFHR2vVjIiUhYq7AhL/9V90XH31IUsbIHv00Uv6+u5OMpmkt7dXBS5SA1TcFdB2333gjtfVYdns8/7OzcisWlWS66jARWqDliZUwPBf/iU93/424xdf/IK/86YmssuXl/R68wu8p6eHmcOM9EUknlTclWBG6vd+j6HPfvbAh6bPOotsUxOWzZa8uPPcnZmZmQMFri31ItVBUyUV1PT44wA8e//9pE88kfrBQVr//d+ZPOcc6O8v23XzBd7d3U1bWxurVq3SMkKRGNOIu4LWXXABAOkTTwQg09HB+CWX4AWs4y6F/G7Mzs5ORkZGyB403y4i8aDirhCbnsZmZxn4/OdDR8Hd2b9/P88+++yBJ9qLSHyouCuk46qrAJi48MLASea4O9lslsHBQbq6unQDUyRGVNwVsuz732fi/PNDx3gBd2d2dpbe3l76+vpIp9OhI4nIAlTcFbBs2zYABm+8MXCSw8vvwuzs7GR0dFTTJyIRpuKugGM+9jGyiQQeg80w7s6+ffvo7u7W8kGRiFJxl1nDb38LQPdddwVOUrj8UbLd3d1kMhmtPhGJGBV3ma1761sBmD311MBJiufuZDIZOjs7mT7oREMRCUfFXU6pFPUjIwx+5jOhkyxJJpOhr6+Pvr4+MplM6DgiNU/FXUYd114LwPif/mngJEs3/+bl1NRU6DgiNU3FXUbt27Yx9Yd/CLmH/1aDbDZLf38/g4ODmvsWCUTFXSZt3/seAP1f/GLgJKU3f+t8MpkMHUek5qi4y2TNBz4AgB/hQcBxlr9x2dPTw759+7TuW6SCVNxlUN/TA0DX9u2Bk5SfuzM6Okp3dzezs7Oh44jUBBV3GazN3YxMveQlgZNURn7dd1dXF2NjY6HjiFQ9FXeppdM0dnYy/NGPhk5Sce7O8PAwAwMDunEpUkYq7hI7+nOfA2B08+bAScJwdyYnJzV1IlJGKu4SW3HrrcyccUZVLQEsVv7Ewa6uLiYnJ0PHEak6Ku4SavnRjwDo27IlcJJocHcGBgYYHh7WqhORElJxl9Bx73oXANmVKwMniQ53Z2xsjJ6eHm2XFykRFXeJ1A8NAdBz++2Bk0SPu5NMJuns7NSTdkRKoODiNrN6M/uFmd1bzkBxdezllwMwc9ZZYYNEWDabpbe3l4mJidBRRGKtmBH3h4A95QoSa9kszXv2MPK+94VOEnnuzuDgoHZbiixBQcVtZscD5wM3lzdOPB11000AjFx5ZeAk8ZDfbTkwMKDyFlkEK+QHx8y2AZ8D2oGr3P2Nh3jNZmAzQEdHxxlbt24tcdSlm5mZIVGGx4c1PfIInkgw++IXL/prlCvbUpU7l5nR2Ni4qM+dmJhgWQTPglGu4kU1WyVzbdq0aZe7n1nIaxsWeoGZvREYcPddZrbxcK9z9y3AFoANGzb4+vXrC4xbOXv37qXUuRIPPsjaq67imZ/+lExHx6K/TjmylUIlctXX17N27dqiC3znzp1s3LixPKGWQLmKF9VsUc1VyFTJq4E3mdnTwB3A2WYWveF0IGsvuQRgSaVd6zKZDF1dXVpxIlKgBYvb3a929+Pd/STgEuB+d7+s7MlioG50FIDeW24JnCT+3J3e3l7ttBQpgNZxL8Ga3Hkk0xH8VSqO8jstdcKgyJEtOMc9n7vvBHaWJUncuNPy0EOMvuMdoZNUlfwJg9lslpXagSpySBpxL9KK3Hkkw5/8ZOAk1cfdGRkZ0RknIoeh4l6ko2+8kdnjj4eGon5pkQLlzzgZGhpSeYscRMW9CE2PPAJA7223BU5S3fIPJR4cHFR5i8yj4l6E4y+4AID0unWBk1S//IMZ+vv7Vd4iOSruIlnugKT+L30pcJLa4e5MT0+rvEVyVNxFOuaKKwCYPO+8wElqS768NW0iouIujjttO3YwftFFoZPUpPy0iW5YSq3TkogitOcekjB4/fWBk9Su/A1Lq+FneopoxF2EjmuvJbNyJTQ3h45S09yd8fFxPQpNapaKu0CNTz4JQM+2bYGTCMyVdyaTYWRkJHQUkYpTcRdoXW5ee/bkkwMnkfn279/PaO6wL5FaoeIugCWT1E1NMfDXfx06ihzE3dm3bx/j4+Oho4hUjIq7AB0f/SgAE295S+AkcijuztDQENPT06GjiFSEirsAy+69l8lzzw0dQ47A3enr6yOVSoWOIlJ2Ku4FLPvudwEY+MIXAieRheQfxqDVJlLtVNwLOObDH8YbG/GWltBRpACZTIaenh6y2WzoKCJlo+I+goZnngGg++67AyeRYqTTaZ1rIlVNxX0E+QcBp373dwMnkWK4OzMzMwwNDYWOIlIWKu7DmZ2lYWCAoWuuCZ1EFiG/NV5rvKUaqbgPY/V11wEwdvnlQXPI4uXXeGuZoFQbFfdhLL/jDqZe/WrQYUax5u709/eTTqdDRxEpGRX3IbTedx8A/V/+cuAkUgrZbJa+vj7drJSqoeI+hGPf8x4AvL09cBIpldnZWd2slKqh4j5IfV8fAN06BbCq5G9Wjo2NhY4ismQq7oMcd9llACRf/vLASaTU3J3h4WGSyWToKCJLouKeL5Oh6be/Zd+VV4ZOImWSP9NE2+IlzlTc86y68UYA9r///YGTSDllMhntrJRYW7C4zSxhZj83s1+Z2WNm9ulKBAth5c03M/PSl2oJYA1IJpOa75bYKmTEnQTOdveXAi8D3mBmv1/eWJWX+M//BKDv5psDJ5FKyG/O0TGwEkcLFrfPmci925h7q7rfMde+4x0AZFetCpxEKiW/OUdTJhI3Vsg3rZnVA7uA9cCX3f1jh3jNZmAzQEdHxxlbt24tcdSlm5mZIZFIvPAv0mma9uxh9uST8WXLKh+MI2QLLKq5oHTZ6urqaGhoKEGiORMTEywL9H10JFHNBdHNVslcmzZt2uXuZxby2oK+W909A7zMzFYC283sdHd/9KDXbAG2AGzYsMHXr19fZOzy27t3L4fKtfaCC0g88ghP/eY3AVLNOVy20KKaC0qXzcw49thjaSnRmes7d+5k48aNJflapRTVXBDdbFHNVdSqEnffD+wA3lCeOAG4k3jkEfZv3hw6iQSSnzLREkGJi0JWlXTkRtqYWQvwOuDxcgerlJVf+hIA+666KnASCSmbzTI4OKj5bomFQqZKjgO+npvnrgPudPd7yxurclbddBOpU06B+vrQUSSw6elpJiYmaNcZNRJxCxa3u/8/oCr3fzfv2gVA7ze+ETiJRIG7MzQ0RGtrK/X6h1wirKZ3Tq67+GIAMmvWBE4iUZEvb5Eoq9nitvFxAPq2bAmcRKJmampKT82RSKvZ4j72ve8FYOqccwInkahxdwYGBshms6GjiBxSbRa3Oy0/+Qmjb3tb6CQSUdlslpGRkdAxRA6pJot7+a23AjB87bVhg0hkuTtjY2M6u1siqSaLe/X115NeswYaG0NHkQjLT5lobbdETc0Vd9Pu3QD0fOtbgZNIHKTTaR3/KpFTc8W97oILAEifcELgJBIH+eNftR1eoqS2ijubxTIZ+v/u70InkRjJl7dIVNRUcTd0dgIw+aY3BU4icTMxMaGHLkhk1FRx142NMa7SlkXQjkqJkpop7vY77wRg6IYbAieRuEomk9pRKZFQM8XdcfXVUFeHNzeHjiIx5e46+lUioSaKuzH3ZJvZU04JnETiLpPJaHmgBFcTxb32LW8BwCP67ESJj/wKE51jIiFVf3Enk9SPjTF4/fWhk0gV2b9/f+gIUsOqvrg7PvlJAMYvuSRwEqkW7s7o6Kg25UgwVV/c7du3M3n22WAWOopUmdHR0dARpEZVdXG3/eu/AjBw002Bk0i10ahbQqrq4l7zF38BgLe1BU4i1Uqjbgmhaou7oasLgK7vfjdwEqlWGnVLKFVb3MddeikAqdNPD5xEqp1WmEilVWdxp9M09vQw/IlPhE4iVS7/pByNuqWSqrK4j86t2R5917sCJ5FaoVG3VFJVFveKb3yD6bPO0hJAqYj8qFu7KaVSqq64W3bsAKD/H/8xcBKpNRMTE6EjSI2ouuI+7s//HIDsihWBk0gtcXf279+vkwOlIhYsbjM7wcx2mNluM3vMzD5UiWCLUT8wAEC3HgQsAWQyGZ3XLRXRUMBr0sD/cveHzawd2GVm97n77jJnK9pxf/ZnACTPPDNwEqlF+VG3SLktOOJ29153fzj353FgD7Cu3MGKls3S9MQTjOR2S4qEkEwmNV0iZWfFfJOZ2UnAA8Dp7j520N9tBjYDdHR0nLF169bSpSxAfV8f9YODcxtuDrOaZGZmhkREz+SOarao5oLoZksmk7S3t4eO8QITExMsW7YsdIxDimq2SubatGnTLncvaLqgkKkSAMxsGfAd4IqDSxvA3bcAWwA2bNjg69evL/RLl8SLzj2X5Gmn0f0v/3LY1+zdu5dK5ypUVLNFNRdEN9vevXt5zWteQ319fegoz7Nz5042btwYOsYhRTVbVHMVtKrEzBqZK+1vuvtd5Y1UvMRPfwpA79e+FjiJyJzx8fHQEaSKFbKqxIBbgD3u/rflj1S8tW97GwDZ1asDJxGZMzo6qrluKZtCRtyvBt4OnG1mv8y9/Y8y5ypY3cgIAL233ho2iMg82WyWZDIZOoZUqQXnuN39P4DI7h0/dvNmAKZf85rASUSek98GH8WbpxJ/8d456U7i4YfZr8OkJIImJyd1fomURayLe8U//RMA+z7+8cBJRA5tcnIydASpQrEu7qP/5m+YPfFEiNiyKxF4brpEpNRiW9zNv/oVAD0V3ugjUoxUKsXs7GzoGFJlYlvc6y66CIDM2rWBk4gcnrtrTbeUXCyL23LnHvd95SuBk4gsbGxsTGu6paRiWdxrPvhBAKbOPTdwEpGFuTupVCp0DKki8Stud1ofeICxiy8OnUSkIO6up+NIScWuuJfnbkYOffrTgZOIFE7FLaUUu+Jefd11pFevhqam0FFECpbNZjVdIiUTq+Ju/PWvAejRo8kkZtxdm3GkZGJV3OsuvBCA9EknhQ0isghaFiilEpvitpkZ6pJJBj7/+dBRRBYlk8loM46URGyKu+OqqwCYyI26ReJI0yVSCrEp7mXf+x4T550XOobIomkXpZRKLIp72V1zT0sb1DSJxNzs7KyOepUli0VxH/ORj5BNJHAdSi8xZ2ZMT0+HjiExF/nibnj6aQC674rcM4pFiubuTE1NhY4hMRf54l6b29o+e+qpgZOIlIaKW5Yq2sWdStEwPMzQddeFTiJSMtlslnQ6HTqGxFiki3v1pz4FwNhllwVOIlJamueWpYh0cS+/806mXvtasMg+ZF6kaNr+LksV2eJu/cEPAOj/4hcDJxEpvenpaT1cQRYtssV97PveB4C3twdOIlIe2v4uixXJ4q7v6QGge/v2wElEymdmZiZ0BImpSBb32tzNyORLXhI4iUh5uLtuUMqiRa+4Mxkan3mG4Y98JHQSkbJKJpOhI0hMLVjcZvZVMxsws0crEWjVDTcAMPrud1ficiLBpNNpnVsii1LIiPtW4A1lznHAyq9+lZkzztASQKl6ZqZRtyzKgsXt7g8A+yqQhZYf/xiAvi1bKnE5kaDcXTcoZVEiNcd93OWXA5BduTJsEJEKUXHLYlghmwDM7CTgXnc//Qiv2QxsBujo6Dhj69atxSVJp2nas4fZk0/Gly0r7nMLNDMzQyKiR8NGNVtUc0F0sxWbq6mpqYxpnjMxMcGyMv1sLVVUs1Uy16ZNm3a5+5mFvLahVBd19y3AFoANGzb4+vXri/r8dX/0RzTv3s1Tv/lNqSK9wN69eyk2V6VENVtUc0F0sxWTy8w44YQTaGgo2Y/iYe3cuZONGzeW/TqLEdVsUc0VjamSbJbm3bsZee97QycRqbhUKhU6gsRMIcsBbwd+ApxqZl1m9j9LHeKo3HkkIx/+cKm/tEikubuKW4q24O9n7n5puUMc9Q//QPLUU6EuGr8AiFSSlgRKsYI3ZfNDDwHQ9/WvB04iEoZG3FKs4MW97k/+BIBMR0fgJCJhpNNpHfEqRQla3HWjowD03nxzyBgiQbm7tr5LUYIW95r3vAeA6U2bQsYQCaqurk7TJVKUcMXtTsvPf87o298eLIJIFLi7HqogRQlW3CtuuQWA4WuuCRVBJBLcXStLpCjBivvoz32O2XXroAI7xkSiTlMlUowgxd306NzR3r233x7i8iKRk06nQ0eQGAlS3Me/+c0ApNetC3F5kcjRqhIpRsWL2yYnAejPbXMXES0JlOJUvLiPueIKACbPP7/SlxaJLDPTdIkUrLLF7U7b/fczftFFFb2sSBxkMpnQESQmKlrc7XfcAcDg9ddX8rIisaARtxSqosXdcc01ZFasgObmSl5WJPLcXSNuKVjFirvxyScB6Nm2rVKXFIkV7Z6UQlWsuNf98R8DMPuiF1XqkiKxouKWQlWkuC2ZpG5yksEbbqjE5URiSVMlUqiKFPfqj38cgPG3vrUSlxOJJZ3JLYWqSHG333MPk69/fSUuJRJb2oAjhSp7cbfdcw8AA1/4QrkvJRJrGnFLocpe3GuuvBKvr8dbW8t9KZFYc3eVtxSkrMXd0NkJQPfdd5fzMiJVQ8UthShrca/NPQg4ddpp5byMSFUwMxW3FKR8xT07S0N/P0N6wo1IwXSDUgpRtuJe/ZnPADB2+eXluoRI1VFxSyHKVtzLb7uNqVe9CszKdQmRqqPilkKUp7hzO8D6v/KVsnx5kWqlOW4pRFmK23IPPvX29nJ8eRGRmlZQcZvZG8zs12a218w+XsjndH/720tLJlJjTNOKUqAFi9vM6oEvA+cBpwGXmtmC6/uSr3jF0tOJ1BBtwJFCFTLiPgvY6+5PuXsKuAN48xE/o7GxBNFERORQGgp4zTqgc977XcArD36RmW0GNufeTZ5yyimPLj1eya0GhkKHOIyoZotqLohuNuUqXlSzVTLXiYW+sJDiLoi7bwG2AJjZQ+5+Zqm+dqlENRdEN1tUc0F0sylX8aKaLaq5Cpkq6QZOmPf+8bmPiYhIAIUU94PAi83sZDNrAi4B7ilvLBEROZwFp0rcPW1mHwB+ANQDX3X3xxb4tC2lCFcGUc0F0c0W1VwQ3WzKVbyoZotkLtPyIxGReKnYU95FRKQ0VNwiIjFT0uJezNb4SjCzr5rZgJlFam25mZ1gZjvMbLeZPWZmHwqdKc/MEmb2czP7VS7bp0Nnms/M6s3sF2Z2b+gs85nZ02b2iJn90sweCp0nz8xWmtk2M3vczPaY2R+EzgRgZqfm/l/l38bM7IrQuQDM7Mrc9/6jZna7mSVCZ8or2Rx3bmv8E8DrmNuk8yBwqbvvLskFlsDMXgtMAP/H3U8PnSfPzI4DjnP3h82sHdgFXBCR/2cGtLn7hJk1Av8BfMjdfxo4GgBm9mHgTGC5u78xdJ48M3saONPdI7WZxMy+DvzY3W/OrQ5rdff9oXPNl+uQbuCV7v5M4CzrmPueP83dp83sTuDf3P3WkLnySjniLn5rfIW4+wPAvtA5Dubuve7+cO7P48Ae5naqBudzJnLvNubeInEn28yOB84Hbg6dJQ7MbAXwWuAWAHdPRa20c84BfhO6tOdpAFrMrAFoBXoC5zmglMV9qK3xkSihODCzk4CXAz8Lm+Q5uemIXwIDwH3uHpVsNwEfBaL41AEHfmhmu3LHQETBycAg8LXc9NLNZtYWOtQhXALcHjoEgLt3A58HngV6gVF3/2HYVM/RzckIMLNlwHeAK9x9LHSePHfPuPvLmNste5aZBZ9mMrM3AgPuvit0lsP4b+7+CuZO03x/bpoutAbgFcD/dveXA5NAZO5BAeSmb94EROI8aDM7irkZg5OBtUCbmV0WNtVzSlnc2hq/CLn54+8A33T3u0LnOZTcr9U7gDeEzgK8GnhTbi75DuBsM9saNtJzciM13H0A2M7cFGJoXUDXvN+YtjFX5FFyHvCwu/eHDpLz34Hfuvugu88CdwGvCpzpgFIWt7bGFyl3A/AWYI+7/23oPPOZWYeZrcz9uYW5m86Ph00F7n61ux/v7icx9z12v7tHYiRkZm25m8zkpiJeDwRfyeTufUCnmZ2a+9A5QPAb4Ae5lIhMk+Q8C/y+mbXmfk7PYe4eVCSU8nTAxWyNrwgzux3YCKw2sy7gU+5+S9hUwNzo8e3AI7m5ZIBPuPu/BcyUdxzw9dyd/jrgTneP1NK7CFoDbM89yaYBuM3dvx820gEfBL6ZG1Q9BbwzcJ4Dcv/IvQ54d+gsee7+MzPbBjwMpIFfEKHt79ryLiISM7o5KSISMypuEZGYUXGLiMSMiltEJGZU3CIiMaPiFhGJGRW3iEjM/H+RA6AvZrnUKwAAAABJRU5ErkJggg==\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "radius = LA.norm(u)\n",
    "plt.gca().add_artist(plt.Circle((0,0), radius, color=\"#DDDDDD\"))\n",
    "plot_vector2d(u, color=\"red\")\n",
    "plt.axis([0, 8.7, 0, 6])\n",
    "plt.grid()\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Adición\n",
    "\n",
    "Dos vectores del mismo tamaño pueden ser sumados elemento a elemento."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 33,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "  [2 5]\n",
      "+ [3 1]\n",
      "----------\n"
     ]
    },
    {
     "data": {
      "text/plain": [
       "array([5, 6])"
      ]
     },
     "execution_count": 33,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "print(\" \", u)\n",
    "print(\"+\", v)\n",
    "print(\"-\"*10)\n",
    "u + v"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Text(2.4,2.5,'u+v')"
      ]
     },
     "execution_count": 14,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plot_vector2d(u, color=\"r\")\n",
    "plot_vector2d(v, color=\"b\")\n",
    "plot_vector2d(v, origin=u, color=\"b\", linestyle=\"dotted\")\n",
    "plot_vector2d(u, origin=v, color=\"r\", linestyle=\"dotted\")\n",
    "plot_vector2d(u+v, color=\"g\")\n",
    "plt.axis([0, 6, 0, 7])\n",
    "plt.grid()\n",
    "plt.text(0.7, 3, \"u\", color=\"r\", fontsize=18)\n",
    "plt.text(4, 3, \"u\", color=\"r\", fontsize=18)\n",
    "plt.text(1.8, 0.2, \"v\", color=\"b\", fontsize=18)\n",
    "plt.text(3.1, 5.6, \"v\", color=\"b\", fontsize=18)\n",
    "plt.text(2.4, 2.5, \"u+v\", color=\"g\", fontsize=18)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$\\textbf{Propiedades}$:\n",
    "\n",
    "1. Conmutatividad: $\\textbf{u} + \\textbf{v} = \\textbf{v} + \\textbf{u}$\n",
    "\n",
    "2. Asociatividad: $\\textbf{u} + (\\textbf{v} + \\textbf{w}) = (\\textbf{u} + \\textbf{v}) + \\textbf{w}$.\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Si se tiene una figura definida por ciertos puntos (vectores) y se añade un vector $\\textbf{v}$ a cada uno de esos puntos, la figura resultante estará trasladada por $\\textbf{v}$. Esto se llama traslación geométrica."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "t1 = np.array([2, 0.25])\n",
    "t2 = np.array([2.5, 3.5])\n",
    "t3 = np.array([1, 2])\n",
    "\n",
    "x_coords, y_coords = zip(t1, t2, t3, t1)\n",
    "plt.plot(x_coords, y_coords, x_coords, y_coords,  linestyle='--', color='b')\n",
    "plot_vector2d(v, t1, color=\"r\")\n",
    "plot_vector2d(v, t2, color=\"r\")\n",
    "plot_vector2d(v, t3, color=\"r\")\n",
    "\n",
    "t1b = t1 + v\n",
    "t2b = t2 + v\n",
    "t3b = t3 + v\n",
    "\n",
    "x_coords_b, y_coords_b = zip(t1b, t2b, t3b, t1b)\n",
    "plt.plot(x_coords_b, y_coords_b, x_coords_b, y_coords_b, linestyle='-', color='b')\n",
    "\n",
    "plt.text(4, 4.2, \"v\", color=\"r\", fontsize=18)\n",
    "plt.text(3, 2.3, \"v\", color=\"r\", fontsize=18)\n",
    "plt.text(3.5, 0.4, \"v\", color=\"r\", fontsize=18)\n",
    "\n",
    "plt.axis([0, 6, 0, 5])\n",
    "plt.grid()\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Multiplicación por un escalar\n",
    "\n",
    "Cuando multiplicamos por un escalar, todos los elementos de nuestro vector serán multiplicados por este número."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 64,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "2.5 * [2 5] = [ 5.  12.5]\n"
     ]
    }
   ],
   "source": [
    "print(\"2.5 * {} = {}\".format(u, 2.5*u))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Gráficamente, la multiplicación por un escalar resulta en un cambio de escala en la figura (recordar \"escalar\"). Además, la distancia del vector respecto al origen también ser incrementada por este escalar."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 86,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "k = 2.5\n",
    "t1c = t1 * k\n",
    "t2c = t2 * k\n",
    "t3c = t3 * k\n",
    "\n",
    "x_coords, y_coords = zip(t1, t2, t3, t1)\n",
    "plt.plot(x_coords, y_coords,  linestyle='--', color='b')\n",
    "plot_vector2d(t1, color=\"r\", linestyle=':')\n",
    "plot_vector2d(t2, color=\"r\", linestyle=':')\n",
    "plot_vector2d(t3, color=\"r\", linestyle=':')\n",
    "plt.axis([0, 7, 0, 9])\n",
    "\n",
    "\n",
    "plot_vector2d(t1 * k, color=\"r\", linestyle=':')\n",
    "plot_vector2d(t2 * k, color=\"r\", linestyle=':')\n",
    "plot_vector2d(t3 * k, color=\"r\", linestyle=':')\n",
    "\n",
    "\n",
    "x_coords, y_coords = zip(t1c, t2c, t3c, t1c)\n",
    "plt.plot(x_coords, y_coords,  linestyle='-', color='b')\n",
    "plt.grid()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$\\textbf{Propiedades}$:\n",
    "\n",
    "0. $\\dfrac{\\textbf{u}}{\\lambda} = \\dfrac{1}{\\lambda} \\times \\textbf{u}$\n",
    "\n",
    "1. Conmutatividad: $\\lambda \\times \\textbf{u} = \\textbf{u} \\times \\lambda$.\n",
    "\n",
    "2. Asociatividad: $\\lambda_1 \\times (\\lambda_2 \\times \\textbf{u}) = (\\lambda_1 \\times \\lambda_2) \\times \\textbf{u}$.\n",
    "\n",
    "3. Distributividad: $\\lambda \\times (\\textbf{u} + \\textbf{v}) = \\lambda \\times \\textbf{u} + \\lambda \\times \\textbf{v}$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Otros tipos de vectores:\n",
    "\n",
    "1. Un vector-cero es un vector cuyos elementos son todos 0.\n",
    "\n",
    "2. Un vector unitario es aquel cuya norma es igual a 1.\n",
    "\n",
    "3. Un vector normalizado de un vector no nulo $\\textbf{u}$, denotado como $\\hat{\\textbf{u}}$, es el vector unitario que apunta en la misma dirección de $\\textbf{u}$. Se calcula en la forma: $\\hat{\\textbf{u}} = \\frac{\\textbf{u}}{||\\textbf{u}||}$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 90,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.gca().add_artist(plt.Circle((0,0),1,color='c'))\n",
    "plt.plot(0, 0, \"ko\")\n",
    "plot_vector2d(v / LA.norm(v), color=\"k\")\n",
    "plot_vector2d(v, color=\"b\", linestyle=\":\")\n",
    "plt.text(0.3, 0.3, \"$\\hat{u}$\", color=\"k\", fontsize=18)\n",
    "plt.text(1.5, 0.7, \"$u$\", color=\"b\", fontsize=18)\n",
    "plt.axis([-1.5, 5.5, -1.5, 3.5])\n",
    "plt.grid()\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Producto punto\n",
    "\n",
    "$\\textbf{Definición}$: El producto punto entre dos vectores $\\textbf{u}$ y $\\textbf{v}$ es una operación muy utilizada en algebra lineal. Su notación está dada por $\\textbf{u} \\cdot \\textbf{v}$ y es definida como:\n",
    "\n",
    "\n",
    "$\\textbf{u} \\cdot \\textbf{v} = ||\\textbf{u}|| \\times ||\\textbf{v}|| \\times \\cos{(\\theta)}$, \n",
    "\n",
    "donde $\\theta$ es el ángulo entre estos dos vectores. Otra manera de cálcular el producto punto es:\n",
    "\n",
    "$\\textbf{u} \\cdot \\textbf{v} = \\sum_i{\\textbf{u}_i \\times \\textbf{v}_i}$\n",
    "\n",
    "\n",
    "Podemos implementarlo de una manera muy fácil en Python:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 91,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "11"
      ]
     },
     "execution_count": 91,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "def dot_product(u,v):\n",
    "    return sum(ui * vi for ui, vi in zip(u,v))\n",
    "\n",
    "dot_product(u,v)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "O simplemente utilizar la implementación del producto punto de NumPy utilizando la función `dot`:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 92,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "11"
      ]
     },
     "execution_count": 92,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "np.dot(u,v)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$\\textbf{Propiedades}$:\n",
    "\n",
    "1. Conmutatividad: $\\textbf{u} \\cdot \\textbf{v} = \\textbf{v} \\cdot \\textbf{u}$.\n",
    "\n",
    "2. La operación producto punto está definida para dos vectores. No hay manera de calcular el producto punto entre un escalar y un vector.\n",
    "\n",
    "3. El producto punto $\\textbf{NO}$ es asociativo: $\\textbf{u} \\cdot (\\textbf{v} \\cdot \\textbf{w}) \\neq (\\textbf{u} \\cdot \\textbf{v}) \\cdot \\textbf{w}$.\n",
    "\n",
    "4. El producto punto es asociativo respecto a un escalar en la forma: $\\lambda \\times (\\textbf{u} \\cdot \\textbf{v}) = (\\lambda \\times \\textbf{u}) \\cdot \\textbf{v} = \\textbf{u} \\cdot (\\lambda \\times \\textbf{v})$.\n",
    "\n",
    "5. Distributividad: $\\textbf{u} \\cdot (\\textbf{v} + \\textbf{w}) = \\textbf{u} \\cdot \\textbf{v} + \\textbf{u} \\cdot \\textbf{w}$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Angulo entre vectores:\n",
    "\n",
    "Un posible uso del producto punto es el cálculo del ángulo entre dos vectores distintos de cero. Podemos encontrar en ángulo entre estos vectores a través de la formula:\n",
    "\n",
    "$\\theta = \\arccos{\\left ( \\dfrac{\\textbf{u} \\cdot \\textbf{v}}{\\left \\Vert \\textbf{u} \\right \\| \\times \\left \\Vert \\textbf{v} \\right \\|} \\right ) }$\n",
    "\n",
    "$\\textbf{Obs}$: Si $\\textbf{u} \\cdot \\textbf{v} = 0$, entonces $\\theta = \\dfrac{\\pi}{2} = 90º$. Entonces sigue que si el producto punto entre dos vectores no-nulos es cero, ellos son ortogonales.\n",
    "\n",
    "Esto lo podemos implementar en Python:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 95,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "El ángulo entre v y u medido es radianes es de:  0.8685393952858896\n",
      "El ángulo entre v y u medido es grados es de:  49.76364169072618\n"
     ]
    }
   ],
   "source": [
    "def vector_angle(u, v):\n",
    "    cos = np.dot(u,v)/(LA.norm(u)*LA.norm(v))\n",
    "    return np.arccos(np.clip(cos, -1, 1))\n",
    "\n",
    "theta = vector_angle(u,v)\n",
    "\n",
    "print('El ángulo entre v y u medido es radianes es de: ', theta)\n",
    "print('El ángulo entre v y u medido es grados es de: ', theta * 180/np.pi)\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$\\textbf{Obs}$: Debido al error que se puede producir cuando se tienen flotantes muy pequeños, los que hacen que $\\cos$ caiga fuera del intervalo $[-1,1]$, tenemos que aplicar la función clip para hacer caer el valor de $\\cos$ dentro del rango."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Proyección Ortogonal.\n",
    "\n",
    "El producto punto es muy útil para proyectar puntos dentro de un eje. La proyección del vector $\\textbf{v}$ en el eje $\\textbf{u}$ es dado por la fórmula:\n",
    "\n",
    "$\\textbf{proj}_{\\textbf{u}}{\\textbf{v}} = \\dfrac{\\textbf{u} \\cdot \\textbf{v}}{\\left \\Vert \\textbf{u} \\right \\| ^2} \\times \\textbf{u}$\n",
    "\n",
    "O equivalentemente:\n",
    "\n",
    "$\\textbf{proj}_{\\textbf{u}}{\\textbf{v}} = (\\textbf{v} \\cdot \\hat{\\textbf{u}}) \\times \\hat{\\textbf{u}}$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 117,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "def projection(u, v):\n",
    "    normalized_u = u / LA.norm(u)\n",
    "    return np.dot(v, normalized_u) * normalized_u\n",
    "\n",
    "proj = projection(u, v)\n",
    "\n",
    "plot_vector2d(u, color=\"r\")\n",
    "plot_vector2d(v, color=\"b\")\n",
    "\n",
    "\n",
    "plot_vector2d(proj, color=\"k\", linestyle=\":\")\n",
    "plt.plot(proj[0], proj[1], \"ko\")\n",
    "plt.plot([proj[0], v[0]], [proj[1], v[1]], \":\", color='b')\n",
    "\n",
    "\n",
    "plt.text(1, 2, \"$proj_u v$\", color=\"k\", fontsize=15)\n",
    "plt.text(1.8, 0.2, \"$v$\", color=\"b\", fontsize=15)\n",
    "plt.text(0.8, 3, \"$u$\", color=\"r\", fontsize=15)\n",
    "\n",
    "plt.axis([0, 8, 0, 5.5])\n",
    "plt.grid()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Matrices\n",
    "\n",
    "Una matriz es un arreglo rectangular de escalares, dispuestos en filas y columnas:\n",
    "\n",
    "$M = \\begin{bmatrix} 10 & 20 & 30 \\\\ 40 & 50 & 60 \\end{bmatrix}$\n",
    "\n",
    "\n",
    "Una matriz puede ser vista como una lista de vectores.\n",
    "\n",
    "Las matrices son eficientes porque podemos operar con muchos vectores al mismo tiempo. Veremos además que las matrices son muy convenientes a la hora de realizar transformaciones lineales tales como escalamiento, rotación y traslación. \n",
    "\n",
    "\n",
    "En Python podemos representar las matrices de dos formas:\n",
    "\n",
    "1. Como una lista de listas:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 116,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[[10, 20, 30], [40, 50, 60]]"
      ]
     },
     "execution_count": 116,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "[\n",
    "    [10, 20, 30],\n",
    "    [40, 50, 60]\n",
    "]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "O más eficientemente, a través de NumPy, la cual provee una implementación muy eficiente a la hora de operar con matrices."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[10, 20, 30],\n",
       "       [40, 50, 60]])"
      ]
     },
     "execution_count": 18,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "M = np.array([\n",
    "    [10, 20, 30],\n",
    "    [40, 50, 60]\n",
    "    ])\n",
    "\n",
    "M"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Tamaño\n",
    "\n",
    "El tamaño de una matriz está definidia por la cantidad de filas y columnas de ésta. Esto lo notaremos como $Filas \\times Columnas$. Por ejemplo la matriz M es una matriz de $2 \\times 3$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 120,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(2, 3)"
      ]
     },
     "execution_count": 120,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "M.shape"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Indexación:\n",
    "\n",
    "El número ubicado en la $i^{ésima}$ fila y $j^{ésima}$ columna de una matriz $\\textbf{X}$, es comúnmente denotada por $\\textbf{x}_{ij}$. Esto quiere decir que la matriz $\\textbf{X}$ está compuesta por los elementos:\n",
    "\n",
    "\n",
    "$X = \\begin{bmatrix}\n",
    "  x_{1,1} & x_{1,2} & x_{1,3} & \\cdots & x_{1,n}\\\\\n",
    "  x_{2,1} & x_{2,2} & x_{2,3} & \\cdots & x_{2,n}\\\\\n",
    "  x_{3,1} & x_{3,2} & x_{3,3} & \\cdots & x_{3,n}\\\\\n",
    "  \\vdots & \\vdots & \\vdots & \\ddots & \\vdots \\\\\n",
    "  x_{m,1} & x_{m,2} & x_{m,3} & \\cdots & x_{m,n}\\\\\n",
    "\\end{bmatrix}$\n",
    "\n",
    "$\\textbf{Obs}$: A pesar de que matemáticamente el primer elemento de una matriz está denotado por $\\textbf{x}_{1,1}$, hay que recordar que la indexación en Python comienza desde 0. Entonces, si queremos acceder al elemento $\\textbf{m}_{2,3}$, debiesemos acceder de la forma:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "10"
      ]
     },
     "execution_count": 19,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "M[0,0]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Se puede además acceder a la $i^{ésima}$ fila de una matriz, denotada por $\\textbf{M}_{i,*}$ de la siguiente forma:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 123,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([40, 50, 60])"
      ]
     },
     "execution_count": 123,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "M[1,:] #Segunda fila."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Podemos acceder a la $j^{ésima}$ columna de una matriz, denotada por $\\textbf{M}_{*,j}$ de la siguiente forma:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 126,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([30, 60])"
      ]
     },
     "execution_count": 126,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "M[:,2] #tercera columna."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Podemos acceder a más de una columna a la vez a través de la indexación de la siguiente forma:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 129,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[20, 30],\n",
       "       [50, 60]])"
      ]
     },
     "execution_count": 129,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "M[:, 1:3] #Esto es, acceder desde la columna 2 a la 4 (excluyendo la 4)."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Tipos de matrices:\n",
    "\n",
    "\n",
    "\n",
    "1. $\\textbf{Matriz cuadrada}$: Posee el mismo número de filas como de columnas.\n",
    "\n",
    "\\begin{bmatrix}\n",
    "  4 & 9 & 2 \\\\\n",
    "  3 & 5 & 7 \\\\\n",
    "  8 & 1 & 6\n",
    "\\end{bmatrix}\n",
    "\n",
    "2. $\\textbf{Matriz triangular superior}$: Los elementos bajo la diagonal tienen valor 0.\n",
    "\n",
    "\\begin{bmatrix}\n",
    "  4 & 9 & 2 \\\\\n",
    "  0 & 5 & 7 \\\\\n",
    "  0 & 0 & 6\n",
    "\\end{bmatrix}\n",
    "\n",
    "\n",
    "3. $\\textbf{Matriz triangular inferior}$: Los elementos sobre la diagonal tienen valor 0.\n",
    "\n",
    "\\begin{bmatrix}\n",
    "  4 & 0 & 0 \\\\\n",
    "  3 & 5 & 0 \\\\\n",
    "  8 & 1 & 6\n",
    "\\end{bmatrix}\n",
    "\n",
    "4. $\\textbf{Matriz diagonal}$: Es una matriz triangular superior e inferior.\n",
    "\n",
    "\\begin{bmatrix}\n",
    "  4 & 0 & 0 \\\\\n",
    "  0 & 5 & 0 \\\\\n",
    "  0 & 0 & 6\n",
    "\\end{bmatrix}\n",
    "\n",
    "\n",
    "\n",
    "Podemos construir matrices diagonales utilizando NumPy:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 130,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[1, 0, 0],\n",
       "       [0, 2, 0],\n",
       "       [0, 0, 3]])"
      ]
     },
     "execution_count": 130,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "np.diag([1,2,3])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Además podemos obtener la diagonal de una matriz utilizando la misma función:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 131,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([1, 5, 9])"
      ]
     },
     "execution_count": 131,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "D = np.array([\n",
    "        [1, 2, 3],\n",
    "        [4, 5, 6],\n",
    "        [7, 8, 9],\n",
    "    ])\n",
    "np.diag(D)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "5. $\\textbf{Matriz Identidad}$: Una matriz identidad de tamaño $n$ denotada como $I_{n}$, es una matriz diagonal de tamaño $n \\times n$ con 1's en la diagonal. Por ejemplo:\n",
    "\n",
    "$  I_{3} = \n",
    "\\begin{bmatrix}\n",
    "  1 & 0 & 0 \\\\\n",
    "  0 & 1 & 0 \\\\\n",
    "  0 & 0 & 1\n",
    "\\end{bmatrix} $\n",
    "\n",
    "Podemos obtener una matriz identidad en NumPY a través de la función `eye`."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 132,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[1., 0., 0.],\n",
       "       [0., 1., 0.],\n",
       "       [0., 0., 1.]])"
      ]
     },
     "execution_count": 132,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "I_3 = np.eye(3)\n",
    "I_3"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Adición\n",
    "\n",
    "Si dos matrices Q y R tienen el mismo tamaño $m \\times n$, pueden ser sumadas. La adición de matrices se realia elemento a elemento y su resultado es una matriz de tamaño $m \\times n$ donde: $S_{i,j} = Q_{i,j} + R_{i,j}$\n",
    "\n",
    "$S =\n",
    "\\begin{bmatrix}\n",
    "  Q_{11} + R_{11} & Q_{12} + R_{12} & Q_{13} + R_{13} & \\cdots & Q_{1n} + R_{1n} \\\\\n",
    "  Q_{21} + R_{21} & Q_{22} + R_{22} & Q_{23} + R_{23} & \\cdots & Q_{2n} + R_{2n}  \\\\\n",
    "  Q_{31} + R_{31} & Q_{32} + R_{32} & Q_{33} + R_{33} & \\cdots & Q_{3n} + R_{3n}  \\\\\n",
    "  \\vdots & \\vdots & \\vdots & \\ddots & \\vdots \\\\\n",
    "  Q_{m1} + R_{m1} & Q_{m2} + R_{m2} & Q_{m3} + R_{m3} & \\cdots & Q_{mn} + R_{mn}  \\\\\n",
    "\\end{bmatrix}$\n",
    "\n",
    "\n",
    "Por ejemplo:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[[10 20 30]\n",
      " [40 50 60]]\n",
      "[[1 2 3]\n",
      " [4 5 6]]\n"
     ]
    }
   ],
   "source": [
    "A = np.array([[10,20,30], [40, 50, 60]])\n",
    "B = np.array([[1,2,3], [4, 5, 6]])\n",
    "print(A)\n",
    "print(B)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 137,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[11, 22, 33],\n",
       "       [44, 55, 66]])"
      ]
     },
     "execution_count": 137,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "A+B"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$\\textbf{Propiedades}$:\n",
    "\n",
    "1. $\\textbf{Conmutatividad}$: $A + B = B + A$\n",
    "\n",
    "\n",
    "2. $\\textbf{Asociatividad}$: $A + (B + C) = (A + B) + C$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Multiplicación Escalar\n",
    "\n",
    "La multiplicación de una matriz por un escalar se realiza elemento a elemento dentro de la matriz de la siguiente forma:\n",
    "\n",
    "\n",
    "$\\lambda M =\n",
    "\\begin{bmatrix}\n",
    "  \\lambda \\times M_{11} & \\lambda \\times M_{12} & \\lambda \\times M_{13} & \\cdots & \\lambda \\times M_{1n} \\\\\n",
    "  \\lambda \\times M_{21} & \\lambda \\times M_{22} & \\lambda \\times M_{23} & \\cdots & \\lambda \\times M_{2n} \\\\\n",
    "  \\lambda \\times M_{31} & \\lambda \\times M_{32} & \\lambda \\times M_{33} & \\cdots & \\lambda \\times M_{3n} \\\\\n",
    "  \\vdots & \\vdots & \\vdots & \\ddots & \\vdots \\\\\n",
    "  \\lambda \\times M_{m1} & \\lambda \\times M_{m2} & \\lambda \\times M_{m3} & \\cdots & \\lambda \\times M_{mn} \\\\\n",
    "\\end{bmatrix}$\n",
    "\n",
    "\n",
    "\n",
    "En NumPy basta con utilizar el operador `*`"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 138,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[ 30,  60,  90],\n",
       "       [120, 150, 180]])"
      ]
     },
     "execution_count": 138,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "3 * A "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$\\textbf{Propiedades}$:\n",
    "\n",
    "1. $\\textbf{Conmutatividad}$: $ \\alpha A = A \\alpha$\n",
    "\n",
    "\n",
    "2. $\\textbf{Asociatividad}$: $\\alpha (\\beta A) = (\\alpha \\times \\beta) A$\n",
    "\n",
    "\n",
    "3. $\\textbf{Distributividad}$: $\\alpha (A + B) = \\alpha A + \\alpha B$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 139,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[ 60, 120, 180],\n",
       "       [240, 300, 360]])"
      ]
     },
     "execution_count": 139,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "2 * (3 * A)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 140,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[ 60, 120, 180],\n",
       "       [240, 300, 360]])"
      ]
     },
     "execution_count": 140,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "(2 * 3) * A"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Multiplicación de matrices\n",
    "\n",
    "Una matriz A de tamaño $m \\times n$, puede ser multiplicada por una matriz B de tamaño $n \\times q$. El resultado es una matriz C cuyos elementos son calculados como la suma de sus productos:\n",
    "\n",
    "$P_{i,j} = \\sum_{k=1}^n{Q_{i,k} \\times R_{k,j}}$\n",
    "\n",
    "Así, P está dado por: \n",
    "\n",
    "$P =\n",
    "\\begin{bmatrix}\n",
    "Q_{11} R_{11} + Q_{12} R_{21} + \\cdots + Q_{1n} R_{n1} &\n",
    "  Q_{11} R_{12} + Q_{12} R_{22} + \\cdots + Q_{1n} R_{n2} &\n",
    "    \\cdots &\n",
    "      Q_{11} R_{1q} + Q_{12} R_{2q} + \\cdots + Q_{1n} R_{nq} \\\\\n",
    "Q_{21} R_{11} + Q_{22} R_{21} + \\cdots + Q_{2n} R_{n1} &\n",
    "  Q_{21} R_{12} + Q_{22} R_{22} + \\cdots + Q_{2n} R_{n2} &\n",
    "    \\cdots &\n",
    "      Q_{21} R_{1q} + Q_{22} R_{2q} + \\cdots + Q_{2n} R_{nq} \\\\\n",
    "  \\vdots & \\vdots & \\ddots & \\vdots \\\\\n",
    "Q_{m1} R_{11} + Q_{m2} R_{21} + \\cdots + Q_{mn} R_{n1} &\n",
    "  Q_{m1} R_{12} + Q_{m2} R_{22} + \\cdots + Q_{mn} R_{n2} &\n",
    "    \\cdots &\n",
    "      Q_{m1} R_{1q} + Q_{m2} R_{2q} + \\cdots + Q_{mn} R_{nq}\n",
    "\\end{bmatrix}$\n",
    "\n",
    "\n",
    "Notar que cada elemento $P_{i,j}$ es el producto punto entre el vector fila $Q_{i,*}$ y el vector columna $R_{*,j}$, entonces podemos calcular cada elemento como:\n",
    "\n",
    "$P_{i,j} = Q_{i,*} \\cdot R_{*,j}$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "La multiplicación de dos matrices en NumPy puede ser realizada utilizando el método `dot`:\n",
    "\n",
    "$E = AD = \\begin{bmatrix}\n",
    "  10 & 20 & 30 \\\\\n",
    "  40 & 50 & 60\n",
    "\\end{bmatrix} \n",
    "\\begin{bmatrix}\n",
    "  2 & 3 & 5 & 7 \\\\\n",
    "  11 & 13 & 17 & 19 \\\\\n",
    "  23 & 29 & 31 & 37\n",
    "\\end{bmatrix} = \n",
    "\\begin{bmatrix}\n",
    "  930 & 1160 & 1320 & 1560 \\\\\n",
    "  2010 & 2510 & 2910 & 3450\n",
    "\\end{bmatrix}$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[ 10,  40,  90],\n",
       "       [160, 250, 360]])"
      ]
     },
     "execution_count": 22,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "A * B"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 149,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[ 930, 1160, 1320, 1560],\n",
       "       [2010, 2510, 2910, 3450]])"
      ]
     },
     "execution_count": 149,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "D = np.array([\n",
    "        [ 2,  3,  5,  7],\n",
    "        [11, 13, 17, 19],\n",
    "        [23, 29, 31, 37]\n",
    "    ])\n",
    "E = np.dot(A,D)\n",
    "E"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$\\textbf{Propiedades}$:\n",
    "\n",
    "1. $\\textbf{NO Conmutatividad}$: $ AB \\neq BA$\n",
    "\n",
    "\n",
    "2. $\\textbf{Asociatividad}$: $A(BC) = (AB)C$\n",
    "\n",
    "\n",
    "3. $\\textbf{Distributividad en adición}$: $(A + B) C = AC + BC$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 147,
   "metadata": {},
   "outputs": [
    {
     "ename": "ValueError",
     "evalue": "shapes (3,4) and (2,3) not aligned: 4 (dim 1) != 2 (dim 0)",
     "output_type": "error",
     "traceback": [
      "\u001b[0;31m---------------------------------------------------------------------------\u001b[0m",
      "\u001b[0;31mValueError\u001b[0m                                Traceback (most recent call last)",
      "\u001b[0;32m<ipython-input-147-f91e55252a46>\u001b[0m in \u001b[0;36m<module>\u001b[0;34m()\u001b[0m\n\u001b[0;32m----> 1\u001b[0;31m \u001b[0mnp\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mdot\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mD\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0mA\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m",
      "\u001b[0;31mValueError\u001b[0m: shapes (3,4) and (2,3) not aligned: 4 (dim 1) != 2 (dim 0)"
     ]
    }
   ],
   "source": [
    "np.dot(D,A)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 150,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[21640, 28390, 27320, 31140],\n",
       "       [47290, 62080, 60020, 68580]])"
      ]
     },
     "execution_count": 150,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "F = np.array([\n",
    "        [8,  7,  4,  2],\n",
    "        [2,  5,  1,  0],\n",
    "        [9, 11, 17, 21],\n",
    "        [0,  1,  0,  1]])\n",
    "\n",
    "np.dot(E,F)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 154,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[21640, 28390, 27320, 31140],\n",
       "       [47290, 62080, 60020, 68580]])"
      ]
     },
     "execution_count": 154,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "np.dot(A, np.dot(D,F))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$\\textbf{Observación}$: El operador `*` realiza una multiplicación elemento a elemento, no es una multiplicación de matrices."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 155,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[ 10,  40,  90],\n",
       "       [160, 250, 360]])"
      ]
     },
     "execution_count": 155,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "A * B"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Transposición\n",
    "\n",
    "La transpuesta de una matriz $M$ (denotada por $M^{T}$), es una matriz tal que la $i^{ésima}$ fila de M, corresponde a la $i^{ésima}$ columna de $M^{T}$.\n",
    "\n",
    "$ A^T =\n",
    "\\begin{bmatrix}\n",
    "  10 & 20 & 30 \\\\\n",
    "  40 & 50 & 60\n",
    "\\end{bmatrix}^T =\n",
    "\\begin{bmatrix}\n",
    "  10 & 40 \\\\\n",
    "  20 & 50 \\\\\n",
    "  30 & 60\n",
    "\\end{bmatrix}$\n",
    "\n",
    "O en otras palabras: ($A^T)_{i,j}$ = $A_{j,i}$\n",
    "\n",
    "En NumPy, podemos transponer una matriz utilizando el atributo `T`:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 156,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[10, 20, 30],\n",
       "       [40, 50, 60]])"
      ]
     },
     "execution_count": 156,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "A"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 157,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[10, 40],\n",
       "       [20, 50],\n",
       "       [30, 60]])"
      ]
     },
     "execution_count": 157,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "A.T"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 158,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[10, 20, 30],\n",
       "       [40, 50, 60]])"
      ]
     },
     "execution_count": 158,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "A.T.T"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Una matriz simétrica es aquella en que $M = M^{T}$. Por definición, esta matriz debe ser cuadrada.\n",
    "\n",
    "$\\textbf{Observación}$: El producto punto de una matriz por su transpuesta es siempre simétrica."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 161,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[133,  55, 259,   9],\n",
       "       [ 55,  30,  90,   5],\n",
       "       [259,  90, 932,  32],\n",
       "       [  9,   5,  32,   2]])"
      ]
     },
     "execution_count": 161,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "np.dot(F, F.T)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 162,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[133,  55, 259,   9],\n",
       "       [ 55,  30,  90,   5],\n",
       "       [259,  90, 932,  32],\n",
       "       [  9,   5,  32,   2]])"
      ]
     },
     "execution_count": 162,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "np.dot(F, F.T).T"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 26,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[ 8,  2,  9,  0],\n",
       "       [ 7,  5, 11,  1],\n",
       "       [ 4,  1, 17,  0],\n",
       "       [ 2,  0, 21,  1]])"
      ]
     },
     "execution_count": 26,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "T = np.array([\n",
    "        [8,  7,  4,  2],\n",
    "        [2,  5,  1,  0],\n",
    "        [9, 11, 17, 21],\n",
    "        [0,  1,  0,  1]])\n",
    "\n",
    "T.T"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Convirtiendo vectores 1D a 2D con NumPy.\n",
    "\n",
    "En NumPy, cuando tenemos vectores 1D, realmente tenemos una dimensión, por lo cual no tenemos vectores fila o columna. Por lo cual, si queremos convertir un vector $\\textbf{u}$ en una fila, tenemos dos formas de hacerlo:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 27,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([2, 5])"
      ]
     },
     "execution_count": 27,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "u"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 28,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([2, 5])"
      ]
     },
     "execution_count": 28,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "u.T"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 165,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[2, 5]])"
      ]
     },
     "execution_count": 165,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "u_row = np.array([u])\n",
    "u_row"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Otra opción es añadir un nuevo eje:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 166,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[2, 5]])"
      ]
     },
     "execution_count": 166,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "u[np.newaxis, :]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Ahora somos capaces de transponer un vector fila:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 167,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[2],\n",
       "       [5]])"
      ]
     },
     "execution_count": 167,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "u_column = u_row.T\n",
    "u_column"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Otra manera de obtener un vector columna es añadiendo un nuevo eje en las columnas:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 168,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[2],\n",
       "       [5]])"
      ]
     },
     "execution_count": 168,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "u[:, np.newaxis]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Graficando una matriz\n",
    "\n",
    "\n",
    "Si queremos darle una representación gráfica a una matriz, podemos pensar en que estas son listas de vectores. Por esto, podemos tomar cada uno de estos vectores y graficarlos como vectores independientes. Por ejemplo, si queremos graficar una matriz de $2 \\times 4$:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 29,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "P = np.array([\n",
    "        [3.0, 4.0, 1.0, 4.6],\n",
    "        [0.2, 3.5, 2.0, 0.5]\n",
    "    ])\n",
    "x_coords_P, y_coords_P = P\n",
    "plt.scatter(x_coords_P, y_coords_P)\n",
    "plt.axis([0, 5, 0, 4])\n",
    "plt.grid()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$\\textbf{Observación}$: Podríamos haber almacenado nuestros 4 vectores bidimensionales como una matriz de $4 \\times 2$ ($P^{T}$).\n",
    "\n",
    "Como en las matrices los vectores están ordenados en posiciones específicas, podemos verlos como caminos, pudiendo representarla como puntos conectados:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 170,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.plot(x_coords_P, y_coords_P, \"bo\")\n",
    "plt.plot(x_coords_P, y_coords_P, \"b--\")\n",
    "plt.axis([0, 5, 0, 4])\n",
    "plt.grid()\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "También podemos representarlas como polígonos:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 32,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "from matplotlib.patches import Polygon\n",
    "plt.gca().add_artist(Polygon(P.T))\n",
    "plt.axis([0, 5, 0, 4])\n",
    "plt.grid()\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Adición de matrices\n",
    "\n",
    "La adición en matrices puede ser vista como multiples traslaciones geométricas.\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 173,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "H = np.array([\n",
    "        [ 0.5, -0.2, 0.2, -0.1],\n",
    "        [ 0.4,  0.4, 1.5, 0.6]\n",
    "    ])\n",
    "P_moved = P + H\n",
    "\n",
    "plt.gca().add_artist(Polygon(P.T, alpha=0.2))\n",
    "plt.gca().add_artist(Polygon(P_moved.T, alpha=0.3, color=\"r\"))\n",
    "for vector, origin in zip(H.T, P.T):\n",
    "    plot_vector2d(vector, origin=origin)\n",
    "\n",
    "plt.text(2.2, 1.8, \"$P$\", color=\"b\", fontsize=18)\n",
    "plt.text(2.0, 3.2, \"$P+H$\", color=\"r\", fontsize=18)\n",
    "plt.text(2.5, 0.5, \"$H_{*,1}$\", color=\"k\", fontsize=18)\n",
    "plt.text(4.1, 3.5, \"$H_{*,2}$\", color=\"k\", fontsize=18)\n",
    "plt.text(0.4, 2.6, \"$H_{*,3}$\", color=\"k\", fontsize=18)\n",
    "plt.text(4.4, 0.2, \"$H_{*,4}$\", color=\"k\", fontsize=18)\n",
    "\n",
    "plt.axis([0, 5, 0, 4])\n",
    "plt.grid()\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 174,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "H2 = np.array([\n",
    "        [-0.5, -0.5, -0.5, -0.5],\n",
    "        [ 0.4,  0.4,  0.4,  0.4]\n",
    "    ])\n",
    "P_translated = P + H2\n",
    "\n",
    "plt.gca().add_artist(Polygon(P.T, alpha=0.2))\n",
    "plt.gca().add_artist(Polygon(P_translated.T, alpha=0.3, color=\"r\"))\n",
    "for vector, origin in zip(H2.T, P.T):\n",
    "    plot_vector2d(vector, origin=origin)\n",
    "\n",
    "plt.axis([0, 5, 0, 4])\n",
    "plt.grid()\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Multiplicación Escalar\n",
    "\n",
    "La multiplicación de una matriz por un escalar, al igual que con los vectores, resulta en una reescalación de la figura. Entonces supongamos que queremos reescalar una matriz por un factor de 0.6:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 33,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "def plot_transformation(P_before, P_after, text_before, text_after, axis = [0, 5, 0, 4], arrows=False):\n",
    "    if arrows:\n",
    "        for vector_before, vector_after in zip(P_before.T, P_after.T):\n",
    "            plot_vector2d(vector_before, color=\"blue\", linestyle=\"--\")\n",
    "            plot_vector2d(vector_after, color=\"red\", linestyle=\"-\")\n",
    "    plt.gca().add_artist(Polygon(P_before.T, alpha=0.2))\n",
    "    plt.gca().add_artist(Polygon(P_after.T, alpha=0.3, color=\"r\"))\n",
    "    plt.text(P_before[0].mean(), P_before[1].mean(), text_before, fontsize=18, color=\"blue\")\n",
    "    plt.text(P_after[0].mean(), P_after[1].mean(), text_after, fontsize=18, color=\"red\")\n",
    "    plt.axis(axis)\n",
    "    plt.grid()\n",
    "\n",
    "P_rescaled = 0.60 * P\n",
    "plot_transformation(P, P_rescaled, \"$P$\", \"$0.6 P$\", arrows=True)\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Producto punto: Proyección en un eje.\n",
    "\n",
    "Esta es una de las herramientas más utiles, pero al mismo tiempo más difíciles de visualizar. Partiremos por un caso simple en el cual tenemos una matriz de tamaño $1 \\times 2$, definida por: $U = \\begin{bmatrix} 1 & 0 \\end{bmatrix}$. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 39,
   "metadata": {},
   "outputs": [],
   "source": [
    "U = np.array([[1, 0]])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 40,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[3. , 4. , 1. , 4.6]])"
      ]
     },
     "execution_count": 40,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "np.dot(U,P)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Estas son las coordenadas horizontales del vector P: proyección de P sobre el eje horizontal."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 41,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "def plot_projection(U, P):\n",
    "    U_P = U.dot(P)\n",
    "    \n",
    "    axis_end = 100 * U\n",
    "    plot_vector2d(axis_end[0], color=\"black\")\n",
    "\n",
    "    plt.gca().add_artist(Polygon(P.T, alpha=0.2))\n",
    "    for vector, proj_coordinate in zip(P.T, U_P.T):\n",
    "        proj_point = proj_coordinate * U\n",
    "        plt.plot(proj_point[0][0], proj_point[0][1], \"ro\")\n",
    "        plt.plot([vector[0], proj_point[0][0]], [vector[1], proj_point[0][1]], \"r--\")\n",
    "\n",
    "    plt.axis([0, 5, 0, 4])\n",
    "    plt.grid()\n",
    "    plt.show()\n",
    "\n",
    "plot_projection(U, P)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Podemos proyectar esta matriz dentro de cualquier otro eje, reemplazando $U$ por cualqueir otro vector unitario. Por ejemplo, si queremos proyectar la matriz en un eje que tiene una inclinación de $30º$ respecto al eje horizontal:\n",
    "\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 42,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "angle30 = 30 * np.pi / 180  # angle in radians\n",
    "U_30 = np.array([[np.cos(angle30), np.sin(angle30)]])\n",
    "\n",
    "plot_projection(U_30, P)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Rotación\n",
    "\n",
    "Vamos a crear una matriz $V$ de tamaño $2 \\times 2$, que contiene dos vectores unitarios que forman un ángulo de $30º$ y $120º$ respecto al eje horizontal.\n",
    "\n",
    "$V = \\begin{bmatrix} \\cos(30°) & \\sin(30°) \\\\ \\cos(120°) & \\sin(120°) \\end{bmatrix}$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 43,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[ 0.8660254,  0.5      ],\n",
       "       [-0.5      ,  0.8660254]])"
      ]
     },
     "execution_count": 43,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "angle120 = 120 * np.pi / 180\n",
    "V = np.array([\n",
    "        [np.cos(angle30), np.sin(angle30)],\n",
    "        [np.cos(angle120), np.sin(angle120)]\n",
    "    ])\n",
    "V"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 44,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[ 2.69807621,  5.21410162,  1.8660254 ,  4.23371686],\n",
       "       [-1.32679492,  1.03108891,  1.23205081, -1.8669873 ]])"
      ]
     },
     "execution_count": 44,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "np.dot(V,P)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "La primera fila corresponde a las coordenadas de la proyección de P en el eje de $30º$. La segunda fila corresponde a las coordenadas de la proyeccción de P en el eje de $120º$. Finalmente obtenemos las coordenadas de P después de rotar el eje vertical y horizontal en $30º$ respecto al origen."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 45,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "P_rotated = V.dot(P)\n",
    "plot_transformation(P, P_rotated, \"$P$\", \"$RP$\", [-2, 6, -2, 4], arrows=True)\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Matriz inversa\n",
    "\n",
    "La matriz inversa de una matriz $A$, denotada por $A^{-1}$ es aquella que cumple con la siguiente propiedad: \n",
    "\n",
    "$A * A^{-1} = I$\n",
    "\n",
    "Podemos calcular la inversa de una matriz (si es que existe) a través de la función `inv` de `linalg` de la siguiente forma:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 48,
   "metadata": {},
   "outputs": [],
   "source": [
    "import numpy.linalg as LA\n",
    "\n",
    "inv_V = LA.inv(V)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 49,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[1, 0],\n",
       "       [0, 1]], dtype=uint64)"
      ]
     },
     "execution_count": 49,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "np.dot(V, inv_V).astype('uint')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "La matriz inversa representa una forma de volver desde una transformación a la transformación inicial. De esta forma, podemos volver desde la matriz rotada $P\\_rotated$ hacia  $P$ a través de la matriz inversa $inv\\_V$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 50,
   "metadata": {},
   "outputs": [],
   "source": [
    "p_unrotated = np.dot(inv_V, P_rotated)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 52,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[<matplotlib.lines.Line2D at 0x10a1cdf98>]"
      ]
     },
     "execution_count": 52,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plot_transformation(P_rotated, p_unrotated, \"$P_{unrotated}$\", \"$P_{rotated}$\",[-2, 6, -2, 4])\n",
    "plt.plot(P[0], P[1], \"b--\")\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Determinante\n",
    "\n",
    "El determinante de una matriz cuadrada $A$ (denotado por $\\det(A)$ o $|A|$) es un valor que puede ser calculado a través de sus elementos. Una de las maneras más simples de calcularla es utilizando la siguiente recursividad: \n",
    "\n",
    "$|M| = M_{1,1}\\times|M^{(1,1)}| - M_{2,1}\\times|M^{(2,1)}| + M_{3,1}\\times|M^{(3,1)}| - M_{4,1}\\times|M^{(4,1)}| + \\cdots ± M_{n,1}\\times|M^{(n,1)}|$\n",
    "\n",
    "\n",
    "Por ejemplo, el determinante de la siguiente matriz de tamaño $3 \\times 3$:\n",
    "\n",
    "$M = \\begin{bmatrix}\n",
    "  1 & 2 & 3 \\\\\n",
    "  4 & 5 & 6 \\\\\n",
    "  7 & 8 & 0\n",
    "\\end{bmatrix}$\n",
    "\n",
    "Puede ser calculado como:\n",
    "\n",
    "\n",
    "$|M| = 1 \\times \\left | \\begin{bmatrix} 5 & 6 \\\\ 8 & 0 \\end{bmatrix} \\right |\n",
    "     - 2 \\times \\left | \\begin{bmatrix} 4 & 6 \\\\ 7 & 0 \\end{bmatrix} \\right |\n",
    "     + 3 \\times \\left | \\begin{bmatrix} 4 & 5 \\\\ 7 & 8 \\end{bmatrix} \\right |$\n",
    "\n",
    "\n",
    "$\\left | \\begin{bmatrix} 5 & 6 \\\\ 8 & 0 \\end{bmatrix} \\right | = 5 \\times 0 - 6 \\times 8 = -48$\n",
    "\n",
    "$\\left | \\begin{bmatrix} 4 & 6 \\\\ 7 & 0 \\end{bmatrix} \\right | = 4 \\times 0 - 6 \\times 7 = -42$\n",
    "\n",
    "$\\left | \\begin{bmatrix} 4 & 5 \\\\ 7 & 8 \\end{bmatrix} \\right | = 4 \\times 8 - 5 \\times 7 = -3$\n",
    "\n",
    "Y así podemos calcular el resultado final:\n",
    "\n",
    "$|M| = 1 \\times (-48) - 2 \\times (-42) + 3 \\times (-3) = 27$\n",
    "\n",
    "\n",
    "El determinante podemos calculando utilizando la función `det` implementada en `linalg`:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 230,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "27.0"
      ]
     },
     "execution_count": 230,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "M = np.array([\n",
    "        [1, 2, 3],\n",
    "        [4, 5, 6],\n",
    "        [7, 8, 0]\n",
    "    ])\n",
    "LA.det(M)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "El determinante puede ser utilizado para saber si una matriz es invertible o no.\n",
    "Una matriz es invertible si $\\det(A) \\neq 0$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Autovalores y autovectores\n",
    "\n",
    "Un **autovector** de una matriz cuadrada $M$ es un vector no-nulo que permanece en la misma linea después de una transformación lineal asociada a $M$. Formalmente, es un vector $v$ tal que:\n",
    "\n",
    "$M \\cdot v = \\lambda \\times v$\n",
    "\n",
    "Donde $\\lambda$ es un valor escalar llamado **autovalor** asociado al vector $v$.\n",
    "\n",
    "\n",
    "Otra definición puede ser: Las transformaciones lineales rotan y escalan a los vectores, pero solo escalan a los autovectores. Este escalamiento del autovector es proporcional a su autovalor.\n",
    "\n",
    "Por ejemplo:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 283,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 576x576 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "x = np.linspace(-1,1,100)\n",
    "y1 = np.sqrt(1-x**2)\n",
    "y2 = -1 * y1\n",
    "\n",
    "plt.figure(figsize=(8,8))\n",
    "plt.plot(x,y1, 'b')\n",
    "plt.plot(x,y2, 'b')\n",
    "plt.axis([-2,2,-2,2])\n",
    "plt.grid()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Definimos una transformación que está dada por la siguiente matriz:\n",
    "\n",
    "$M = \\begin{bmatrix}\n",
    "9 & 4  \\\\\n",
    "4 & 3 \n",
    "\\end{bmatrix}$\n",
    "    "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 286,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 576x576 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "def transformation(x,y):\n",
    "    return 9*x + 4*y, 4*x + 3*y\n",
    "\n",
    "x_new1, y_new1 = transformation(x, y1)\n",
    "x_new2, y_new2 = transformation(x, y2)\n",
    "\n",
    "plt.figure(figsize=(8,8))\n",
    "plt.plot(x_new1,y_new1, 'b')\n",
    "plt.plot(x_new2,y_new2, 'b')\n",
    "plt.axis([-10,10,-10,10])\n",
    "plt.grid()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 288,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Eigen value:  [11.  1.]\n",
      "Eigen vetor:  [[ 0.89442719 -0.4472136 ]\n",
      " [ 0.4472136   0.89442719]]\n"
     ]
    }
   ],
   "source": [
    "eigen_val, eigen_vect = LA.eig(np.array([[9,4],[4,3]]))\n",
    "\n",
    "print('Eigen value: ', eigen_val)\n",
    "print('Eigen vetor: ', eigen_vect)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 306,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 576x576 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "soa = np.array([\n",
    "                 eigen_val[0] * eigen_vect[0][0],\n",
    "                 eigen_val[0] * eigen_vect[1][0]])\n",
    "\n",
    "soa1 = np.array([\n",
    "                 eigen_val[1] * eigen_vect[0][1],\n",
    "                 eigen_val[1] * eigen_vect[1][1]])\n",
    "\n",
    "plt.figure(figsize=(8,8))\n",
    "plt.plot(x,y1, 'b')\n",
    "plt.plot(x,y2, 'b')\n",
    "plt.plot(x_new1,y_new1, 'r')\n",
    "plt.plot(x_new2,y_new2, 'r')\n",
    "\n",
    "plot_vector2d(soa, origin=[0,0])\n",
    "plot_vector2d(soa1, origin=[0,0])\n",
    "\n",
    "plt.axis([-10,10,-10,10])\n",
    "plt.grid()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Gracias.\n",
    "\n",
    "\n",
    "Cualquier duda o consulta: maperezc@udec.cl"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  }
 ],
 "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": "3.6.7"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 2
}