introduccion-sympy.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Introducción a `SymPy`\n",
    "\n",
    "Esta introducción está basada en las [transparencias de SymPy](https://minireference.com/static/tutorials/sympy_tutorial.pdf) de [J. Gross](http://www.jonathangross.de)\n",
    "\n",
    "Véase también [Taming math and physics using SymPy](https://minireference.com/static/tutorials/sympy_tutorial.pdf), basado en [No bullshit guide to Math and Physics](https://minireference.com)\n",
    "\n",
    "Una forma fácil de usar `SymPy` en línea es en la página de [SymPy Live](http://live.sympy.org) o bien utilizando los servicios de [sage en la nube](https://cloud.sagemath.com)\n",
    "\n",
    "\n",
    "\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Para que los gráficos se incrusten en este bloc necesitamos usar el siguiente comando"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {
    "scrolled": true
   },
   "outputs": [],
   "source": [
    "%matplotlib inline"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Con las siguientes instrucciones importamos todas las variables y métodos de `SymPy`(en algunos casos sólo importaremos los métodos y variables que nos interesen)\n",
    "\n",
    "También se puede ejecutar en línea con `isympy`\n",
    "\n",
    "```python\n",
    "~$ isympy \n",
    "IPython console for SymPy 0.7.6.1 (Python 3.5.1-64-bit) (ground types: python)\n",
    "\n",
    "These commands were executed:\n",
    ">>> from __future__ import division\n",
    ">>> from sympy import *\n",
    ">>> x, y, z, t = symbols('x y z t')\n",
    ">>> k, m, n = symbols('k m n', integer=True)\n",
    ">>> f, g, h = symbols('f g h', cls=Function)\n",
    ">>> init_printing()\n",
    "\n",
    "Documentation can be found at http://www.sympy.org\n",
    "\n",
    "In [1]: \n",
    "```\n",
    "\n",
    "A partir de `Python 3` la divisón se denota por `/` y la división entera por `//`; de ahí que se use \n",
    "```\n",
    "from __future__ import division\n",
    "``` \n",
    "para evitar confusiones si usamos `Python 2`"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [],
   "source": [
    "from sympy import *"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Enteros, racionales, reales y complejos"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "División entera"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {
    "scrolled": true
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0"
      ]
     },
     "execution_count": 3,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "1//7"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "División real"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.14285714285714285"
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "1/7"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "float"
      ]
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "type(1/7)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [],
   "source": [
    "r=S(\"1/7\")\n",
    "s=S(\"2/3\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\frac{17}{21}$"
      ],
      "text/plain": [
       "17/21"
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "r+s"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "sympy.core.numbers.Rational"
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "type(_)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "1125899906842624"
      ]
     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "2**50"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Si queremos trabajar con un racional $a/b$, podemos \"sympificar\" la expresión \"a/b\""
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [],
   "source": [
    "a=S(\"1/7\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle 0.14285714285714285714$"
      ],
      "text/plain": [
       "0.14285714285714285714"
      ]
     },
     "execution_count": 11,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "a.evalf(20)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "True"
      ]
     },
     "execution_count": 12,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "a==Rational(1,7)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.14285714285714285"
      ]
     },
     "execution_count": 13,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "float(a)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle 0.142857142857143$"
      ],
      "text/plain": [
       "0.142857142857143"
      ]
     },
     "execution_count": 14,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "a.n()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle 1 + 2 i$"
      ],
      "text/plain": [
       "1 + 2*I"
      ]
     },
     "execution_count": 15,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "1+2*I"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle 1$"
      ],
      "text/plain": [
       "1"
      ]
     },
     "execution_count": 16,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "(I)**4"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle 3.14159265358979323846264338327950288419716939937510582097494459230781640628620899862803482534211706798214808651328230664709384460955058223172535940813$"
      ],
      "text/plain": [
       "3.14159265358979323846264338327950288419716939937510582097494459230781640628620899862803482534211706798214808651328230664709384460955058223172535940813"
      ]
     },
     "execution_count": 17,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "pi.n(150)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Símbolos\n",
    "\n",
    "Si intentamos evaluar alguna expresión con `x` o `y` dará un error de nombre no definido; tenemos primero que declararlos como símbolos\n",
    "\n",
    "Esto lo hacemos con `Symbol` para una variable, o `symbols` para varios"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {},
   "outputs": [],
   "source": [
    "x,y =symbols(\"x, y\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {},
   "outputs": [],
   "source": [
    "p = (x+y)**10"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\left(x + y\\right)^{10}$"
      ],
      "text/plain": [
       "(x + y)**10"
      ]
     },
     "execution_count": 20,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "p"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle x^{10} + 10 x^{9} y + 45 x^{8} y^{2} + 120 x^{7} y^{3} + 210 x^{6} y^{4} + 252 x^{5} y^{5} + 210 x^{4} y^{6} + 120 x^{3} y^{7} + 45 x^{2} y^{8} + 10 x y^{9} + y^{10}$"
      ],
      "text/plain": [
       "x**10 + 10*x**9*y + 45*x**8*y**2 + 120*x**7*y**3 + 210*x**6*y**4 + 252*x**5*y**5 + 210*x**4*y**6 + 120*x**3*y**7 + 45*x**2*y**8 + 10*x*y**9 + y**10"
      ]
     },
     "execution_count": 21,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "expand(p)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\left(x - y\\right) \\left(x + y\\right)$"
      ],
      "text/plain": [
       "(x - y)*(x + y)"
      ]
     },
     "execution_count": 22,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "factor(x**2-y**2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(x + y, x - y)"
      ]
     },
     "execution_count": 23,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "_.args"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle x^{2} + y$"
      ],
      "text/plain": [
       "x**2 + y"
      ]
     },
     "execution_count": 24,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "simplify(x** 2+2*x+y-2*x)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 25,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle 1$"
      ],
      "text/plain": [
       "1"
      ]
     },
     "execution_count": 25,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "simplify(cos(x)**2+sin(x)**2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 26,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle x^{2} + x \\left(2 y + 2\\right)$"
      ],
      "text/plain": [
       "x**2 + x*(2*y + 2)"
      ]
     },
     "execution_count": 26,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "collect(x**2+2*x*y+2*x,x)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 27,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\log{\\left(2 \\right)} + 1$"
      ],
      "text/plain": [
       "log(2) + 1"
      ]
     },
     "execution_count": 27,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "(ln(x)+y).subs({x:2,y:1})"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 28,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle 1.69314718055995$"
      ],
      "text/plain": [
       "1.69314718055995"
      ]
     },
     "execution_count": 28,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "_.n()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 29,
   "metadata": {},
   "outputs": [],
   "source": [
    "a,b,c,d = symbols(\"a b c d\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 30,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\frac{a d + b c}{b d}$"
      ],
      "text/plain": [
       "(a*d + b*c)/(b*d)"
      ]
     },
     "execution_count": 30,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "together(a/b+c/d)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 31,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle a + \\frac{1}{a}$"
      ],
      "text/plain": [
       "a + 1/a"
      ]
     },
     "execution_count": 31,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "apart((a**2+1)/a)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 32,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "True"
      ]
     },
     "execution_count": 32,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sin(x)==cos(x-pi/2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 33,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\sin{\\left(2 x \\right)}$"
      ],
      "text/plain": [
       "sin(2*x)"
      ]
     },
     "execution_count": 33,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "expand(sin(2*x))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 34,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle 2 \\sin{\\left(x \\right)} \\cos{\\left(x \\right)}$"
      ],
      "text/plain": [
       "2*sin(x)*cos(x)"
      ]
     },
     "execution_count": 34,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "expand_trig(sin(2*x))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 35,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\sin{\\left(2 x \\right)}$"
      ],
      "text/plain": [
       "sin(2*x)"
      ]
     },
     "execution_count": 35,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "trigsimp(_)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Resolución de ecuaciones"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 36,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[-1 + sqrt(2), -sqrt(2) - 1]"
      ]
     },
     "execution_count": 36,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "solve(x**2+2*x-1)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 37,
   "metadata": {},
   "outputs": [],
   "source": [
    "l=solve(a*x**2+b*x+c,x)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 38,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[(-b - sqrt(-4*a*c + b**2))/(2*a), (-b + sqrt(-4*a*c + b**2))/(2*a)]"
      ]
     },
     "execution_count": 38,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "solve(a*x**2+b*x+c,x)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 39,
   "metadata": {
    "scrolled": true
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle -1 + \\sqrt{2}$"
      ],
      "text/plain": [
       "-1 + sqrt(2)"
      ]
     },
     "execution_count": 39,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "l[1].subs({a:1,b:2,c:-1})"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 40,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[(-sqrt(-1/2 + sqrt(5)/2), -1/2 + sqrt(5)/2),\n",
       " (sqrt(-1/2 + sqrt(5)/2), -1/2 + sqrt(5)/2),\n",
       " (-sqrt(-sqrt(5)/2 - 1/2), -sqrt(5)/2 - 1/2),\n",
       " (sqrt(-sqrt(5)/2 - 1/2), -sqrt(5)/2 - 1/2)]"
      ]
     },
     "execution_count": 40,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "solve([x**2+y**2-1, y-x**2],[x,y])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 41,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 640x480 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/plain": [
       "<sympy.plotting.plot.Plot at 0x7f1b88251610>"
      ]
     },
     "execution_count": 41,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "plot_implicit(Eq(x**2+y**2-1,0) | Eq(x**2-y,0),(x,-2,2),(y,-2,2))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Límites"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "El infinito se denota por `oo` (dos oes en minúscula seguidos)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 42,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\infty$"
      ],
      "text/plain": [
       "oo"
      ]
     },
     "execution_count": 42,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "oo+1"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 43,
   "metadata": {},
   "outputs": [],
   "source": [
    "n=Symbol(\"n\", integer=true)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 44,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle e$"
      ],
      "text/plain": [
       "E"
      ]
     },
     "execution_count": 44,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "limit((1+1/n)**n, n, oo)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 45,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle -\\infty$"
      ],
      "text/plain": [
       "-oo"
      ]
     },
     "execution_count": 45,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "limit(1/x,x,0,dir=\"-\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 46,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\infty$"
      ],
      "text/plain": [
       "oo"
      ]
     },
     "execution_count": 46,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "limit(1/x,x,0,dir=\"+\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 47,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 640x480 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/plain": [
       "<sympy.plotting.plot.Plot at 0x7f1b881ca190>"
      ]
     },
     "execution_count": 47,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "plot((1/x,(x,.1,10)),(1/x,(x,-10,-.1)))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 48,
   "metadata": {},
   "outputs": [],
   "source": [
    "a_n=1/n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 49,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle 1$"
      ],
      "text/plain": [
       "1"
      ]
     },
     "execution_count": 49,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "limit(1+a_n,n,oo)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 50,
   "metadata": {
    "scrolled": true
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[1, 1/2, 1/3, 1/4, 1/5, 1/6, 1/7, 1/8, 1/9]"
      ]
     },
     "execution_count": 50,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "[a_n.subs({n:i}) for i in range(1,10)]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Derivadas"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 51,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\frac{2 x^{4} \\cos{\\left(x^{2} \\right)}}{\\cos{\\left(x \\right)}} + \\frac{x^{3} \\sin{\\left(x \\right)} \\sin{\\left(x^{2} \\right)}}{\\cos^{2}{\\left(x \\right)}} + \\frac{3 x^{2} \\sin{\\left(x^{2} \\right)}}{\\cos{\\left(x \\right)}}$"
      ],
      "text/plain": [
       "2*x**4*cos(x**2)/cos(x) + x**3*sin(x)*sin(x**2)/cos(x)**2 + 3*x**2*sin(x**2)/cos(x)"
      ]
     },
     "execution_count": 51,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "diff((x**3*sin(x**2))/cos(x))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 52,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle 0$"
      ],
      "text/plain": [
       "0"
      ]
     },
     "execution_count": 52,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "diff((x**3*sin(x**2))/cos(x),y)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 53,
   "metadata": {},
   "outputs": [],
   "source": [
    "f=symbols(\"f\", cls=Function)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 54,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle f{\\left(x \\right)} = C_{1} e^{x}$"
      ],
      "text/plain": [
       "Eq(f(x), C1*exp(x))"
      ]
     },
     "execution_count": 54,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "dsolve(f(x)-diff(f(x),x),f(x))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Integrales"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 55,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\sin{\\left(x \\right)}$"
      ],
      "text/plain": [
       "sin(x)"
      ]
     },
     "execution_count": 55,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "integrate(cos(x))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 56,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle 2 \\sin{\\left(1 \\right)}$"
      ],
      "text/plain": [
       "2*sin(1)"
      ]
     },
     "execution_count": 56,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "integrate(cos(x),(x,-1,1))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Series"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 57,
   "metadata": {},
   "outputs": [],
   "source": [
    "a_n = 1/n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 58,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\infty$"
      ],
      "text/plain": [
       "oo"
      ]
     },
     "execution_count": 58,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "summation(a_n,(n,1,oo))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 59,
   "metadata": {},
   "outputs": [],
   "source": [
    "b_n=1/factorial(n)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 60,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle e$"
      ],
      "text/plain": [
       "E"
      ]
     },
     "execution_count": 60,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "summation(b_n,(n,0,oo))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 61,
   "metadata": {},
   "outputs": [],
   "source": [
    "exp = x**n/factorial(n)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 62,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\frac{9864101}{3628800}$"
      ],
      "text/plain": [
       "9864101/3628800"
      ]
     },
     "execution_count": 62,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "summation(exp.subs({x:1}),(n,0,10))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 63,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle 2.71828180114638$"
      ],
      "text/plain": [
       "2.71828180114638"
      ]
     },
     "execution_count": 63,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "_.n()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 64,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle e^{5}$"
      ],
      "text/plain": [
       "exp(5)"
      ]
     },
     "execution_count": 64,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "summation(exp.subs({x:5}),(n,0,oo))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Series de Taylor (en 0, MacLaurin)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 65,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle x - \\frac{x^{3}}{6} + \\frac{x^{5}}{120} - \\frac{x^{7}}{5040} + O\\left(x^{8}\\right)$"
      ],
      "text/plain": [
       "x - x**3/6 + x**5/120 - x**7/5040 + O(x**8)"
      ]
     },
     "execution_count": 65,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "series(sin(x),x,0,8)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "O bien en el 1"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 66,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\sin{\\left(1 \\right)} + x \\cos{\\left(1 \\right)} - \\frac{x^{2} \\sin{\\left(1 \\right)}}{2} - \\frac{x^{3} \\cos{\\left(1 \\right)}}{6} + \\frac{x^{4} \\sin{\\left(1 \\right)}}{24} + \\frac{x^{5} \\cos{\\left(1 \\right)}}{120} - \\frac{x^{6} \\sin{\\left(1 \\right)}}{720} - \\frac{x^{7} \\cos{\\left(1 \\right)}}{5040} + O\\left(x^{8}\\right)$"
      ],
      "text/plain": [
       "sin(1) + x*cos(1) - x**2*sin(1)/2 - x**3*cos(1)/6 + x**4*sin(1)/24 + x**5*cos(1)/120 - x**6*sin(1)/720 - x**7*cos(1)/5040 + O(x**8)"
      ]
     },
     "execution_count": 66,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "series(sin(x+1),x,0,8)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Matrices (y vectores)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Las matrices se introducen for filas como una lista de listas con el comando `Matrix`\n",
    "Los vectores se consideran matrices con una sola fila"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 67,
   "metadata": {},
   "outputs": [],
   "source": [
    "a=Matrix([[1,1,2],[1,-1,0]])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 68,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}1 & 1 & 2\\\\1 & -1 & 0\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "Matrix([\n",
       "[1,  1, 2],\n",
       "[1, -1, 0]])"
      ]
     },
     "execution_count": 68,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "a"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 69,
   "metadata": {},
   "outputs": [],
   "source": [
    "a=a[0:2,0:2]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 70,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle -2$"
      ],
      "text/plain": [
       "-2"
      ]
     },
     "execution_count": 70,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "a.det()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 71,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}1 & 1\\\\1 & -1\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "Matrix([\n",
       "[1,  1],\n",
       "[1, -1]])"
      ]
     },
     "execution_count": 71,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "a.transpose()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 72,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "⎡1  1 ⎤\n",
      "⎢     ⎥\n",
      "⎣1  -1⎦\n"
     ]
    }
   ],
   "source": [
    "pretty_print(a)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 73,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "'\\\\left[\\\\begin{matrix}1 & 1\\\\\\\\1 & -1\\\\end{matrix}\\\\right]'"
      ]
     },
     "execution_count": 73,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "latex(a)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 74,
   "metadata": {
    "scrolled": true
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle 1$"
      ],
      "text/plain": [
       "1"
      ]
     },
     "execution_count": 74,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "a[0,1]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 75,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle 1$"
      ],
      "text/plain": [
       "1"
      ]
     },
     "execution_count": 75,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "a[0,0]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 76,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}\\frac{1}{2} & \\frac{1}{2}\\\\\\frac{1}{2} & - \\frac{1}{2}\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "Matrix([\n",
       "[1/2,  1/2],\n",
       "[1/2, -1/2]])"
      ]
     },
     "execution_count": 76,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "a.inv()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 77,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}1 & 0\\\\0 & 1\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "Matrix([\n",
       "[1, 0],\n",
       "[0, 1]])"
      ]
     },
     "execution_count": 77,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "a.inv()*a"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 78,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}1 & -1\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "Matrix([[1, -1]])"
      ]
     },
     "execution_count": 78,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "a.row(1)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 79,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(Matrix([\n",
       " [1, 0],\n",
       " [0, 1]]),\n",
       " (0, 1))"
      ]
     },
     "execution_count": 79,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "a.rref()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Valores propios, vectores propios y diagonalización"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 80,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "{-sqrt(2): 1, sqrt(2): 1}"
      ]
     },
     "execution_count": 80,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "a.eigenvals()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 81,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[(-sqrt(2),\n",
       "  1,\n",
       "  [Matrix([\n",
       "   [1 - sqrt(2)],\n",
       "   [          1]])]),\n",
       " (sqrt(2),\n",
       "  1,\n",
       "  [Matrix([\n",
       "   [1 + sqrt(2)],\n",
       "   [          1]])])]"
      ]
     },
     "execution_count": 81,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "a.eigenvects()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 82,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[Matrix([\n",
       " [-1/(1 - sqrt(2))],\n",
       " [               1]])]"
      ]
     },
     "execution_count": 82,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "(a-sqrt(2)*eye(2)).nullspace()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 83,
   "metadata": {},
   "outputs": [],
   "source": [
    "Q,D=a.diagonalize()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 84,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}- \\sqrt{2} & 0\\\\0 & \\sqrt{2}\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "Matrix([\n",
       "[-sqrt(2),       0],\n",
       "[       0, sqrt(2)]])"
      ]
     },
     "execution_count": 84,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "D"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 85,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}1 - \\sqrt{2} & 1 + \\sqrt{2}\\\\1 & 1\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "Matrix([\n",
       "[1 - sqrt(2), 1 + sqrt(2)],\n",
       "[          1,           1]])"
      ]
     },
     "execution_count": 85,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Q"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 86,
   "metadata": {
    "scrolled": true
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}- \\frac{\\sqrt{2} \\left(-1 + \\sqrt{2}\\right) \\left(1 - \\sqrt{2}\\right)}{-4 + 2 \\sqrt{2}} + \\frac{1}{2} + \\frac{\\sqrt{2}}{2} & - \\frac{\\sqrt{2} \\cdot \\left(1 - \\sqrt{2}\\right)}{4 - 2 \\sqrt{2}} + \\sqrt{2} \\cdot \\left(\\frac{1}{2} - \\frac{\\sqrt{2}}{4}\\right) \\left(1 + \\sqrt{2}\\right)\\\\- \\frac{\\sqrt{2} \\left(-1 + \\sqrt{2}\\right)}{-4 + 2 \\sqrt{2}} + \\frac{1}{2} & - \\frac{\\sqrt{2}}{4 - 2 \\sqrt{2}} + \\sqrt{2} \\cdot \\left(\\frac{1}{2} - \\frac{\\sqrt{2}}{4}\\right)\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "Matrix([\n",
       "[-sqrt(2)*(-1 + sqrt(2))*(1 - sqrt(2))/(-4 + 2*sqrt(2)) + 1/2 + sqrt(2)/2, -sqrt(2)*(1 - sqrt(2))/(4 - 2*sqrt(2)) + sqrt(2)*(1/2 - sqrt(2)/4)*(1 + sqrt(2))],\n",
       "[                          -sqrt(2)*(-1 + sqrt(2))/(-4 + 2*sqrt(2)) + 1/2,                             -sqrt(2)/(4 - 2*sqrt(2)) + sqrt(2)*(1/2 - sqrt(2)/4)]])"
      ]
     },
     "execution_count": 86,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "(Q*D*Q.inv())"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 87,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}1 & 1\\\\1 & -1\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "Matrix([\n",
       "[1,  1],\n",
       "[1, -1]])"
      ]
     },
     "execution_count": 87,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "simplify(_)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 88,
   "metadata": {},
   "outputs": [],
   "source": [
    "b=Matrix([[1,3],[0,1]])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 89,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "no es diagonalizable\n"
     ]
    }
   ],
   "source": [
    "try:\n",
    "    b.diagonalize()\n",
    "except:\n",
    "    print(\"no es diagonalizable\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 90,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[(1,\n",
       "  2,\n",
       "  [Matrix([\n",
       "   [1],\n",
       "   [0]])])]"
      ]
     },
     "execution_count": 90,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "b.eigenvects()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 91,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(Matrix([\n",
       " [3, 0],\n",
       " [0, 1]]),\n",
       " Matrix([\n",
       " [1, 1],\n",
       " [0, 1]]))"
      ]
     },
     "execution_count": 91,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "b.jordan_form()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "La matriz indentidad tiene un función propia"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 92,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}1 & 0 & 0\\\\0 & 1 & 0\\\\0 & 0 & 1\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "Matrix([\n",
       "[1, 0, 0],\n",
       "[0, 1, 0],\n",
       "[0, 0, 1]])"
      ]
     },
     "execution_count": 92,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "eye(3)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Vamos a discutir el siguiente sistema en función del parámetro $a$. \n",
    "\n",
    "$$ \\begin{array}{c} x+y+z=a,\\\\  2x+ay+z=-2,\\\\  -2x-2y+az=4.  \\end{array} $$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 93,
   "metadata": {},
   "outputs": [],
   "source": [
    "x,y,z,a=symbols(\"x y z a\")\n",
    "C=Matrix([[1,1,1],[2,a,1],[-2,-2,a]])\n",
    "b=Matrix([a,-2,4])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 94,
   "metadata": {},
   "outputs": [],
   "source": [
    "sol=solve(C.det())"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 95,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[-2, 2]"
      ]
     },
     "execution_count": 95,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sol"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 96,
   "metadata": {},
   "outputs": [],
   "source": [
    "Ca=C.col_insert(-1,b)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 97,
   "metadata": {
    "scrolled": true
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "2"
      ]
     },
     "execution_count": 97,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "C.subs({a:2}).rank()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 98,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "3"
      ]
     },
     "execution_count": 98,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Ca.subs({a:2}).rank()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 99,
   "metadata": {
    "scrolled": true
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\emptyset$"
      ],
      "text/plain": [
       "EmptySet"
      ]
     },
     "execution_count": 99,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "linsolve((C.subs({a:2}),b.subs({a:2})),[x,y,z])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 100,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "2"
      ]
     },
     "execution_count": 100,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "C.subs({a:-2}).rank()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 101,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "2"
      ]
     },
     "execution_count": 101,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Ca.subs({a:-2}).rank()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 102,
   "metadata": {
    "scrolled": false
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\left\\{\\left( - \\frac{3 z}{4} - \\frac{3}{2}, \\  - \\frac{z}{4} - \\frac{1}{2}, \\  z\\right)\\right\\}$"
      ],
      "text/plain": [
       "{(-3*z/4 - 3/2, -z/4 - 1/2, z)}"
      ]
     },
     "execution_count": 102,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "linsolve((C.subs({a:-2}),b.subs({a:-2})),[x,y,z])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 103,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\left\\{\\left( \\frac{a^{2} - 2 a + 4}{a - 2}, \\  - \\frac{2 a}{a - 2}, \\  2\\right)\\right\\}$"
      ],
      "text/plain": [
       "{((a**2 - 2*a + 4)/(a - 2), -2*a/(a - 2), 2)}"
      ]
     },
     "execution_count": 103,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "linsolve((C,b),[x,y,z])"
   ]
  }
 ],
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