vectors.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Verwendung von Numpy für die Vektorgeometrie #\n",
    "Numpy ist eine Bibliothek für die lineare Algebra. Insbesondere können auch die Vektoroperationen, die wir bisher behandelt haben, mit Numpy durchgeführt werden."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "import numpy.linalg as la"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Definition von Vektoren und Vektorraum-Operationen ###"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([1, 2, 3])"
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "a = np.array([1,2,3])\n",
    "a"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([10, 20, 30])"
      ]
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "b = 10 * a\n",
    "b"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([11, 22, 33])"
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "a + b"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([ 9, 18, 27])"
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "b - a"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Produkte ###"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "140"
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Skalarprodukt\n",
    "a @ b"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([0, 0, 0])"
      ]
     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Vektorprodukt\n",
    "np.cross(a, b)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([0, 0, 1])"
      ]
     },
     "execution_count": 10,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "np.cross([1,0,0], [0,1,0])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[[2 0]\n",
      " [0 3]]\n",
      "[[2 3]\n",
      " [5 7]]\n",
      "[[ 4  6]\n",
      " [15 21]]\n"
     ]
    }
   ],
   "source": [
    "# Matrizenprodukt\n",
    "D = np.array([[2,0],[0,3]])\n",
    "M = np.array([[2,3],[5,7]])\n",
    "print(D)\n",
    "print(M)\n",
    "print(D @ M)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[[0.5        0.        ]\n",
      " [0.         0.33333333]]\n",
      "[[-7.  3.]\n",
      " [ 5. -2.]]\n"
     ]
    }
   ],
   "source": [
    "# Inverse Matrix\n",
    "print(la.inv(D))\n",
    "print(la.inv(M))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Gleichungssysteme und Matrizenrang"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[[2 3]\n",
      " [5 7]]\n",
      "[1 2]\n",
      "[-1.  1.]\n"
     ]
    }
   ],
   "source": [
    "# Lösung eines Gleichungssystems \n",
    "b = np.array([1,2])\n",
    "print(M)\n",
    "print(b)\n",
    "print(la.solve(M, b))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "2"
      ]
     },
     "execution_count": 14,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Matrizenrang\n",
    "la.matrix_rank(M)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Determinanten ###"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[[1 2]\n",
      " [3 4]]\n",
      "-2.0000000000000004\n",
      "-2.0000000000000004\n"
     ]
    }
   ],
   "source": [
    "# 2*2-Determinanten\n",
    "f = np.array([1,2])\n",
    "g = np.array([3,4])\n",
    "M2 = np.array([f,g])\n",
    "print(M2)\n",
    "print(la.det([f, g]))\n",
    "print(la.det(M2))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[[ 1  2  3]\n",
      " [ 2  1  4]\n",
      " [ 1 -1  1]]\n",
      "0.0\n",
      "[[ 1  2  3]\n",
      " [ 1 -1  1]\n",
      " [ 2  1  0]]\n",
      "11.999999999999995\n"
     ]
    }
   ],
   "source": [
    "# 3*3-Determinanten \n",
    "a = np.array([1,2,3])\n",
    "c = np.array([1, -1, 1])\n",
    "b = a + c\n",
    "d = np.array([2,  1, 0])\n",
    "Mabc = np.array([a, b, c])\n",
    "print(Mabc)\n",
    "print(la.det(Mabc))\n",
    "Macd = np.array([a, c, d])\n",
    "print(Macd)\n",
    "print(la.det(Macd))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Indizierung ###"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[10 11 12 13 14 15 16]\n",
      "10\n",
      "11\n",
      "[12 13 15]\n",
      "[12 13 14 15]\n",
      "14\n",
      "[10 11 12 13]\n",
      "[14 15 16]\n"
     ]
    }
   ],
   "source": [
    "# Indizierung\n",
    "v = np.array([10, 11, 12, 13, 14, 15, 16])\n",
    "print(v)\n",
    "print(v[0])\n",
    "print(v[1])\n",
    "print(v[[2, 3, 5]])\n",
    "print(v[2:6])\n",
    "print(v[4])\n",
    "print(v[:4])\n",
    "print(v[4:])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Zeilen und Spaltenvektoren ###\n",
    "Bei eindimensionalen Vektoren wird nicht zwischen Zeilen und Spaltenvektoren unterschieden.\n",
    "\n",
    "Es ist jedoch möglich, Vektoren als Matrizen aufzufassen mit nur einer Zeile bzw. nur einer Spalte."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[[10 11 12 13 14 15 16]]\n",
      "[[10 11 12 13 14 15 16]]\n"
     ]
    }
   ],
   "source": [
    "# Zeilenvektoren\n",
    "v_zeile_a = v.reshape(1,-1)\n",
    "v_zeile_b = v.reshape(1,len(v))\n",
    "print(v_zeile_a)\n",
    "print(v_zeile_b)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[[10]\n",
      " [11]\n",
      " [12]\n",
      " [13]\n",
      " [14]\n",
      " [15]\n",
      " [16]]\n"
     ]
    }
   ],
   "source": [
    "# Spaltenvektoren\n",
    "v_spalte = v.reshape(-1,1)\n",
    "print(v_spalte)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[10, 20, 30],\n",
       "       [40, 50, 60]])"
      ]
     },
     "execution_count": 20,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "vek = np.array([10, 20, 30, 40, 50 , 60])\n",
    "vek.reshape(2,-1)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Vergleich von Vektoren ###"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "metadata": {},
   "outputs": [],
   "source": [
    "# Vergleich von Vektoren\n",
    "u = np.array([1, 1, 1])\n",
    "v = np.array([0, 1, 2])\n",
    "w = np.array([0, 1, 2])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([False,  True, False])"
      ]
     },
     "execution_count": 22,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Elementweiser Vergleich\n",
    "u == v"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "False\n",
      "True\n"
     ]
    }
   ],
   "source": [
    "# Vergleich der Strukturen\n",
    "print(np.array_equal(u, v))\n",
    "print(np.array_equal(v, w))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Erzeugung von Bereichen ###"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([0, 1, 2, 3, 4])"
      ]
     },
     "execution_count": 24,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Erzeugung\n",
    "np.arange(5)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 25,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([ 3,  4,  5,  6,  7,  8,  9, 10])"
      ]
     },
     "execution_count": 25,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "np.arange(3, 11)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 26,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([ 0,  2,  4,  6,  8, 10, 12, 14, 16, 18])"
      ]
     },
     "execution_count": 26,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "np.arange(0, 20, 2) # Schrittweite = 2"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 27,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([ 0.,  2.,  4.,  6.,  8., 10., 12., 14., 16., 18., 20.])"
      ]
     },
     "execution_count": 27,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "np.linspace(0, 20, 11) # Anzahl Elemente = 11"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Matplotlib"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 28,
   "metadata": {},
   "outputs": [],
   "source": [
    "import matplotlib.pyplot as plt"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 29,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[<matplotlib.lines.Line2D at 0x10f2b7520>]"
      ]
     },
     "execution_count": 29,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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",
      "text/plain": [
       "<Figure size 640x480 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "xx = np.linspace(-1, 1, 101)\n",
    "yy = np.sqrt(1 - xx**2)\n",
    "plt.plot(xx,yy)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 30,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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",
      "text/plain": [
       "<Figure size 640x480 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "xx = np.linspace(-1, 1, 101)\n",
    "yy = np.sqrt(1 - xx**2)\n",
    "plt.plot(xx,yy)\n",
    "plt.gca().set_aspect(1)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 31,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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",
      "text/plain": [
       "<Figure size 640x480 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.arrow(1,2,3,4, head_width=0.3, length_includes_head=True)\n",
    "plt.xlim((0,5))\n",
    "plt.ylim((0,7))\n",
    "plt.text(0.7,1.6,'A')\n",
    "plt.text(4.1,6.1,'B')\n",
    "plt.text(3,4,'$\\overrightarrow{AB}$') # LaTeX Syntax ist möglich\n",
    "plt.gca().set_aspect(1)"
   ]
  }
 ],
 "metadata": {
  "anaconda-cloud": {},
  "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": "3.9.6 (default, Oct 18 2022, 12:41:40) \n[Clang 14.0.0 (clang-1400.0.29.202)]"
  },
  "vscode": {
   "interpreter": {
    "hash": "31f2aee4e71d21fbe5cf8b01ff0e069b9275f58929596ceb00d14d90e3e16cd6"
   }
  }
 },
 "nbformat": 4,
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