ZHAWNotes/Notes/Semester 2/LA - Lineare Algebra/Zusammenfassung.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Lineare Algebra Zusammenfassung"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 30,
   "metadata": {},
   "outputs": [],
   "source": [
    "from sympy import Add, Matrix, Eq, Expr, MatMul, Mul, Equality, Equivalent, Function, Symbol, init_printing, latex\n",
    "from IPython.display import Markdown, Math\n",
    "init_printing(latex_printer=lambda *args, **kwargs: latex(*args, mul_symbol='dot', **kwargs))\n",
    "\n",
    "def calculate(vars: dict, calculation: Expr, precision: int = None):\n",
    "    for key in vars:\n",
    "        display(Eq(key, vars[key], evaluate=False))\n",
    "\n",
    "        if isinstance(vars[key], Expr):\n",
    "            display(Eq(key, vars[key].subs(vars, simultaneous=True), evaluate=False))\n",
    "\n",
    "    display(Eq(calculation, calculation.subs(vars, simultaneous=True), evaluate=False))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Linearer Spann im $\\mathbb{R}^2$\n",
    "  - 2 Kollineare Vektoren  \n",
    "    Ursprungsgerade\n",
    "  - 2 nicht-kollineare Vektoren $\\mathbb{R}^2$\n",
    "\n",
    "Im $\\mathbb{R}^3$\n",
    "  - 3 kollineare Vektoren Ursprungsgerade\n",
    "  - 3 komplanare, nicht-kollineare Vektoren  \n",
    "    Ursprungsebene\n",
    "  - 3 linear unabhängige Vektoren  \n",
    "    $\\mathbb{R}^3$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 33,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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",
      "text/latex": [
       "$\\displaystyle M = \\left[\\begin{matrix}1 & 2 & 3\\\\4 & 5 & 6\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "    ⎡1  2  3⎤\n",
       "M = ⎢       ⎥\n",
       "    ⎣4  5  6⎦"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": "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",
      "text/latex": [
       "$\\displaystyle N = \\left[\\begin{matrix}1 & 2\\\\3 & 4\\\\5 & 6\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "    ⎡1  2⎤\n",
       "    ⎢    ⎥\n",
       "N = ⎢3  4⎥\n",
       "    ⎢    ⎥\n",
       "    ⎣5  6⎦"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": "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",
      "text/latex": [
       "$\\displaystyle M \\cdot N = \\left[\\begin{matrix}9 & 12 & 15\\\\19 & 26 & 33\\\\29 & 40 & 51\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "      ⎡9   12  15⎤\n",
       "      ⎢          ⎥\n",
       "M⋅N = ⎢19  26  33⎥\n",
       "      ⎢          ⎥\n",
       "      ⎣29  40  51⎦"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "M = Symbol(\"M\")\n",
    "N = Symbol(\"N\")\n",
    "calculate({M: Matrix([[1, 2, 3], [4, 5, 6]]), N: Matrix([[1, 2], [3, 4], [5, 6]])}, M*N)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Die Funktionen $f: \\mathbb{R} \\rightarrow \\mathbb{R}: x \\mapsto m \\cdot x$ ($m \\in \\mathbb{R}$) sind lineare Abbildungen."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAMcAAAAVCAYAAADy8/riAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAFqElEQVR4Ae2b61HcMBDHLwwFkJRw6YCEDqADIBUkdACTT/CNgQ6AChjoADrg0QF0kOM6IP+fkIysk3UPn3U+w84I2auVtA/triQfvdfX117dcnh42K87Rpv6d02epnTbJT3FZFnp1YSjo6N9DbFec5i2de9budrGV2v46aDdR2z+hahSBVLAmdqGtv1Y7+7ZoPS+rYcN1QeWpjOVZMLph6rPOyPUnATpqt1Dm1dmDhGeSJffVC5V/qhsqhSg9jW9/FXdOcdASMl1qmrPygnqE6QBq49O2j20eaVzSA84BI6Bg9yr3Kr4gPOQWbKBmN9Uga9cgHzI+QnvGshq90XaPOocMCRdkBlu9UzZUiltqdS2K1zuLQc8UbKAlQ85s82ZRbB6k+S2+8JsHnUO6W5L5TniEEatwnPWeDYv3f+DnLvdF3O8hB/I7sbmq75KJDxbFhyDzDHQ+5Wt91T7AE24zSra1Q9vd9ufDT3/Vumr/FIB7kRz/faY56/H03fN+KD3IuvZtgvVOxFukBN5C/oITWMo8YQtuA10emSbiz3+qcDXjWhOVaBx+uX5RLhKG6l9Flgqu0t+tw5nsnnoHCyAcw3KFRa3UxxKY/BTyNR5A8MYh1LNrc+FCg5xoOL28VmdQ/ObQ6TmJ+vBj7/YyQzgY/AkJAs0CVausXTBII/qF3NIn4wtLXq7EbLQIwTCIcOL6qFqMr25HFGNLAS2ryrzhGWzey2bl5wDLUqxRHggFXXwyIGhCv6oPxnDP8RiOIxF9gCIfH67QTb5RzwRSe/sHES/kPdURITW6cQOMVppjjC7jhJNidGYOBtOAcADDuAHLHQJrAnv2ws8Npo3LI3dpY/aNh9xDmmTQXsa/DGhWZTPoo/Bvfr655EfIiJCGnrV4yJlTzRkl1gUZl7aYwsxFYVZVE4essQx43jAXCHONSNLEwvNjZ+q0SXfWpgf5wjlNrYS3s+CjGd0zkMK7LgvokF39BkHjdld87fO5jHnYG/rL+5xCiu1S0i3CB0+thhdW7TWGOEiMHTCk4H4kulHz+gYPlL0zjFxAhZasZjUxgID50devRaQWhAFURMPjm+NbQKF3kMeyXhF4PF4QOeFjB6+9Mj4Ktiaq/paoHFq2V39W2fzmHOwWEJBQ8UNhGBBJUECu8VYnC+EM/1UD5Odm2kka4WLCR5ZJFUywy/yJkH9qyJfql8q2/n9OGjH+IP3Qrd0EB9O5/DDO1uuSl2rjcPqpLCMdp/Z5qsRrXDoqtpiOHKizcg+HEMIz0HQ3ZQQDVx00qOBRX5dNft2x4itU+cNSMgcYzOpZI9GPjtH3YoFX8oEVtfI484kbg74wOnYStI+0tcRzlAvo91ntvmKryCrTBZ4LEr5pLSz/QoBQ1AG1nhEmgKEo80djAt8xofSIrf8wFO4wHyWJtq/+x3m+ezZ5DIYl61TT+3hVgv7ORw/fyk5VTDGtK/LaPeZbb4aaIctVUzhAZn5WQkZIgSMgjFYcIyDcbiHJ8U/qOA0pW0AdBmBq06+Z3BbxncCt6VwiynGCrI0mRVic/o4E/nEcxiwcILYwke2HdHvqzZbK3+wms846LLZfXab+7/112/az1SufFzVs+ieVNar2pvAa75tlf15ja2xTlQeqsZTW1/lqar9I+LRh0o2u2uuhdl8RRFmW8VFA84bYfquCjZEqNwRdag5KVODZOQcRPYyoGciL99kjt8w0b9EHeT8hHcN5Lb7wmzOmQNhudFgsfRUT7TtER0pnWtV0n4W0Fz8CDK2lZhkfq6BLz1Cvjbza4CovFYu5Jt1Pm+q7jxafWSzu+ZbmM05cxjnsDXXXtMA9GQdbnzaDmSBb1I2e3HOGmcoPsE0+/XcmTHBTqualsXutWye/E/AScyhBUbmYGs21Ye5ScZeFI11oGvVpZuORfHTxnm7ZveYzf8DaygUuvdtqZkAAAAASUVORK5CYII=",
      "text/latex": [
       "$\\displaystyle f{\\left(x + y \\right)} = m \\cdot \\left(x + y\\right)$"
      ],
      "text/plain": [
       "f(x + y) = m⋅(x + y)"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "f = Function('f')\n",
    "m = Symbol('m')\n",
    "x = Symbol('x')\n",
    "y = Symbol('y')\n",
    "la = Symbol('\\lambda')\n",
    "display(Eq(f(x + y), m * (x + y)))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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",
      "text/latex": [
       "$\\displaystyle f{\\left(\\left[\\begin{matrix}x\\\\y\\\\z\\\\w\\end{matrix}\\right] \\right)} = \\left[\\begin{matrix}w + x\\\\- y + z\\\\- 2 \\cdot w - x + y\\\\2 \\cdot x - y\\\\w + z\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "         ⎡   w + x    ⎤\n",
       " ⎛⎡x⎤⎞   ⎢            ⎥\n",
       " ⎜⎢ ⎥⎟   ⎢   -y + z   ⎥\n",
       " ⎜⎢y⎥⎟   ⎢            ⎥\n",
       "f⎜⎢ ⎥⎟ = ⎢-2⋅w - x + y⎥\n",
       " ⎜⎢z⎥⎟   ⎢            ⎥\n",
       " ⎜⎢ ⎥⎟   ⎢  2⋅x - y   ⎥\n",
       " ⎝⎣w⎦⎠   ⎢            ⎥\n",
       "         ⎣   w + z    ⎦"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": 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pAJcm7KGT/eC0U9pkn8buejQgUmOXxkV4ZR08LhOdXjSq+lcPwh1Cti+JpXoWW8QQXY0yRqwJVU7np4FRfPsFiPMDxqgcVz1Ldt/+qF3zqBoslUVqAN8+fv17ipmcEIf69gsQpdES2jVQjZubJpDthTveLV1zR0WVbHk1UICYV7+FekcNFCB2VFTJllcDBYh59Vuod9RAAWJHRZVseTVQZs159TsV6oP89hFCaObd6vsvQIzQ8nRpRPntIyRs9f3/DwZm6Vpkvz70AAAAAElFTkSuQmCC",
      "text/latex": [
       "$\\displaystyle f{\\left(\\left[\\begin{matrix}1\\\\0\\\\0\\\\0\\end{matrix}\\right] \\right)} = \\left[\\begin{matrix}1\\\\0\\\\-1\\\\2\\\\0\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "         ⎡1 ⎤\n",
       " ⎛⎡1⎤⎞   ⎢  ⎥\n",
       " ⎜⎢ ⎥⎟   ⎢0 ⎥\n",
       " ⎜⎢0⎥⎟   ⎢  ⎥\n",
       "f⎜⎢ ⎥⎟ = ⎢-1⎥\n",
       " ⎜⎢0⎥⎟   ⎢  ⎥\n",
       " ⎜⎢ ⎥⎟   ⎢2 ⎥\n",
       " ⎝⎣0⎦⎠   ⎢  ⎥\n",
       "         ⎣0 ⎦"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": "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",
      "text/latex": [
       "$\\displaystyle f{\\left(\\left[\\begin{matrix}0\\\\1\\\\0\\\\0\\end{matrix}\\right] \\right)} = \\left[\\begin{matrix}0\\\\-1\\\\1\\\\-1\\\\0\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "         ⎡0 ⎤\n",
       " ⎛⎡0⎤⎞   ⎢  ⎥\n",
       " ⎜⎢ ⎥⎟   ⎢-1⎥\n",
       " ⎜⎢1⎥⎟   ⎢  ⎥\n",
       "f⎜⎢ ⎥⎟ = ⎢1 ⎥\n",
       " ⎜⎢0⎥⎟   ⎢  ⎥\n",
       " ⎜⎢ ⎥⎟   ⎢-1⎥\n",
       " ⎝⎣0⎦⎠   ⎢  ⎥\n",
       "         ⎣0 ⎦"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": "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",
      "text/latex": [
       "$\\displaystyle f{\\left(\\left[\\begin{matrix}0\\\\0\\\\1\\\\0\\end{matrix}\\right] \\right)} = \\left[\\begin{matrix}0\\\\1\\\\0\\\\0\\\\1\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "         ⎡0⎤\n",
       " ⎛⎡0⎤⎞   ⎢ ⎥\n",
       " ⎜⎢ ⎥⎟   ⎢1⎥\n",
       " ⎜⎢0⎥⎟   ⎢ ⎥\n",
       "f⎜⎢ ⎥⎟ = ⎢0⎥\n",
       " ⎜⎢1⎥⎟   ⎢ ⎥\n",
       " ⎜⎢ ⎥⎟   ⎢0⎥\n",
       " ⎝⎣0⎦⎠   ⎢ ⎥\n",
       "         ⎣1⎦"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": 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6k2i49isI43A1+uosY8TRXNULNqeBCsTNmTxK4GSx/axdc5QK6sWzaQDDleSx/QrE2cy7nIYBxKCVihiJatcco716rZkGKhDNVFkJxWigAjFGe/VaMw1UIJqpshKK0UAFYoz26rVmGqizZjNVFk0oKm5vIRlm3r2x/wpECy2XS8Mqbm8hYW/s//8yFOqY0Fs/5gAAAABJRU5ErkJggg==",
      "text/latex": [
       "$\\displaystyle f{\\left(\\left[\\begin{matrix}0\\\\0\\\\0\\\\1\\end{matrix}\\right] \\right)} = \\left[\\begin{matrix}1\\\\0\\\\-2\\\\0\\\\1\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "         ⎡1 ⎤\n",
       " ⎛⎡0⎤⎞   ⎢  ⎥\n",
       " ⎜⎢ ⎥⎟   ⎢0 ⎥\n",
       " ⎜⎢0⎥⎟   ⎢  ⎥\n",
       "f⎜⎢ ⎥⎟ = ⎢-2⎥\n",
       " ⎜⎢0⎥⎟   ⎢  ⎥\n",
       " ⎜⎢ ⎥⎟   ⎢0 ⎥\n",
       " ⎝⎣1⎦⎠   ⎢  ⎥\n",
       "         ⎣1 ⎦"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "x = Symbol('x')\n",
    "y = Symbol('y')\n",
    "z = Symbol('z')\n",
    "w = Symbol('w')\n",
    "f = Function('f')\n",
    "\n",
    "def realF(x, y, z, w):\n",
    "    return Matrix([x + w, z - y, y - x - 2 * w, 2 * x - y, z + w])\n",
    "\n",
    "display(Eq(f(Matrix([x, y, z, w])), realF(x, y, z, w), evaluate=False))\n",
    "input = 1, 0, 0, 0\n",
    "display(Eq(f(Matrix(input)), realF(*input), evaluate=False))\n",
    "input = 0, 1, 0, 0\n",
    "display(Eq(f(Matrix(input)), realF(*input), evaluate=False))\n",
    "input = 0, 0, 1, 0\n",
    "display(Eq(f(Matrix(input)), realF(*input), evaluate=False))\n",
    "input = 0, 0, 0, 1\n",
    "display(Eq(f(Matrix(input)), realF(*input), evaluate=False))"
   ]
  }
 ],
 "metadata": {
  "interpreter": {
   "hash": "184a40b3dca75cecb3b866e2d98ad88cc8239a23b64498ef3e7e2e1f5f9f2635"
  },
  "kernelspec": {
   "display_name": "Python 3.10.2 ('ZHAW-Pb88adSo')",
   "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": "3.10.2"
  },
  "orig_nbformat": 4
 },
 "nbformat": 4,
 "nbformat_minor": 2
}