{
 "cells": [
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "Zadanie 4.6"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "from sympy import symbols, Matrix\n",
    "from numpy.linalg import eig\n",
    "\n",
    "A=np.matrix(QQ,5,3,[2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 31])\n",
    "print(A.transpose()*A)\n",
    "print((A.transpose()*A)^(-1))\n",
    "mm=(A.transpose()*A)^(-1)\n",
    "mm=(A.transpose()*A)^(-1)*A.transpose()\n",
    "print(mm)\n",
    "\n",
    "b1=np.vector([-1,0,1,0,1])\n",
    "mm1=mm*b1\n",
    "print(mm1)\n",
    "print((b1-A*mm1))\n",
    "b2=np.vector([1,1,1,1,1])\n",
    "mm2=mm*b2\n",
    "print(mm2)\n",
    "print((b2-A*mm2))\n",
    "\n",
    "b2 in (m.transpose()).image()"
   ]
  },
  {
   "attachments": {},
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Zadanie 4.7\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "attachments": {},
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Zadanie 4.9"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  }
 ],
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   "display_name": "base",
   "language": "python",
   "name": "python3"
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   "name": "python",
   "version": "3.10.9"
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