{
 "cells": [
  {
   "attachments": {},
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Zadanie 4.6"
   ]
  },
  {
   "attachments": {},
   "cell_type": "markdown",
   "metadata": {},
   "source": []
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [
    {
     "ename": "NameError",
     "evalue": "name 'QQ' is not defined",
     "output_type": "error",
     "traceback": [
      "\u001b[1;31m---------------------------------------------------------------------------\u001b[0m",
      "\u001b[1;31mNameError\u001b[0m                                 Traceback (most recent call last)",
      "Cell \u001b[1;32mIn[1], line 5\u001b[0m\n\u001b[0;32m      2\u001b[0m \u001b[39mfrom\u001b[39;00m \u001b[39msympy\u001b[39;00m \u001b[39mimport\u001b[39;00m symbols, Matrix\n\u001b[0;32m      3\u001b[0m \u001b[39mfrom\u001b[39;00m \u001b[39mnumpy\u001b[39;00m\u001b[39m.\u001b[39;00m\u001b[39mlinalg\u001b[39;00m \u001b[39mimport\u001b[39;00m eig\n\u001b[1;32m----> 5\u001b[0m A\u001b[39m=\u001b[39mnp\u001b[39m.\u001b[39mmatrix(QQ,\u001b[39m5\u001b[39m,\u001b[39m3\u001b[39m,[\u001b[39m2\u001b[39m, \u001b[39m4\u001b[39m, \u001b[39m6\u001b[39m, \u001b[39m8\u001b[39m, \u001b[39m10\u001b[39m, \u001b[39m12\u001b[39m, \u001b[39m14\u001b[39m, \u001b[39m16\u001b[39m, \u001b[39m18\u001b[39m, \u001b[39m20\u001b[39m, \u001b[39m22\u001b[39m, \u001b[39m24\u001b[39m, \u001b[39m26\u001b[39m, \u001b[39m28\u001b[39m, \u001b[39m31\u001b[39m])\n\u001b[0;32m      6\u001b[0m \u001b[39mprint\u001b[39m(A\u001b[39m.\u001b[39mtranspose()\u001b[39m*\u001b[39mA)\n\u001b[0;32m      7\u001b[0m \u001b[39mprint\u001b[39m((A\u001b[39m.\u001b[39mtranspose()\u001b[39m*\u001b[39mA)\u001b[39m^\u001b[39m(\u001b[39m-\u001b[39m\u001b[39m1\u001b[39m))\n",
      "\u001b[1;31mNameError\u001b[0m: name 'QQ' is not defined"
     ]
    }
   ],
   "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": [
    "zb1=[(1,1),(2,3),(4,5)]\n",
    "zb2=[(1,1),(2,3),(3,4),(4,5),(5,7),(6,9)]\n",
    "\n",
    "m1=matrix(3,2,[1,exp(1.0),1,exp(2.0),1,exp(4.0)])\n",
    "m2=matrix(6,2,[1,exp(1.0),1,exp(2.0),1,exp(3.0),1,exp(4.0),1,exp(5.0),1,exp(6.0)])\n",
    "\n",
    "a,b,t=var('a,b,t')\n",
    "\n",
    "m1*vector([a,b])-vector([1,3,5])\n",
    "m2*vector([a,b])-vector([1,3,4,5,7,9])\n",
    "\n",
    "\n",
    "\n",
    "M1=m1.transpose()*m1\n",
    "M1.det()\n",
    "\n",
    "\n",
    "\n",
    "M2=m2.transpose()*m2\n",
    "M2.det()\n",
    "\n",
    "\n",
    "\n",
    "M1^(-1)*m1.transpose()*vector([1,3,5])\n",
    "\n",
    "M2^(-1)*m2.transpose()*vector([1,3,4,5,7,9])\n",
    "\n",
    "\n",
    "plot(1.64148598265947+ 0.0629860338045423*exp(t),(t,0,4))+sum([point(x) for x in zb1])\n",
    "\n",
    "plot(3.10041190358990+ 0.0163320609303546*exp(t),(t,0,6))+sum([point(x) for x in zb2])"
   ]
  },
  {
   "attachments": {},
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Zadanie 4.9"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "m=matrix(3,3,[1,1,0,1,2,2,0,2,3])\n",
    "\n",
    "eigenvalues = np.m.eigvals(matrix)\n",
    "\n",
    "eigen=m.right_eigenvectors()\n",
    "e1=eigen[0][1][0]\n",
    "e2=eigen[1][1][0]\n",
    "print(e1.dot_product(e2))\n",
    "e3=eigen[2][1][0]\n",
    "print(e3.dot_product(e1))\n",
    "print(e2.dot_product(e3))"
   ]
  }
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