NumPy_Linear_Algebra -checkpoint.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "id": "b09a7167",
   "metadata": {},
   "source": [
    "# NumPy: Linear Algebra"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "id": "2fc7ee6e",
   "metadata": {},
   "outputs": [],
   "source": [
    "import numpy as np"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "id": "2c31df9e",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[1, 2],\n",
       "       [3, 4]])"
      ]
     },
     "execution_count": 2,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "#Q1\n",
    "p =[[1,0], [0,1]]\n",
    "q =[[1,2], [3,4]]\n",
    "r = np.dot(p,q)\n",
    "r"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "id": "ea4fb4a3",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[1, 2, 3, 4],\n",
       "       [0, 0, 0, 0],\n",
       "       [0, 0, 0, 0],\n",
       "       [1, 2, 3, 4]])"
      ]
     },
     "execution_count": 3,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "#Q2\n",
    "p = [[1, 0], [0, 1]]\n",
    "q = [[1, 2], [3, 4]]\n",
    "o = np.outer(p,q)\n",
    "o"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "id": "7a847119",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([ 2, -3])"
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "#Q3\n",
    "# Cross product of 2 vectors is perpendicular to both the vectors\n",
    "# and has a magnitude equal to the area of the parallelogram formed by them\n",
    "p = [[1, 0], [0, 1]]\n",
    "q = [[1, 2], [3, 4]]\n",
    "c = np.cross(p,q)\n",
    "c"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "id": "a69a1bca",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([-2,  3])"
      ]
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "c = np.cross(q,p)\n",
    "c"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "id": "6e1d3aae",
   "metadata": {},
   "outputs": [],
   "source": [
    "#Q4 Finding determinant\n",
    "from numpy import linalg as LA"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "id": "79047669",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "1.0"
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "np.linalg.det(p)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "id": "31d5d613",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "-2.0000000000000004"
      ]
     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "LA.det(q)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "id": "1562fcdd",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[0, 1, 2],\n",
       "       [3, 4, 5],\n",
       "       [6, 7, 8]])"
      ]
     },
     "execution_count": 3,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "#Q5\n",
    "#Einstein's summation convention is a notation that simplifies the expression of\n",
    "#sums involving tensors or multidimensional arrays.\n",
    "#It uses repeated indices to indicate summation over a range of values.\n",
    "x = np.arange(9).reshape(3, 3)\n",
    "x"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "id": "9e402c78",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[ 3,  4,  5],\n",
       "       [ 6,  7,  8],\n",
       "       [ 9, 10, 11]])"
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "y = np.arange(3, 12).reshape(3, 3)\n",
    "y"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "id": "20c532ad",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[ 24,  27,  30],\n",
       "       [ 78,  90, 102],\n",
       "       [132, 153, 174]])"
      ]
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "z = np.einsum(\"mk,kn\", x, y)\n",
    "z"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "id": "a3cf715b",
   "metadata": {},
   "outputs": [],
   "source": [
    "#Q6 inner product\n",
    "a = np.array([2,3,7])\n",
    "b = np.array([1,2,6])\n",
    "c = np.inner(a, b)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "id": "4700be7b",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "50"
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "c"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "id": "18905e78",
   "metadata": {},
   "outputs": [],
   "source": [
    "x = np.arange(9).reshape(3, 3)\n",
    "y = np.arange(3, 12).reshape(3, 3)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "id": "8394989c",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[ 14,  23,  32],\n",
       "       [ 50,  86, 122],\n",
       "       [ 86, 149, 212]])"
      ]
     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "z =np.inner(x, y)\n",
    "z"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "id": "c13b655e",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Eigenvalues: [-0.23606798  4.23606798]\n",
      "Eigenvectors: [[-0.85065081 -0.52573111]\n",
      " [ 0.52573111 -0.85065081]]\n"
     ]
    }
   ],
   "source": [
    "#Q7\n",
    "A = np.array([[1, 2], [2, 3]])\n",
    "eigenvalues, eigenvectors = np.linalg.eig(A)\n",
    "print(\"Eigenvalues:\", eigenvalues)\n",
    "print(\"Eigenvectors:\", eigenvectors)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "id": "0a510f90",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[ 1,  0, -1],\n",
       "       [ 0,  1,  0],\n",
       "       [ 1,  0,  1]])"
      ]
     },
     "execution_count": 11,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "#Q9\n",
    "#the condition number of a function with respect to an argument measures \n",
    "#how much the output value\n",
    "#of the function can change for a small change in the input argument.\n",
    "a = np.array([[1, 0, -1], [0, 1, 0], [1, 0, 1]])\n",
    "a"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "id": "068ba219",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "1.4142135623730951"
      ]
     },
     "execution_count": 14,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "LA.cond(a)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "id": "f7ded832",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "9.539392014169456"
      ]
     },
     "execution_count": 15,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "#Q10\n",
    "v = np.arange(7)\n",
    "u = np.linalg.norm(v)\n",
    "u"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "id": "2a3b4a35",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([0, 1, 2, 3, 4, 5, 6])"
      ]
     },
     "execution_count": 16,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "#Q12 inverse of matrix\n",
    "v"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "id": "75f2b90a",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[ 0.5,  0. ,  0.5],\n",
       "       [ 0. ,  1. ,  0. ],\n",
       "       [-0.5,  0. ,  0.5]])"
      ]
     },
     "execution_count": 18,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "np.linalg.inv(a)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "id": "067150ac",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[ 1,  0, -1],\n",
       "       [ 0,  1,  0],\n",
       "       [ 1,  0,  1]])"
      ]
     },
     "execution_count": 19,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "a"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "id": "0d1a8253",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[0, 1, 2],\n",
       "       [3, 4, 5],\n",
       "       [6, 7, 8]])"
      ]
     },
     "execution_count": 23,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "#Q15 sum of the diagonal element\n",
    "x = np.arange(9).reshape(3,3)\n",
    "x"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "id": "7a580db6",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "12"
      ]
     },
     "execution_count": 24,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "z = np.trace(x)\n",
    "z"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 29,
   "id": "b66c3182",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[ 2.,  0.,  0.],\n",
       "       [ 6.,  1.,  0.],\n",
       "       [-8., -5.,  3.]])"
      ]
     },
     "execution_count": 29,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "#Q16\n",
    "a = np.array([[4, 12, -16], [12, 37, -53], [-16, -53, 98]])\n",
    "c=np.linalg.cholesky(a)\n",
    "c"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 30,
   "id": "51ce113b",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "14.2828568570857"
      ]
     },
     "execution_count": 30,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "#Q19\n",
    "np.linalg.norm(x)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 31,
   "id": "58043b38",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "2.857326358417597e+16"
      ]
     },
     "execution_count": 31,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "np.linalg.cond(x)"
   ]
  }
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