matrix-multiplication-numpy.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "- Creating and Manipulating 1D & 2D Arrays\n",
    "- Array Operations"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 1D Numpy Array"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Create array"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "import numpy as np  "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [],
   "source": [
    "a = np.array([0,1,2,3,4])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Numpy Array a \n",
      "[0 1 2 3 4]\n",
      "Type of Numpy Array a <class 'numpy.ndarray'>\n",
      "Elements Type of Numpy Array a int64\n",
      "Size of Numpy Array a 5\n",
      "Dimensions of Numpy Array a 1\n",
      "Shape of Numpy Array a (5,)\n"
     ]
    }
   ],
   "source": [
    "print(f'Numpy Array a \\n{a}')\n",
    "print(f'Type of Numpy Array a {type(a)}')\n",
    "print(f'Elements Type of Numpy Array a {a.dtype}')\n",
    "print(f'Size of Numpy Array a {a.size}')\n",
    "print(f'Dimensions of Numpy Array a {a.ndim}')\n",
    "print(f'Shape of Numpy Array a {a.shape}')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Manipulate Array"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Indexing"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0"
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "a[0]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "index 0 element 0\n",
      "index 1 element 1\n",
      "index 2 element 2\n",
      "index 3 element 3\n",
      "index 4 element 4\n"
     ]
    }
   ],
   "source": [
    "for index,element in enumerate(a):\n",
    "    print(f'index {index} element {element}')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [],
   "source": [
    "b=np.array([3.1,11.02,6.2,231.2,5.2])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Numpy Array b [  3.1   11.02   6.2  231.2    5.2 ]\n",
      "Type of Numpy Array b <class 'numpy.ndarray'>\n",
      "Elements Type of Numpy Array b float64\n",
      "Size of Numpy Array b 5\n",
      "Dimensions of Numpy Array b 1\n",
      "Shape of Numpy Array b (5,)\n"
     ]
    }
   ],
   "source": [
    "print(f'Numpy Array b {b}')\n",
    "print(f'Type of Numpy Array b {type(b)}')\n",
    "print(f'Elements Type of Numpy Array b {b.dtype}')\n",
    "print(f'Size of Numpy Array b {b.size}')\n",
    "print(f'Dimensions of Numpy Array b {b.ndim}')\n",
    "print(f'Shape of Numpy Array b {b.shape}')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "index 0 element 3.1\n",
      "index 1 element 11.02\n",
      "index 2 element 6.2\n",
      "index 3 element 231.2\n",
      "index 4 element 5.2\n"
     ]
    }
   ],
   "source": [
    "for index,element in enumerate(b):\n",
    "    print(f'index {index} element {element}')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([20,  1,  2,  3,  4])"
      ]
     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "c=np.array([20,1,2,3,4])\n",
    "c"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([100,   1,   2,   3,   4])"
      ]
     },
     "execution_count": 10,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "c[0]=100\n",
    "c"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([100,   1,   2,   3,   0])"
      ]
     },
     "execution_count": 11,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "c[4]=0\n",
    "c"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Slice"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([1, 2, 3])"
      ]
     },
     "execution_count": 12,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "d = c[1:4]\n",
    "d"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "3"
      ]
     },
     "execution_count": 13,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "d.size"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([100,   1,   2, 300, 400])"
      ]
     },
     "execution_count": 14,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "c[3:5]=300,400\n",
    "c"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Array Operations"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Vector Addition & Subtraction"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([1, 0])"
      ]
     },
     "execution_count": 15,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "u = np.array([1,0])\n",
    "u"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([0, 1])"
      ]
     },
     "execution_count": 16,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "v = np.array([0,1])\n",
    "v"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([1, 1])"
      ]
     },
     "execution_count": 17,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "z = u + v\n",
    "z"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "numpy.ndarray"
      ]
     },
     "execution_count": 18,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "type(z)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([ 1, -1])"
      ]
     },
     "execution_count": 19,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "z = u-v\n",
    "z"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Array multiplication with a Scalar"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([1, 2])"
      ]
     },
     "execution_count": 20,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "y = np.array([1,2])\n",
    "y"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([2, 4])"
      ]
     },
     "execution_count": 21,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "z = 2*y\n",
    "z"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Product of two Numpy arrays"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([1, 2])"
      ]
     },
     "execution_count": 22,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "u = np.array([1,2])\n",
    "u"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([3, 1])"
      ]
     },
     "execution_count": 23,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "v = np.array([3,1])\n",
    "v             "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Hadamard Product"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([3, 2])"
      ]
     },
     "execution_count": 24,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "z = u*v\n",
    "z"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Dot Product"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 25,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([1, 2])"
      ]
     },
     "execution_count": 25,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "u.T"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 26,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "5"
      ]
     },
     "execution_count": 26,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "z= np.dot(u,v)\n",
    "z"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 27,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "5"
      ]
     },
     "execution_count": 27,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "z = u@v\n",
    "z"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Broadcasting"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Adding Constant to an Numpy Array"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 28,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([ 1,  2,  3, -1])"
      ]
     },
     "execution_count": 28,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "u = np.array([1,2,3,-1])\n",
    "u"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 29,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([2, 3, 4, 0])"
      ]
     },
     "execution_count": 29,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "z = u + 1\n",
    "z"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Universal Functions"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 30,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([ 1, -1,  1, -1])"
      ]
     },
     "execution_count": 30,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "a = np.array([1,-1,1,-1])\n",
    "a"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 31,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.0"
      ]
     },
     "execution_count": 31,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "mean_a = a.mean()\n",
    "mean_a"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 32,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([ 1, -2,  3,  4,  5])"
      ]
     },
     "execution_count": 32,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "b = np.array([1,-2,3,4,5])\n",
    "b"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 33,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "5"
      ]
     },
     "execution_count": 33,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "max_b = b.max()\n",
    "max_b"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 34,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "3.141592653589793"
      ]
     },
     "execution_count": 34,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "np.pi"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 35,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([0.        , 1.57079633, 3.14159265])"
      ]
     },
     "execution_count": 35,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "x = np.array([0,np.pi/2,np.pi])\n",
    "x"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 36,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([0.0000000e+00, 1.0000000e+00, 1.2246468e-16])"
      ]
     },
     "execution_count": 36,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "y = np.sin(x)\n",
    "y"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 37,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([-2., -1.,  0.,  1.,  2.])"
      ]
     },
     "execution_count": 37,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "np.linspace(-2,2,num=5)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 38,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([-2. , -1.5, -1. , -0.5,  0. ,  0.5,  1. ,  1.5,  2. ])"
      ]
     },
     "execution_count": 38,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "np.linspace(-2,2,num=9)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Plotting "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 39,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([0.        , 0.06346652, 0.12693304, 0.19039955, 0.25386607,\n",
       "       0.31733259, 0.38079911, 0.44426563, 0.50773215, 0.57119866,\n",
       "       0.63466518, 0.6981317 , 0.76159822, 0.82506474, 0.88853126,\n",
       "       0.95199777, 1.01546429, 1.07893081, 1.14239733, 1.20586385,\n",
       "       1.26933037, 1.33279688, 1.3962634 , 1.45972992, 1.52319644,\n",
       "       1.58666296, 1.65012947, 1.71359599, 1.77706251, 1.84052903,\n",
       "       1.90399555, 1.96746207, 2.03092858, 2.0943951 , 2.15786162,\n",
       "       2.22132814, 2.28479466, 2.34826118, 2.41172769, 2.47519421,\n",
       "       2.53866073, 2.60212725, 2.66559377, 2.72906028, 2.7925268 ,\n",
       "       2.85599332, 2.91945984, 2.98292636, 3.04639288, 3.10985939,\n",
       "       3.17332591, 3.23679243, 3.30025895, 3.36372547, 3.42719199,\n",
       "       3.4906585 , 3.55412502, 3.61759154, 3.68105806, 3.74452458,\n",
       "       3.8079911 , 3.87145761, 3.93492413, 3.99839065, 4.06185717,\n",
       "       4.12532369, 4.1887902 , 4.25225672, 4.31572324, 4.37918976,\n",
       "       4.44265628, 4.5061228 , 4.56958931, 4.63305583, 4.69652235,\n",
       "       4.75998887, 4.82345539, 4.88692191, 4.95038842, 5.01385494,\n",
       "       5.07732146, 5.14078798, 5.2042545 , 5.26772102, 5.33118753,\n",
       "       5.39465405, 5.45812057, 5.52158709, 5.58505361, 5.64852012,\n",
       "       5.71198664, 5.77545316, 5.83891968, 5.9023862 , 5.96585272,\n",
       "       6.02931923, 6.09278575, 6.15625227, 6.21971879, 6.28318531])"
      ]
     },
     "execution_count": 39,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "x = np.linspace(0,2*np.pi,100)\n",
    "x"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 40,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([ 0.00000000e+00,  6.34239197e-02,  1.26592454e-01,  1.89251244e-01,\n",
       "        2.51147987e-01,  3.12033446e-01,  3.71662456e-01,  4.29794912e-01,\n",
       "        4.86196736e-01,  5.40640817e-01,  5.92907929e-01,  6.42787610e-01,\n",
       "        6.90079011e-01,  7.34591709e-01,  7.76146464e-01,  8.14575952e-01,\n",
       "        8.49725430e-01,  8.81453363e-01,  9.09631995e-01,  9.34147860e-01,\n",
       "        9.54902241e-01,  9.71811568e-01,  9.84807753e-01,  9.93838464e-01,\n",
       "        9.98867339e-01,  9.99874128e-01,  9.96854776e-01,  9.89821442e-01,\n",
       "        9.78802446e-01,  9.63842159e-01,  9.45000819e-01,  9.22354294e-01,\n",
       "        8.95993774e-01,  8.66025404e-01,  8.32569855e-01,  7.95761841e-01,\n",
       "        7.55749574e-01,  7.12694171e-01,  6.66769001e-01,  6.18158986e-01,\n",
       "        5.67059864e-01,  5.13677392e-01,  4.58226522e-01,  4.00930535e-01,\n",
       "        3.42020143e-01,  2.81732557e-01,  2.20310533e-01,  1.58001396e-01,\n",
       "        9.50560433e-02,  3.17279335e-02, -3.17279335e-02, -9.50560433e-02,\n",
       "       -1.58001396e-01, -2.20310533e-01, -2.81732557e-01, -3.42020143e-01,\n",
       "       -4.00930535e-01, -4.58226522e-01, -5.13677392e-01, -5.67059864e-01,\n",
       "       -6.18158986e-01, -6.66769001e-01, -7.12694171e-01, -7.55749574e-01,\n",
       "       -7.95761841e-01, -8.32569855e-01, -8.66025404e-01, -8.95993774e-01,\n",
       "       -9.22354294e-01, -9.45000819e-01, -9.63842159e-01, -9.78802446e-01,\n",
       "       -9.89821442e-01, -9.96854776e-01, -9.99874128e-01, -9.98867339e-01,\n",
       "       -9.93838464e-01, -9.84807753e-01, -9.71811568e-01, -9.54902241e-01,\n",
       "       -9.34147860e-01, -9.09631995e-01, -8.81453363e-01, -8.49725430e-01,\n",
       "       -8.14575952e-01, -7.76146464e-01, -7.34591709e-01, -6.90079011e-01,\n",
       "       -6.42787610e-01, -5.92907929e-01, -5.40640817e-01, -4.86196736e-01,\n",
       "       -4.29794912e-01, -3.71662456e-01, -3.12033446e-01, -2.51147987e-01,\n",
       "       -1.89251244e-01, -1.26592454e-01, -6.34239197e-02, -2.44929360e-16])"
      ]
     },
     "execution_count": 40,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "y = np.sin(x)\n",
    "y"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 41,
   "metadata": {},
   "outputs": [],
   "source": [
    "import matplotlib.pyplot as plt"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 42,
   "metadata": {},
   "outputs": [],
   "source": [
    "%matplotlib inline"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 43,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.plot(x,y);"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# 2D Numpy Array"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Create 2D Numpy Array"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 44,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Numpy Array A \n",
      "[[11 12 13]\n",
      " [21 22 23]\n",
      " [31 32 33]]\n",
      "Type of Numpy Array A <class 'numpy.ndarray'>\n",
      "Elements Type of Numpy Array A int64\n",
      "Size of Numpy Array A 9\n",
      "Dimensions of Numpy Array A 2\n",
      "Shape of Numpy Array A (3, 3)\n"
     ]
    }
   ],
   "source": [
    "A = np.array([[11,12,13],\n",
    "             [21,22,23],\n",
    "             [31,32,33]])\n",
    "print(f'Numpy Array A \\n{A}')\n",
    "print(f'Type of Numpy Array A {type(A)}')\n",
    "print(f'Elements Type of Numpy Array A {A.dtype}')\n",
    "print(f'Size of Numpy Array A {A.size}')\n",
    "print(f'Dimensions of Numpy Array A {A.ndim}')\n",
    "print(f'Shape of Numpy Array A {A.shape}')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Indexing"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 45,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([11, 12, 13])"
      ]
     },
     "execution_count": 45,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "A[0]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 46,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "11"
      ]
     },
     "execution_count": 46,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "A[0][0]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 47,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "23"
      ]
     },
     "execution_count": 47,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "A[1][2]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Slicing"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 48,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([11, 12])"
      ]
     },
     "execution_count": 48,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "A[0,0:2]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 49,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([21, 22, 23])"
      ]
     },
     "execution_count": 49,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "A[1,:]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 50,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([13, 23, 33])"
      ]
     },
     "execution_count": 50,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "A[:,2]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 51,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([13, 23])"
      ]
     },
     "execution_count": 51,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "A[0:2,2]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Addition"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 52,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[1, 0],\n",
       "       [0, 1]])"
      ]
     },
     "execution_count": 52,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "X = np.array([[1,0],\n",
    "              [0,1]])\n",
    "X"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 53,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[2, 1],\n",
       "       [1, 2]])"
      ]
     },
     "execution_count": 53,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Y = np.array([[2,1],\n",
    "              [1,2]])\n",
    "Y"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 54,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[3, 1],\n",
       "       [1, 3]])"
      ]
     },
     "execution_count": 54,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Z = X + Y\n",
    "Z"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 55,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[4, 2],\n",
       "       [2, 4]])"
      ]
     },
     "execution_count": 55,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Z = 2*Y\n",
    "Z"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Matrix Multiplication"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Hadamard Matrix Multiplication"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 56,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[2, 0],\n",
       "       [0, 2]])"
      ]
     },
     "execution_count": 56,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Z = X * Y\n",
    "Z"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Dot Matrix Multiplication"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 57,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[2, 1],\n",
       "       [1, 2]])"
      ]
     },
     "execution_count": 57,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Z = X@Y\n",
    "Z"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 58,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[0, 1, 1],\n",
       "       [1, 0, 1]])"
      ]
     },
     "execution_count": 58,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "A = np.array([[0,1,1],\n",
    "              [1,0,1]])\n",
    "A"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 59,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[ 1,  1],\n",
       "       [ 1,  1],\n",
       "       [-1,  1]])"
      ]
     },
     "execution_count": 59,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "B = np.array([[1,1],\n",
    "              [1,1],\n",
    "              [-1,1]])\n",
    "B"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 60,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[0, 2],\n",
       "       [0, 2]])"
      ]
     },
     "execution_count": 60,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "C = A@B\n",
    "C"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Plotting functions"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 61,
   "metadata": {},
   "outputs": [],
   "source": [
    "# Plotting functions Plotvec1,Plotvec2\n",
    "\n",
    "def Plotvec1(u, z, v):\n",
    "    \n",
    "    ax = plt.axes()\n",
    "    ax.arrow(0, 0, *u, head_width=0.05, color='r', head_length=0.1)\n",
    "    plt.text(*(u + 0.1), 'u')\n",
    "    \n",
    "    ax.arrow(0, 0, *v, head_width=0.05, color='b', head_length=0.1)\n",
    "    plt.text(*(v + 0.1), 'v')\n",
    "    ax.arrow(0, 0, *z, head_width=0.05, head_length=0.1)\n",
    "    plt.text(*(z + 0.1), 'z')\n",
    "    plt.ylim(-2, 2)\n",
    "    plt.xlim(-2, 2)\n",
    "\n",
    "def Plotvec2(a,b):\n",
    "    ax = plt.axes()\n",
    "    ax.arrow(0, 0, *a, head_width=0.05, color ='r', head_length=0.1)\n",
    "    plt.text(*(a + 0.1), 'a')\n",
    "    ax.arrow(0, 0, *b, head_width=0.05, color ='b', head_length=0.1)\n",
    "    plt.text(*(b + 0.1), 'b')\n",
    "    plt.ylim(-2, 2)\n",
    "    plt.xlim(-2, 2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 62,
   "metadata": {},
   "outputs": [],
   "source": [
    "u = np.array([1, 0])\n",
    "v = np.array([0, 1])\n",
    "z = u + v"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 63,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "The dot product u@z is 1\n"
     ]
    },
    {
     "data": {
      "image/png": "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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "Plotvec1(u, z, v)\n",
    "print(f\"The dot product u@z is {u@z}\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 64,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "The dot product a@b is 0\n"
     ]
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "a,b = np.array([-1,1]),np.array([1,1])\n",
    "Plotvec2(a,b)\n",
    "print(f\"The dot product a@b is {a@b}\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 65,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "The dot product a@b is 0\n"
     ]
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "a,b = np.array([1,0]),np.array([0,1])\n",
    "Plotvec2(a,b)\n",
    "print(f\"The dot product a@b is {a@b}\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 66,
   "metadata": {},
   "outputs": [],
   "source": [
    "# The vectors are perpendicular. \n",
    "# As a result, the dot product is zero. "
   ]
  }
 ],
 "metadata": {
  "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": "3.6.6"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 4
}