PY0101EN-5-1-Numpy1D.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<center>\n",
    "    <img src=\"https://cf-courses-data.s3.us.cloud-object-storage.appdomain.cloud/IBMDeveloperSkillsNetwork-PY0101EN-SkillsNetwork/IDSNlogo.png\" width=\"300\" alt=\"cognitiveclass.ai logo\"  />\n",
    "</center>\n",
    "\n",
    "# 1D Numpy in Python\n",
    "\n",
    "Estimated time needed: **30** minutes\n",
    "\n",
    "## Objectives\n",
    "\n",
    "After completing this lab you will be able to:\n",
    "\n",
    "*   Import and use the `numpy` library\n",
    "*   Perform operations with `numpy`\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<h2>Table of Contents</h2>\n",
    "<div class=\"alert alert-block alert-info\" style=\"margin-thttps://op/?utm_medium=Exinfluencer&utm_source=Exinfluencer&utm_content=000026UJ&utm_term=10006555&utm_id=NA-SkillsNetwork-Channel-SkillsNetworkCoursesIBMDeveloperSkillsNetworkPY0101ENSkillsNetwork19487395-2021-01-01: 20px\">\n",
    "    <ul>\n",
    "        <li><a href=\"https://pre/?utm_medium=Exinfluencer&utm_source=Exinfluencer&utm_content=000026UJ&utm_term=10006555&utm_id=NA-SkillsNetwork-Channel-SkillsNetworkCoursesIBMDeveloperSkillsNetworkPY0101ENSkillsNetwork19487395-2021-01-01\">Preparation</a></li>\n",
    "        <li>\n",
    "            <a href=\"https://numpy/?utm_medium=Exinfluencer&utm_source=Exinfluencer&utm_content=000026UJ&utm_term=10006555&utm_id=NA-SkillsNetwork-Channel-SkillsNetworkCoursesIBMDeveloperSkillsNetworkPY0101ENSkillsNetwork19487395-2021-01-01\">What is Numpy?</a>\n",
    "            <ul>\n",
    "                <li><a href=\"https://type/?utm_medium=Exinfluencer&utm_source=Exinfluencer&utm_content=000026UJ&utm_term=10006555&utm_id=NA-SkillsNetwork-Channel-SkillsNetworkCoursesIBMDeveloperSkillsNetworkPY0101ENSkillsNetwork19487395-2021-01-01\">Type</a></li>\n",
    "                <li><a href=\"https://val/?utm_medium=Exinfluencer&utm_source=Exinfluencer&utm_content=000026UJ&utm_term=10006555&utm_id=NA-SkillsNetwork-Channel-SkillsNetworkCoursesIBMDeveloperSkillsNetworkPY0101ENSkillsNetwork19487395-2021-01-01\">Assign Value</a></li>\n",
    "                <li><a href=\"https://slice/?utm_medium=Exinfluencer&utm_source=Exinfluencer&utm_content=000026UJ&utm_term=10006555&utm_id=NA-SkillsNetwork-Channel-SkillsNetworkCoursesIBMDeveloperSkillsNetworkPY0101ENSkillsNetwork19487395-2021-01-01\">Slicing</a></li>\n",
    "                <li><a href=\"https://list/?utm_medium=Exinfluencer&utm_source=Exinfluencer&utm_content=000026UJ&utm_term=10006555&utm_id=NA-SkillsNetwork-Channel-SkillsNetworkCoursesIBMDeveloperSkillsNetworkPY0101ENSkillsNetwork19487395-2021-01-01\">Assign Value with List</a></li>\n",
    "                <li><a href=\"https://other/?utm_medium=Exinfluencer&utm_source=Exinfluencer&utm_content=000026UJ&utm_term=10006555&utm_id=NA-SkillsNetwork-Channel-SkillsNetworkCoursesIBMDeveloperSkillsNetworkPY0101ENSkillsNetwork19487395-2021-01-01\">Other Attributes</a></li>\n",
    "            </ul>\n",
    "        </li>\n",
    "        <li>\n",
    "            <a href=\"op\">Numpy Array Operations</a>\n",
    "            <ul>\n",
    "                <li><a href=\"https://add/?utm_medium=Exinfluencer&utm_source=Exinfluencer&utm_content=000026UJ&utm_term=10006555&utm_id=NA-SkillsNetwork-Channel-SkillsNetworkCoursesIBMDeveloperSkillsNetworkPY0101ENSkillsNetwork19487395-2021-01-01\">Array Addition</a></li>\n",
    "                <li><a href=\"https://multi/?utm_medium=Exinfluencer&utm_source=Exinfluencer&utm_content=000026UJ&utm_term=10006555&utm_id=NA-SkillsNetwork-Channel-SkillsNetworkCoursesIBMDeveloperSkillsNetworkPY0101ENSkillsNetwork19487395-2021-01-01\">Array Multiplication</a></li>\n",
    "                <li><a href=\"https://prod/?utm_medium=Exinfluencer&utm_source=Exinfluencer&utm_content=000026UJ&utm_term=10006555&utm_id=NA-SkillsNetwork-Channel-SkillsNetworkCoursesIBMDeveloperSkillsNetworkPY0101ENSkillsNetwork19487395-2021-01-01\">Product of Two Numpy Arrays</a></li>\n",
    "                <li><a href=\"https://dot/?utm_medium=Exinfluencer&utm_source=Exinfluencer&utm_content=000026UJ&utm_term=10006555&utm_id=NA-SkillsNetwork-Channel-SkillsNetworkCoursesIBMDeveloperSkillsNetworkPY0101ENSkillsNetwork19487395-2021-01-01\">Dot Product</a></li>\n",
    "                <li><a href=\"https://cons/?utm_medium=Exinfluencer&utm_source=Exinfluencer&utm_content=000026UJ&utm_term=10006555&utm_id=NA-SkillsNetwork-Channel-SkillsNetworkCoursesIBMDeveloperSkillsNetworkPY0101ENSkillsNetwork19487395-2021-01-01\">Adding Constant to a Numpy Array</a></li>\n",
    "            </ul>\n",
    "        </li>\n",
    "        <li><a href=\"https://math/?utm_medium=Exinfluencer&utm_source=Exinfluencer&utm_content=000026UJ&utm_term=10006555&utm_id=NA-SkillsNetwork-Channel-SkillsNetworkCoursesIBMDeveloperSkillsNetworkPY0101ENSkillsNetwork19487395-2021-01-01\">Mathematical Functions</a></li>\n",
    "        <li><a href=\"https://lin/?utm_medium=Exinfluencer&utm_source=Exinfluencer&utm_content=000026UJ&utm_term=10006555&utm_id=NA-SkillsNetwork-Channel-SkillsNetworkCoursesIBMDeveloperSkillsNetworkPY0101ENSkillsNetwork19487395-2021-01-01\">Linspace</a></li>\n",
    "    </ul>\n",
    "\n",
    "</div>\n",
    "\n",
    "<hr>\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<h2 id=\"pre\">Preparation</h2>\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {
    "tags": []
   },
   "outputs": [],
   "source": [
    "# Import the libraries\n",
    "\n",
    "import time \n",
    "import sys\n",
    "import numpy as np \n",
    "\n",
    "import matplotlib.pyplot as plt\n",
    "%matplotlib inline  "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [],
   "source": [
    "# Plotting functions\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": "markdown",
   "metadata": {},
   "source": [
    "Create a Python List as follows:\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [],
   "source": [
    "# Create a python list\n",
    "\n",
    "a = [\"0\", 1, \"two\", \"3\", 4]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We can access the data via an index:\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<img src=\"https://cf-courses-data.s3.us.cloud-object-storage.appdomain.cloud/IBMDeveloperSkillsNetwork-PY0101EN-SkillsNetwork/labs/Module%205/images/NumOneList.png\" width=\"660\" />\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We can access each element using a square bracket as follows:\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "a[0]: 0\n",
      "a[1]: 1\n",
      "a[2]: two\n",
      "a[3]: 3\n",
      "a[4]: 4\n"
     ]
    }
   ],
   "source": [
    "# Print each element\n",
    "\n",
    "print(\"a[0]:\", a[0])\n",
    "print(\"a[1]:\", a[1])\n",
    "print(\"a[2]:\", a[2])\n",
    "print(\"a[3]:\", a[3])\n",
    "print(\"a[4]:\", a[4])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<hr>\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<h2 id=\"numpy\">What is Numpy?</h2>\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "A numpy array is similar to a list. It's usually fixed in size and each element is of the same type. We can cast a list to a numpy array by first importing `numpy`:\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [],
   "source": [
    "# import numpy library\n",
    "\n",
    "import numpy as np "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We then cast the list as follows:\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([0, 1, 2, 3, 4])"
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Create a numpy array\n",
    "\n",
    "a = np.array([0, 1, 2, 3, 4])\n",
    "a"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Each element is of the same type, in this case integers:\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<img src=\"https://cf-courses-data.s3.us.cloud-object-storage.appdomain.cloud/IBMDeveloperSkillsNetwork-PY0101EN-SkillsNetwork/labs/Module%205/images/NumOneNp.png\" width=\"500\" />\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "As with lists, we can access each element via a square bracket:\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "a[0]: 0\n",
      "a[1]: 1\n",
      "a[2]: 2\n",
      "a[3]: 3\n",
      "a[4]: 4\n"
     ]
    }
   ],
   "source": [
    "# Print each element\n",
    "\n",
    "print(\"a[0]:\", a[0])\n",
    "print(\"a[1]:\", a[1])\n",
    "print(\"a[2]:\", a[2])\n",
    "print(\"a[3]:\", a[3])\n",
    "print(\"a[4]:\", a[4])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<h3 id=\"type\">Type</h3>\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "If we check the type of the array we get <b>numpy.ndarray</b>:\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "numpy.ndarray"
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Check the type of the array\n",
    "\n",
    "type(a)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "As numpy arrays contain data of the same type, we can use the attribute \"dtype\" to obtain the data type of the array’s elements. In this case, it's a 64-bit integer:\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "dtype('int64')"
      ]
     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Check the type of the values stored in numpy array\n",
    "\n",
    "a.dtype"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We can create a numpy array with real numbers:\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [],
   "source": [
    "# Create a numpy array\n",
    "\n",
    "b = np.array([3.1, 11.02, 6.2, 213.2, 5.2])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "When we check the type of the array we get <b>numpy.ndarray</b>:\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "numpy.ndarray"
      ]
     },
     "execution_count": 11,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Check the type of array\n",
    "\n",
    "type(b)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "If we examine the attribute <code>dtype</code> we see float 64, as the elements are not integers:\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "dtype('float64')"
      ]
     },
     "execution_count": 12,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Check the value type\n",
    "\n",
    "b.dtype"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<h3 id=\"val\">Assign value</h3>\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We can change the value of the array. Consider the array <code>c</code>:\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([20,  1,  2,  3,  4])"
      ]
     },
     "execution_count": 13,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Create numpy array\n",
    "\n",
    "c = np.array([20, 1, 2, 3, 4])\n",
    "c"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We can change the first element of the array to 100 as follows:\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([100,   1,   2,   3,   4])"
      ]
     },
     "execution_count": 14,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Assign the first element to 100\n",
    "\n",
    "c[0] = 100\n",
    "c"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We can change the 5th element of the array to 0 as follows:\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([100,   1,   2,   3,   0])"
      ]
     },
     "execution_count": 15,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Assign the 5th element to 0\n",
    "\n",
    "c[4] = 0\n",
    "c"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<h3 id=\"slice\">Slicing</h3>\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Like lists, we can slice the numpy array. We can select the elements from 1 to 3 and assign it to a new numpy array <code>d</code> as follows:\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([1, 2, 3])"
      ]
     },
     "execution_count": 16,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Slicing the numpy array\n",
    "\n",
    "d = c[1:4]\n",
    "d"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We can assign the corresponding indexes to new values as follows:\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([100,   1,   2, 300, 400])"
      ]
     },
     "execution_count": 17,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Set the fourth element and fifth element to 300 and 400\n",
    "\n",
    "c[3:5] = 300, 400\n",
    "c"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<h3 id=\"list\">Assign Value with List</h3>\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Similarly, we can use a list to select more than one specific index.\n",
    "The list `select` contains several values:\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {
    "tags": []
   },
   "outputs": [],
   "source": [
    "# Create the index list\n",
    "\n",
    "select = [0, 2, 3]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We can use the list as an argument in the brackets. The output is the elements corresponding to the particular indexes:\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([100,   2, 300])"
      ]
     },
     "execution_count": 19,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Use List to select elements\n",
    "\n",
    "d = c[select]\n",
    "d"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We can assign the specified elements to a new value. For example, we can assign the values to 100 000 as follows:\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([100000,      1, 100000, 100000,    400])"
      ]
     },
     "execution_count": 20,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Assign the specified elements to new value\n",
    "\n",
    "c[select] = 100000\n",
    "c"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<h3 id=\"other\">Other Attributes</h3>\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Let's review some basic array attributes using the array <code>a</code>:\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([0, 1, 2, 3, 4])"
      ]
     },
     "execution_count": 21,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Create a numpy array\n",
    "\n",
    "a = np.array([0, 1, 2, 3, 4])\n",
    "a"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The attribute <code>size</code> is the number of elements in the array:\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "5"
      ]
     },
     "execution_count": 22,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Get the size of numpy array\n",
    "\n",
    "a.size"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The next two attributes will make more sense when we get to higher dimensions but let's review them. The attribute <code>ndim</code> represents the number of array dimensions, or the rank of the array. In this case, one:\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "1"
      ]
     },
     "execution_count": 23,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Get the number of dimensions of numpy array\n",
    "\n",
    "a.ndim"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The attribute <code>shape</code> is a tuple of integers indicating the size of the array in each dimension:\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(5,)"
      ]
     },
     "execution_count": 24,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Get the shape/size of numpy array\n",
    "\n",
    "a.shape"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 25,
   "metadata": {
    "tags": []
   },
   "outputs": [],
   "source": [
    "# Create a numpy array\n",
    "\n",
    "a = np.array([1, -1, 1, -1])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 26,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.0"
      ]
     },
     "execution_count": 26,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Get the mean of numpy array\n",
    "\n",
    "mean = a.mean()\n",
    "mean"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 27,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "1.0"
      ]
     },
     "execution_count": 27,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Get the standard deviation of numpy array\n",
    "\n",
    "standard_deviation=a.std()\n",
    "standard_deviation"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 28,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([-1,  2,  3,  4,  5])"
      ]
     },
     "execution_count": 28,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Create a numpy array\n",
    "\n",
    "b = np.array([-1, 2, 3, 4, 5])\n",
    "b"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 29,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "5"
      ]
     },
     "execution_count": 29,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Get the biggest value in the numpy array\n",
    "\n",
    "max_b = b.max()\n",
    "max_b"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 30,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "-1"
      ]
     },
     "execution_count": 30,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Get the smallest value in the numpy array\n",
    "\n",
    "min_b = b.min()\n",
    "min_b"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<hr>\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<h2 id=\"op\">Numpy Array Operations</h2>\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<h3 id=\"add\">Array Addition</h3>\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Consider the numpy array <code>u</code>:\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 31,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([1, 0])"
      ]
     },
     "execution_count": 31,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "u = np.array([1, 0])\n",
    "u"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Consider the numpy array <code>v</code>:\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 32,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([0, 1])"
      ]
     },
     "execution_count": 32,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "v = np.array([0, 1])\n",
    "v"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We can add the two arrays and assign it to z:\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 33,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([1, 1])"
      ]
     },
     "execution_count": 33,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Numpy Array Addition\n",
    "\n",
    "z = u + v\n",
    "z"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The operation is equivalent to vector addition:\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 34,
   "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": [
    "# Plot numpy arrays\n",
    "\n",
    "Plotvec1(u, z, v)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<h3 id=\"multi\">Array Multiplication</h3>\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Consider the vector numpy array <code>y</code>:\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 35,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([1, 2])"
      ]
     },
     "execution_count": 35,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Create a numpy array\n",
    "\n",
    "y = np.array([1, 2])\n",
    "y"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We can multiply every element in the array by 2:\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 36,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([2, 4])"
      ]
     },
     "execution_count": 36,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Numpy Array Multiplication\n",
    "\n",
    "z = 2 * y\n",
    "z"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "This is equivalent to multiplying a vector by a scaler:\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<h3 id=\"prod\">Product of Two Numpy Arrays</h3>\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Consider the following array <code>u</code>:\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 37,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([1, 2])"
      ]
     },
     "execution_count": 37,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Create a numpy array\n",
    "\n",
    "u = np.array([1, 2])\n",
    "u"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Consider the following array <code>v</code>:\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 38,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([3, 2])"
      ]
     },
     "execution_count": 38,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Create a numpy array\n",
    "\n",
    "v = np.array([3, 2])\n",
    "v"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The product of the two numpy arrays <code>u</code> and <code>v</code> is given by:\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 39,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([3, 4])"
      ]
     },
     "execution_count": 39,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Calculate the production of two numpy arrays\n",
    "\n",
    "z = u * v\n",
    "z"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "tags": []
   },
   "source": [
    "<h3 id=\"dot\">Dot Product</h3>\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The dot product of the two numpy arrays <code>u</code> and <code>v</code> is given by:\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 40,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "7"
      ]
     },
     "execution_count": 40,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Calculate the dot product\n",
    "\n",
    "np.dot(u, v)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<h3 id=\"cons\">Adding Constant to a Numpy Array</h3>\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Consider the following array:\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 41,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([ 1,  2,  3, -1])"
      ]
     },
     "execution_count": 41,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Create a constant to numpy array\n",
    "\n",
    "u = np.array([1, 2, 3, -1]) \n",
    "u"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Adding the constant 1 to each element in the array:\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 42,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([2, 3, 4, 0])"
      ]
     },
     "execution_count": 42,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Add the constant to array\n",
    "\n",
    "u + 1"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The process is summarised in the following animation:\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<img src=\"https://cf-courses-data.s3.us.cloud-object-storage.appdomain.cloud/IBMDeveloperSkillsNetwork-PY0101EN-SkillsNetwork/labs/Module%205/images/NumOneAdd.gif\" width=\"500\" />\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<hr>\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<h2 id=\"math\">Mathematical Functions</h2>\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We can access the value of <code>pi</code> in numpy as follows :\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 43,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "3.141592653589793"
      ]
     },
     "execution_count": 43,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# The value of pi\n",
    "\n",
    "np.pi"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We can create the following numpy array in Radians:\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 44,
   "metadata": {},
   "outputs": [],
   "source": [
    "# Create the numpy array in radians\n",
    "\n",
    "x = np.array([0, np.pi/2 , np.pi])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We can apply the function <code>sin</code> to the array <code>x</code> and assign the values to the array <code>y</code>; this applies the sine function to each element in the array:\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 45,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([0.0000000e+00, 1.0000000e+00, 1.2246468e-16])"
      ]
     },
     "execution_count": 45,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Calculate the sin of each elements\n",
    "\n",
    "y = np.sin(x)\n",
    "y"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<hr>\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<h2 id=\"lin\">Linspace</h2>\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "A useful function for plotting mathematical functions is <code>linspace</code>.   Linspace returns evenly spaced numbers over a specified interval. We specify the starting point of the sequence and the ending point of the sequence. The parameter \"num\" indicates the Number of samples to generate, in this case 5:\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 46,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([-2., -1.,  0.,  1.,  2.])"
      ]
     },
     "execution_count": 46,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Makeup a numpy array within [-2, 2] and 5 elements\n",
    "\n",
    "np.linspace(-2, 2, num=5)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "If we change the parameter <code>num</code> to 9, we get 9 evenly spaced numbers over the interval from -2 to 2:\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 47,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([-2. , -1.5, -1. , -0.5,  0. ,  0.5,  1. ,  1.5,  2. ])"
      ]
     },
     "execution_count": 47,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Make a numpy array within [-2, 2] and 9 elements\n",
    "\n",
    "np.linspace(-2, 2, num=9)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We can use the function <code>linspace</code> to generate 100 evenly spaced samples from the interval 0 to 2π:\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 48,
   "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": 49,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Make a numpy array within [0, 2π] and 100 elements \n",
    "\n",
    "x = np.linspace(0, 2*np.pi, num=100)\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We can apply the sine function to each element in the array <code>x</code> and assign it to the array <code>y</code>:\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true,
    "jupyter": {
     "outputs_hidden": true
    },
    "tags": []
   },
   "outputs": [],
   "source": [
    "# Calculate the sine of x list\n",
    "\n",
    "y = np.sin(x)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[<matplotlib.lines.Line2D at 0x7f0ac5ba47d0>]"
      ]
     },
     "execution_count": 51,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "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": [
    "# Plot the result\n",
    "\n",
    "plt.plot(x, y)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<hr>\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<h2 id=\"quiz\">Quiz on 1D Numpy Array</h2>\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Implement the following vector subtraction in numpy: u-v\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 55,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([ 1, -1])"
      ]
     },
     "execution_count": 55,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Write your code below and press Shift+Enter to execute\n",
    "\n",
    "u = np.array([1, 0])\n",
    "v = np.array([0, 1])\n",
    "Za = u-v\n",
    "Za"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<details><summary>Click here for the solution</summary>\n",
    "\n",
    "```python\n",
    "u - v\n",
    "```\n",
    "\n",
    "</details>\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<hr>\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Multiply the numpy array z with -2:\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 57,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([-4, -8])"
      ]
     },
     "execution_count": 57,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Write your code below and press Shift+Enter to execute\n",
    "\n",
    "z = np.array([2, 4])\n",
    "z = -2*z\n",
    "z"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<details><summary>Click here for the solution</summary>\n",
    "\n",
    "```python\n",
    "-2 * z\n",
    "```\n",
    "\n",
    "</details>\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<hr>\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Consider the list <code>\\[1, 2, 3, 4, 5]</code> and <code>\\[1, 0, 1, 0, 1]</code>. Cast both lists to a numpy array then multiply them together:\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 59,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([1, 0, 3, 0, 5])"
      ]
     },
     "execution_count": 59,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Write your code below and press Shift+Enter to execute\n",
    "l1 = np.array([1, 2, 3, 4, 5])\n",
    "l2 = np.array([1, 0, 1, 0, 1])\n",
    "l3 = l1 * l2\n",
    "l3"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<details><summary>Click here for the solution</summary>\n",
    "\n",
    "```python\n",
    "a = np.array([1, 2, 3, 4, 5])\n",
    "b = np.array([1, 0, 1, 0, 1])\n",
    "a * b\n",
    "```\n",
    "\n",
    "</details>\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<hr>\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Convert the list <code>\\[-1, 1]</code> and <code>\\[1, 1]</code> to numpy arrays <code>a</code> and <code>b</code>. Then, plot the arrays as vectors using the fuction <code>Plotvec2</code> and find their dot product:\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 61,
   "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": [
    "# Write your code below and press Shift+Enter to execute\n",
    "\n",
    "a = np.array([-1, 1])\n",
    "b = np.array([1, 1])\n",
    "Plotvec2(a,b)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<details><summary>Click here for the solution</summary>\n",
    "\n",
    "```python\n",
    "a = np.array([-1, 1])\n",
    "b = np.array([1, 1])\n",
    "Plotvec2(a, b)\n",
    "print(\"The dot product is\", np.dot(a,b))\n",
    "\n",
    "```\n",
    "\n",
    "</details>\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<hr>\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Convert the list <code>\\[1, 0]</code> and <code>\\[0, 1]</code> to numpy arrays <code>a</code> and <code>b</code>. Then, plot the arrays as vectors using the function <code>Plotvec2</code> and find their dot product:\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 63,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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8IeCDM20kyTgw3mb/JcnuzipcGJcCL426iDmwzm5ZZ7esszv/7nQ/OGsgJHkMuLzPqjur6tutzZ3ASeD+fpvos6xm6q+qJoCJtt3JqhqbrcZROhNqBOvsmnV2yzq7k2TydD87ayBU1Ydn6XwL8DHgt6qq3xf9IWDVlPkrgcPzKVKStPAGvcpoI/BF4D9V1YkZmj0FrEny3iRLgVuBbYP0K0nq3qBXGX0FuBB4NMkzSe4GSHJFku0A7aTz7cDDwF7gr6pqzxy3PzFgfcNwJtQI1tk16+yWdXbntGtM/6M8kqS3Gu9UliQBBoIkqVk0gXCmPAYjySeT7EnyRpIZLz9L8mKSH7RzK6d9Gdjpmkedox7PS5I8muT59v6OGdqNZDxnG5/03NXW70py7bBqm0eNG5Icb2P3TJIvD7vGVsc9SY7OdG/RYhjLVsdsdY58PJOsSvK3Sfa2f+d/2KfN/MezqhbFC/gocF6b/hPgT/q0ORf4J+DXgKXAs8DVQ67zKno3fnwPGDtFuxeBS0c4nrPWuUjG838Ad7TpO/r9dx/VeM5lfIBNwIP07rdZD3x/Eda4Afi/o/pbnFLHfwSuBXbPsH6kYzmPOkc+nsAK4No2fSG9RwcN/Le5aPYQ6gx5DEZV7a2q/cPs83TMsc6Rj2fr7942fS9w85D7P5W5jM9m4L7qeRK4OMmKRVbjolBVO4CXT9Fk1GMJzKnOkauqI1X1dJt+hd4VnCunNZv3eC6aQJjmc/SSbbp+j8GYPgiLRQGPJNnZHsexGC2G8XxXVR2B3h85cNkM7UYxnnMZn1GP4Vz7vyHJs0keTHLNcEqbt1GP5XwsmvFM8h7gN4DvT1s17/Hs8llGsxr2YzBO11zqnIMbq+pwksvo3aexr/2fR2c6qHPk4zmPzSz4ePYxl/EZyhiewlz6fxp4d1W9mmQT8C1gzUIXdhpGPZZztWjGM8kFwF8Df1RVP52+us9HTjmeQw2EOkMegzFbnXPcxuH2fjTJVnq79p1+gXVQ58jHM8mPkqyoqiNtd/boDNtY8PHsYy7jM+pHs8za/9QviqranuRrSS6tqsX2kLZRj+WcLJbxTLKEXhjcX1V/06fJvMdz0Rwyyln0GIwky5Jc+MtpeifMF+NTWxfDeG4DtrTpLcCb9mxGOJ5zGZ9twGfaFR3rgeO/PAQ2JLPWmOTyJGnT6+j9u//xEGucq1GP5ZwshvFs/f85sLeq/nSGZvMfz1GeKZ92RvwAveNdz7TX3W35FcD2aWfO/5HelRV3jqDOj9NL3n8BfgQ8PL1Oeld8PNteexZrnYtkPN8JfBd4vr1fspjGs9/4AJ8HPt+mQ+8HoP4J+AGnuPJshDXe3sbtWXoXbPyHYdfY6ngAOAL8vP1t/v5iG8s51jny8QQ+RO/wz64p35mbBh1PH10hSQIW0SEjSdJoGQiSJMBAkCQ1BoIkCTAQJEmNgSBJAgwESVLz/wGswKsYaRlXhwAAAABJRU5ErkJggg==\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "# Write your code below and press Shift+Enter to execute\n",
    "a = np.array([1, 0])\n",
    "b = np.array([0, 1])\n",
    "Plotvec2(a,b)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<details><summary>Click here for the solution</summary>\n",
    "\n",
    "```python\n",
    "a = np.array([1, 0])\n",
    "b = np.array([0, 1])\n",
    "Plotvec2(a, b)\n",
    "print(\"The dot product is\", np.dot(a, b))\n",
    "\n",
    "```\n",
    "\n",
    "</details>\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<hr>\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Convert the list <code>\\[1, 1]</code> and <code>\\[0, 1]</code> to numpy arrays <code>a</code> and <code>b</code>. Then plot the arrays as vectors using the fuction <code>Plotvec2</code> and find their dot product:\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 64,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAYQAAAD8CAYAAAB3u9PLAAAAOXRFWHRTb2Z0d2FyZQBNYXRwbG90bGliIHZlcnNpb24zLjUuMCwgaHR0cHM6Ly9tYXRwbG90bGliLm9yZy8/fFQqAAAACXBIWXMAAAsTAAALEwEAmpwYAAAWK0lEQVR4nO3de4xd5Xnv8e+DMeVeLuZisA1EsUogNSkZGXOgEVECMlYVkwpOITeURJoGhbaoR9XhBIW2R6qU5lStahEg0wYCFYE0JYATDMZOSx2nmDJ2sA0YsEMNHsbFxgYDMQRsnvPHWtTTYY9nz+w1e+2Z+X6k0ay117vX++j1eH7zrtuOzESSpAPqLkCS1BkMBEkSYCBIkkoGgiQJMBAkSSUDQZIEVBAIETEzIv4lIjZExJMR8UcN2kRELIqITRGxLiLObrVfSVK1DqxgH3uA/5WZayLiCGB1RCzLzKcGtLkYmF1+nQPcVH6XJHWIlmcImbk1M9eUy68DG4CTBzVbCNyehVXAURExvdW+JUnVqWKG8F8i4lTgt4BHB206GdgyYL2vfG1rg310A90Ahx122EdPP/30KkuUpAlt9erVL2fmcaN5b2WBEBGHA3cD12Tma4M3N3hLw2dmZGYP0APQ1dWVvb29VZUoSRNeRDw/2vdWcpVRREylCIM7MvOHDZr0ATMHrM8A+qvoW5JUjSquMgrgO8CGzPzrIZotBr5QXm00D9iVme87XCRJqk8Vh4zOAz4PrI+Ix8vXvgbMAsjMm4ElwAJgE7Ab+GIF/UqSKtRyIGTmShqfIxjYJoGvttqXJGnseKeyJAkwECRJJQNBkgQYCJKkkoEgSQIMBElSyUCQJAEGgiSpZCBIkgADQZJUMhAkSYCBIEkqGQiSJMBAkCSVDARJEmAgSJJKBoIkCTAQJEmlSgIhIm6JiG0R8cQQ2y+IiF0R8Xj5dX0V/UqSqtPyZyqXvgvcANy+nzY/zczfqag/SVLFKpkhZOYKYGcV+5Ik1aOd5xDOjYi1EfFARJzZxn4lSU2o6pDRcNYAp2TmGxGxALgXmN2oYUR0A90As2bNalN5kqS2zBAy87XMfKNcXgJMjYhpQ7TtycyuzOw67rjj2lGeJIk2BUJEnBgRUS7PLfvd0Y6+JUnNqeSQUUTcCVwATIuIPuBPgakAmXkzcClwVUTsAd4ELs/MrKJvSVI1KgmEzLximO03UFyWKknqUN6pLEkCDARJUslAkCQBBoIkqWQgSJIAA0GSVDIQJEmAgSBJKhkIkiTAQJDGzObNm/nwhz9cdxlS0wwESRJgIEhjas+ePVx55ZXMmTOHSy+9lN27d9dd0rh2ySWX8NGPfpQzzzyTnp6eusuZcAwEaQw988wzdHd3s27dOo488khuvPHGuksa12655RZWr15Nb28vixYtYscOn6JfJQNBGkMzZ87kvPPOA+Bzn/scK1eurLmi8W3RokWcddZZzJs3jy1btrBx48a6S5pQ2vURmtKkVH4u1JDrat7DDz/M8uXLeeSRRzj00EO54IILeOutt+oua0JxhiCNoRdeeIFHHnkEgDvvvJPzzz+/5orGr127dnH00Udz6KGH8vTTT7Nq1aq6S5pwDARphD77WbjqqubafuhDH+K2225jzpw57Ny5k6uafaPeZ/78+ezZs4c5c+bw9a9/nXnz5tVd0oQTnfxJll1dXdnb21t3GdJ/eeUVmD4dDjgAXn4ZDj207oomoExYuRJefBEuv7zuasadiFidmV2jea8zBGkE7rgDpkwpvv7pn+quZoJ56SX4xjdgxgz42MeKwVZbVRIIEXFLRGyLiCeG2B4RsSgiNkXEuog4u4p+pXb727+F3bvhjTeKZbVo715YsgQuughOOQX+/M+LYJg7F37wg7qrm3Squsrou8ANwO1DbL8YmF1+nQPcVH6Xxo2f/xz6+/etP/UUbNoEH/xgfTWNa++8AyedBG++Cb/8ZfHa1KkwezYsWwYHH1xvfZNQJTOEzFwB7NxPk4XA7VlYBRwVEdOr6FtqlwcfLH6HvWfvXnjoofrqGfciikB4LwwOOABOOAFWrIAjj6y3tkmqXfchnAxsGbDeV762dXDDiOgGugFmzZrVluKkZvzhH8LHPw7nnlusr1gBH/lIrSWNX2++WYTBq6/C8cfDIYcUx+F+9jM47ri6q5u02nVSudHdOA0vb8rMnszsysyu4/zBUAc57DB470rHww8vlj2qMQrf/W5xedarr8I//3NxzmDjRujrA/8IrFW7Zgh9wMwB6zOA/iHaSpqIXn0Vjj66WP7t34aHHy4OE0Fx7mDq1LoqU6ldM4TFwBfKq43mAbsy832HiyRNUN/85r4wWLOmON52gFe9d5pKZggRcSdwATAtIvqAPwWmAmTmzcASYAGwCdgNfLGKfiV1uK1bi3MFAL/3e3DnncXJZHWkSgIhM68YZnsCX62iL0njxB//MfzN3xTLGzd6fe444NNOJVVr06biXgKAa67ZFwrqeAaCpGpkwmc+A3fdVaz39xcPftK44VkdSa37+c+Lk8R33VU8jyjTMBiHnCFIGr133y3u1luxolh/5RU46qhaS9LoOUOQNDoPP1w89nXFCvjOd4pZgWEwrjlDkDQy77wDZ5xRnDz+9V8vLi095JC6q1IFnCFIat4PfwgHHVSEwX33FXcfGwYThjMEScPbvbt46Nzu3XD66bB+PRzor4+JxhmCpP37u78rnuy3ezf867/Chg2GwQTlv6qkxnbuhGOPLZY/8Yniwx98/tCE5r+upPf7i7/YFwZr18Ly5YbBJOAMQdI+L75YfMg9wOc/D7fd5sPoJhEjX1LhD/5gXxj84hdw++2GwSRjIEiT3bPPFr/4b7gB/uRPihvMPvCBuqtSDTxkJE1WmXDZZXD33cX6f/5n8SH3mrScIUiTUW9vcZL47rvhr/6qCAfDYNJzhiBNJnv3wvnnw6pVxfquXXDkkfXWpI7hDEGaLJYvL24oW7WqOGGcaRjov6kkECJifkQ8ExGbIuLaBtsviIhdEfF4+XV9Ff1KasLbb8Opp8KFF8K0afDmm8UlpdIgLQdCREwBvgVcDJwBXBERZzRo+tPM/Ej59X9b7VdSE37wA/i1X4Pnn4cf/xi2b4eDD667KnWoKs4hzAU2ZeZzABFxF7AQeKqCfUsajTfegGOOKR5V/Zu/CWvW+PwhDauKQ0YnA1sGrPeVrw12bkSsjYgHIuLMoXYWEd0R0RsRvdu3b6+gPGmSufFGOOKIIgxWroR16wwDNaWKn5JGtzLmoPU1wCmZ+UZELADuBWY32llm9gA9AF1dXYP3I2koO3YU5wgA5s+HJUu801gjUsUMoQ+YOWB9BtA/sEFmvpaZb5TLS4CpETGtgr4lAfzZn+0Lg/Xr4YEHDAONWBUzhMeA2RFxGvAicDnwmYENIuJE4KXMzIiYSxFEOyroW5rctmyBWbOK5S99Cf7+7w0CjVrLgZCZeyLiamApMAW4JTOfjIivlNtvBi4FroqIPcCbwOWZ6eEgabQy4aqr4NvfLtb/4z+KS0ulFkQn/17u6urK3t7eusuQ/psIOPxweP31mgrYsKH4kHuAr32t+OwCqRQRqzOzazTv9dIDabzIhIUL4Uc/Kta3bSs+51iqiI+ukMaDRx8tHkb3ox/BokVFOBgGqpgzBKmT7d0L55wDq1cX66+9VtxjII0BZwhSp1q6tLihbPVq+N73ilmBYaAx5AxB6jS/+hWcdhps3QonnQTPPVc8j0gaY84QpE7yve8VD5/burW4uezFFw0DtY0zBKkTvP76vs8m6OoqPrNgypR6a9Kk4wxBqtuiRfvCYNUqeOwxw0C1cIYg1WX7djj++GL5U5+Ce+/1sROqlTMEqQ7XXbcvDJ56Cu67zzBQ7ZwhSO30/PP7njn0+78PN91kEKhjGAhSO2TCl78Mt95arL/wAsycuf/3SG3mISNprD3xRPHYiVtvheuvL8LBMFAHcoYgjZVMWLAAHnywWH/5ZTj22HprkvbDGYI0Fn72s2JW8OCDxWccZxoG6njOEKQq7dkDZ59dfIzl1Kmwc2fx4QnSOOAMQarK/fcXIbB+PXz/+/D224aBxhVnCFKr3noLZsyAHTvglFPg2WfhoIPqrkoasUpmCBExPyKeiYhNEXFtg+0REYvK7esi4uwq+pVq9w//AIccUoTBsmWwebNhoHGr5RlCREwBvgVcCPQBj0XE4sx8akCzi4HZ5dc5wE3ld2l8yoQo/54691z46U99/pDGvSpmCHOBTZn5XGa+DdwFLBzUZiFwexZWAUdFxPQK+pba7qCp73LUL/uKlcceg3/7N8NAE0IV5xBOBrYMWO/j/X/9N2pzMrB18M4iohvoBpg1a1YF5UnVWnHrcxy8/Mdwy7s+dkITShUzhEb/I3IUbYoXM3sysyszu47zQ8TVgc757Ac569ZrDANNOFUEQh8w8D78GUD/KNpIkmpURSA8BsyOiNMi4iDgcmDxoDaLgS+UVxvNA3Zl5vsOF0mS6tPyOYTM3BMRVwNLgSnALZn5ZER8pdx+M7AEWABsAnYDX2y1X0lStSq5MS0zl1D80h/42s0DlhP4ahV9SZLGho+ukCQBBoIkqWQgSJIAA0GSVDIQJEmAgSBJKhkIkiTAQJAklQwESRJgIEiSSgaCJAkwECRJJQNBkgQYCJKkkoEgSQIMBElSyUCQJAEGgiSp1NJHaEbEMcD3gVOBzcD/zMxXGrTbDLwO7AX2ZGZXK/1KkqrX6gzhWuAnmTkb+Em5PpSPZ+ZHDANJ6kytBsJC4LZy+Tbgkhb3J0mqSauBcEJmbgUovx8/RLsEHoqI1RHRvb8dRkR3RPRGRO/27dtbLE+S1KxhzyFExHLgxAabrhtBP+dlZn9EHA8si4inM3NFo4aZ2QP0AHR1deUI+pAktWDYQMjMTw61LSJeiojpmbk1IqYD24bYR3/5fVtE3APMBRoGgiSpHq0eMloMXFkuXwncN7hBRBwWEUe8twxcBDzRYr+SpIq1GgjfAC6MiI3AheU6EXFSRCwp25wArIyItcC/A/dn5oMt9itJqlhL9yFk5g7gEw1e7wcWlMvPAWe10o8kaex5p7IkCTAQJEklA0GSBBgIkqSSgSBJAgwESVLJQJAkAQaCJKlkIEiSAANBklQyECRJgIEgSSoZCJIkwECQJJUMBEkSYCBIkkoGgiQJMBAkSaWWAiEiLouIJyPi3Yjo2k+7+RHxTERsiohrW+lTkjQ2Wp0hPAH8LrBiqAYRMQX4FnAxcAZwRUSc0WK/kqSKHdjKmzNzA0BE7K/ZXGBTZj5Xtr0LWAg81UrfkqRqteMcwsnAlgHrfeVrDUVEd0T0RkTv9u3bx7w4SVJh2BlCRCwHTmyw6brMvK+JPhpNH3KoxpnZA/QAdHV1DdlOklStYQMhMz/ZYh99wMwB6zOA/hb3KUmqWDsOGT0GzI6I0yLiIOByYHEb+pUkjUCrl51+OiL6gHOB+yNiafn6SRGxBCAz9wBXA0uBDcA/ZuaTrZUtSapaq1cZ3QPc0+D1fmDBgPUlwJJW+pIkjS3vVJYkAQaCJKlkIEiSAANBklQyECRJgIEgSSoZCJIkwECQJJUMBEkSYCBIkkoGgiQJMBAkSSUDQZIEGAiSpJKBIEkCDARJUslAkCQBBoIkqdTqZypfFhFPRsS7EdG1n3abI2J9RDweEb2t9ClJGhstfaYy8ATwu8C3m2j78cx8ucX+JEljpKVAyMwNABFRTTWSpNq06xxCAg9FxOqI6G5Tn5KkERh2hhARy4ETG2y6LjPva7Kf8zKzPyKOB5ZFxNOZuWKI/rqBboBZs2Y1uXtJUquGDYTM/GSrnWRmf/l9W0TcA8wFGgZCZvYAPQBdXV3Zat+SpOaM+SGjiDgsIo54bxm4iOJktCSpg7R62emnI6IPOBe4PyKWlq+fFBFLymYnACsjYi3w78D9mflgK/1KkqrX6lVG9wD3NHi9H1hQLj8HnNVKP5KkseedypIkwECQJJUMBEkSYCBIkkoGgiQJMBAkSSUDQZIEGAiSpJKBIEkCDARJUslAkCQBBoIkqWQgSJIAA0GSVDIQJEmAgSBJKhkIkiTAQJAklQwESRLQYiBExP+LiKcjYl1E3BMRRw3Rbn5EPBMRmyLi2lb6lCSNjVZnCMuAD2fmHOBZ4P8MbhARU4BvARcDZwBXRMQZLfYrSapYS4GQmQ9l5p5ydRUwo0GzucCmzHwuM98G7gIWttKvJKl6B1a4ry8B32/w+snAlgHrfcA5Q+0kIrqB7nL1VxHxRGUVjo1pwMt1F9EE66yWdVbLOqvzG6N947CBEBHLgRMbbLouM+8r21wH7AHuaLSLBq/lUP1lZg/QU+63NzO7hquxTuOhRrDOqllntayzOhHRO9r3DhsImfnJYTq/Evgd4BOZ2egXfR8wc8D6DKB/JEVKksZeq1cZzQf+N/CpzNw9RLPHgNkRcVpEHARcDixupV9JUvVavcroBuAIYFlEPB4RNwNExEkRsQSgPOl8NbAU2AD8Y2Y+2eT+e1qsrx3GQ41gnVWzzmpZZ3VGXWM0PsojSZpsvFNZkgQYCJKkUscEwnh5DEZEXBYRT0bEuxEx5OVnEbE5ItaX51ZGfRnYaI2gzrrH85iIWBYRG8vvRw/RrpbxHG58orCo3L4uIs5uV20jqPGCiNhVjt3jEXF9u2ss67glIrYNdW9RJ4xlWcdwddY+nhExMyL+JSI2lP/P/6hBm5GPZ2Z2xBdwEXBgufyXwF82aDMF+AXwAeAgYC1wRpvr/BDFjR8PA137abcZmFbjeA5bZ4eM5zeBa8vlaxv9u9c1ns2MD7AAeIDifpt5wKMdWOMFwI/r+lkcUMfHgLOBJ4bYXutYjqDO2scTmA6cXS4fQfHooJZ/NjtmhpDj5DEYmbkhM59pZ5+j0WSdtY9n2d9t5fJtwCVt7n9/mhmfhcDtWVgFHBUR0zusxo6QmSuAnftpUvdYAk3VWbvM3JqZa8rl1ymu4Dx5ULMRj2fHBMIgX6JItsEaPQZj8CB0igQeiojV5eM4OlEnjOcJmbkVih9y4Pgh2tUxns2MT91j2Gz/50bE2oh4ICLObE9pI1b3WI5Ex4xnRJwK/Bbw6KBNIx7PKp9lNKx2PwZjtJqpswnnZWZ/RBxPcZ/G0+VfHpWpoM7ax3MEuxnz8WygmfFpyxjuRzP9rwFOycw3ImIBcC8we6wLG4W6x7JZHTOeEXE4cDdwTWa+Nnhzg7fsdzzbGgg5Th6DMVydTe6jv/y+LSLuoZjaV/oLrII6ax/PiHgpIqZn5tZyOrttiH2M+Xg20Mz41P1olmH7H/iLIjOXRMSNETEtMzvtIW11j2VTOmU8I2IqRRjckZk/bNBkxOPZMYeMYgI9BiMiDouII95bpjhh3olPbe2E8VwMXFkuXwm8b2ZT43g2Mz6LgS+UV3TMA3a9dwisTYatMSJOjIgol+dS/L/f0cYam1X3WDalE8az7P87wIbM/Oshmo18POs8Uz7ojPgmiuNdj5dfN5evnwQsGXTm/FmKKyuuq6HOT1Mk76+Al4Clg+ukuOJjbfn1ZKfW2SHjeSzwE2Bj+f2YThrPRuMDfAX4SrkcFB8A9QtgPfu58qzGGq8ux20txQUb/6PdNZZ13AlsBd4pfza/3Glj2WSdtY8ncD7F4Z91A35nLmh1PH10hSQJ6KBDRpKkehkIkiTAQJAklQwESRJgIEiSSgaCJAkwECRJpf8PUFrfmFkkuEAAAAAASUVORK5CYII=\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "# Write your code below and press Shift+Enter to execute\n",
    "\n",
    "a = np.array([1, 1])\n",
    "b = np.array([0, 1])\n",
    "Plotvec2(a,b)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<details><summary>Click here for the solution</summary>\n",
    "\n",
    "```python\n",
    "a = np.array([1, 1])\n",
    "b = np.array([0, 1])\n",
    "Plotvec2(a, b)\n",
    "print(\"The dot product is\", np.dot(a, b))\n",
    "\n",
    "```\n",
    "\n",
    "</details>\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<hr>\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Why are the results of the dot product for <code>\\[-1, 1]</code> and <code>\\[1, 1]</code> and the dot product for <code>\\[1, 0]</code> and <code>\\[0, 1]</code> zero, but not zero for the dot product for <code>\\[1, 1]</code> and <code>\\[0, 1]</code>? <p><i>Hint: Study the corresponding figures, pay attention to the direction the arrows are pointing to.</i></p>\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 70,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "2\n",
      "4\n"
     ]
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 2 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "# Write your code below and press Shift+Enter to execute\n",
    "\n",
    "#---- Trying random code ------\n",
    "\n",
    "a = np.array([-1,1])\n",
    "b = np.array([1,1])\n",
    "c = a + b\n",
    "\n",
    "x = np.array([1,0])\n",
    "y = np.array([0,1])\n",
    "z = x + y\n",
    "\n",
    "e = np.array([1,1])\n",
    "f = np.array([0,1])\n",
    "g = e + f\n",
    "\n",
    "print(np.dot(c,z))\n",
    "Plotvec2(c,z)\n",
    "\n",
    "print(np.dot(c,g))\n",
    "Plotvec2(c,g)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<details><summary>Click here for the solution</summary>\n",
    "\n",
    "```python\n",
    "The vectors used for question 4 and 5 are perpendicular. As a result, the dot product is zero. \n",
    "\n",
    "```\n",
    "\n",
    "</details>\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<hr>\n",
    "<h2>The last exercise!</h2>\n",
    "<p>Congratulations, you have completed your first lesson and hands-on lab in Python. However, there is one more thing you need to do. The Data Science community encourages sharing work. The best way to share and showcase your work is to share it on GitHub. By sharing your notebook on GitHub you are not only building your reputation with fellow data scientists, but you can also show it off when applying for a job. Even though this was your first piece of work, it is never too early to start building good habits. So, please read and follow <a href=\"https://cognitiveclass.ai/blog/data-scientists-stand-out-by-sharing-your-notebooks/?utm_medium=Exinfluencer&utm_source=Exinfluencer&utm_content=000026UJ&utm_term=10006555&utm_id=NA-SkillsNetwork-Channel-SkillsNetworkCoursesIBMDeveloperSkillsNetworkPY0101ENSkillsNetwork19487395-2021-01-01\" target=\"_blank\">this article</a> to learn how to share your work.\n",
    "<hr>\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Author\n",
    "\n",
    "<a href=\"https://www.linkedin.com/in/joseph-s-50398b136/?utm_medium=Exinfluencer&utm_source=Exinfluencer&utm_content=000026UJ&utm_term=10006555&utm_id=NA-SkillsNetwork-Channel-SkillsNetworkCoursesIBMDeveloperSkillsNetworkPY0101ENSkillsNetwork19487395-2021-01-01\" target=\"_blank\">Joseph Santarcangelo</a>\n",
    "\n",
    "## Other contributors\n",
    "\n",
    "<a href=\"https://www.linkedin.com/in/jiahui-mavis-zhou-a4537814a?utm_medium=Exinfluencer&utm_source=Exinfluencer&utm_content=000026UJ&utm_term=10006555&utm_id=NA-SkillsNetwork-Channel-SkillsNetworkCoursesIBMDeveloperSkillsNetworkPY0101ENSkillsNetwork19487395-2021-01-01\">Mavis Zhou</a>\n",
    "\n",
    "## Change Log\n",
    "\n",
    "| Date (YYYY-MM-DD) | Version | Changed By | Change Description                 |\n",
    "| ----------------- | ------- | ---------- | ---------------------------------- |\n",
    "| 2020-08-26        | 2.0     | Lavanya    | Moved lab to course repo in GitLab |\n",
    "|                   |         |            |                                    |\n",
    "|                   |         |            |                                    |\n",
    "\n",
    "<hr/>\n",
    "\n",
    "## <h3 align=\"center\"> © IBM Corporation 2020. All rights reserved. <h3/>\n"
   ]
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "Python",
   "language": "python",
   "name": "conda-env-python-py"
  },
  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
    "version": 3
   },
   "file_extension": ".py",
   "mimetype": "text/x-python",
   "name": "python",
   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython3",
   "version": "3.7.12"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 4
}