5-1-Numpy1D.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<h1>1D <code>Numpy</code> in Python</h1>"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<p><strong>Welcome!</strong> This notebook will teach you about using <code>Numpy</code> in the Python Programming Language. By the end of this lab, you'll know what <code>Numpy</code> is and the <code>Numpy</code> operations.</p>"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<h2>Table of Contents</h2>\n",
    "<div class=\"alert alert-block alert-info\" style=\"margin-top: 20px\">\n",
    "    <ul>\n",
    "        <li><a href=\"pre\">Preparation</a></li>\n",
    "        <li>\n",
    "            <a href=\"numpy\">What is Numpy?</a>\n",
    "            <ul>\n",
    "                <li><a href=\"type\">Type</a></li>\n",
    "                <li><a href=\"val\">Assign Value</a></li>\n",
    "                <li><a href=\"slice\">Slicing</a></li>\n",
    "                <li><a href=\"list\">Assign Value with List</a></li>\n",
    "                <li><a href=\"other\">Other Attributes</a></li>\n",
    "            </ul>\n",
    "        </li>\n",
    "        <li>\n",
    "            <a href=\"op\">Numpy Array Operations</a>\n",
    "            <ul>\n",
    "                <li><a href=\"add\">Array Addition</a></li>\n",
    "                <li><a href=\"multi\">Array Multiplication</a></li>\n",
    "                <li><a href=\"prod\">Product of Two Numpy Arrays</a></li>\n",
    "                <li><a href=\"dot\">Dot Product</a></li>\n",
    "                <li><a href=\"cons\">Adding Constant to a Numpy Array</a></li>\n",
    "            </ul>\n",
    "        </li>\n",
    "        <li><a href=\"math\">Mathematical Functions</a></li>\n",
    "        <li><a href=\"lin\">Linspace</a></li>\n",
    "    </ul>\n",
    "    <p>\n",
    "        Estimated time needed: <strong>30 min</strong>\n",
    "    </p>\n",
    "</div>\n",
    "\n",
    "<hr>"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<h2 id=\"pre\">Preparation</h2>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "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": {
    "jupyter": {
     "outputs_hidden": false
    }
   },
   "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:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {
    "jupyter": {
     "outputs_hidden": false
    }
   },
   "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:"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<img src=\"https://s3-api.us-geo.objectstorage.softlayer.net/cf-courses-data/CognitiveClass/PY0101EN/Chapter%205/Images/NumOneList.png\" width=\"660\" />"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We can access each element using a square bracket as follows: "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {
    "jupyter": {
     "outputs_hidden": false
    }
   },
   "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>"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<h2 id=\"numpy\">What is Numpy?</h2>"
   ]
  },
  {
   "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: "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {
    "jupyter": {
     "outputs_hidden": false
    }
   },
   "outputs": [],
   "source": [
    "# import numpy library\n",
    "\n",
    "import numpy as np "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    " We then cast the list as follows:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {
    "jupyter": {
     "outputs_hidden": false
    }
   },
   "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: "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<img src=\"https://s3-api.us-geo.objectstorage.softlayer.net/cf-courses-data/CognitiveClass/PY0101EN/Chapter%205/Images/NumOneNp.png\" width=\"500\" />"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    " As with lists, we can access each element via a square bracket:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {
    "jupyter": {
     "outputs_hidden": false
    }
   },
   "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>"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "If we check the type of the array we get <b>numpy.ndarray</b>:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {
    "jupyter": {
     "outputs_hidden": false
    }
   },
   "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 a 64-bit integer: \n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {
    "jupyter": {
     "outputs_hidden": false
    }
   },
   "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:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {
    "jupyter": {
     "outputs_hidden": false
    }
   },
   "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>:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {
    "jupyter": {
     "outputs_hidden": false
    }
   },
   "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: "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {
    "jupyter": {
     "outputs_hidden": false
    }
   },
   "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>"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We can change the value of the array, consider the array <code>c</code>:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {
    "jupyter": {
     "outputs_hidden": false
    }
   },
   "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:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {
    "jupyter": {
     "outputs_hidden": false
    }
   },
   "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:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {
    "jupyter": {
     "outputs_hidden": false
    }
   },
   "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>"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Like lists, we can slice the numpy array, and we can select the elements from 1 to 3 and assign it to a new numpy array <code>d</code> as follows:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {
    "jupyter": {
     "outputs_hidden": false
    }
   },
   "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: "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {
    "jupyter": {
     "outputs_hidden": false
    }
   },
   "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>"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Similarly, we can use a list to select a specific index.\n",
    "The list ' select ' contains several values:\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {},
   "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 index:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {
    "jupyter": {
     "outputs_hidden": false
    }
   },
   "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:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {
    "jupyter": {
     "outputs_hidden": false
    }
   },
   "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>"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Let's review some basic array attributes using the array <code>a</code>:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "metadata": {
    "jupyter": {
     "outputs_hidden": false
    }
   },
   "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:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "metadata": {
    "jupyter": {
     "outputs_hidden": false
    }
   },
   "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:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "metadata": {
    "jupyter": {
     "outputs_hidden": false
    }
   },
   "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:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "metadata": {
    "jupyter": {
     "outputs_hidden": false
    }
   },
   "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": {},
   "outputs": [],
   "source": [
    "# Create a numpy array\n",
    "\n",
    "a = np.array([1, -1, 1, -1])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 26,
   "metadata": {
    "jupyter": {
     "outputs_hidden": false
    }
   },
   "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": {
    "jupyter": {
     "outputs_hidden": false
    }
   },
   "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": {
    "jupyter": {
     "outputs_hidden": false
    }
   },
   "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": null,
   "metadata": {
    "jupyter": {
     "outputs_hidden": false
    }
   },
   "outputs": [],
   "source": [
    "# Get the biggest value in the numpy array\n",
    "\n",
    "max_b = b.max()\n",
    "max_b"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 29,
   "metadata": {
    "jupyter": {
     "outputs_hidden": false
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "-1"
      ]
     },
     "execution_count": 29,
     "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>"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<h2 id=\"op\">Numpy Array Operations</h2>"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<h3 id=\"add\">Array Addition</h3>"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Consider the numpy array <code>u</code>:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 30,
   "metadata": {
    "jupyter": {
     "outputs_hidden": false
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([1, 0])"
      ]
     },
     "execution_count": 30,
     "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>:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 31,
   "metadata": {
    "jupyter": {
     "outputs_hidden": false
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([0, 1])"
      ]
     },
     "execution_count": 31,
     "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:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 32,
   "metadata": {
    "jupyter": {
     "outputs_hidden": false
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([1, 1])"
      ]
     },
     "execution_count": 32,
     "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:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 33,
   "metadata": {
    "jupyter": {
     "outputs_hidden": false
    }
   },
   "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>"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Consider the vector numpy array <code>y</code>:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 34,
   "metadata": {
    "jupyter": {
     "outputs_hidden": false
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([1, 2])"
      ]
     },
     "execution_count": 34,
     "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:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 35,
   "metadata": {
    "jupyter": {
     "outputs_hidden": false
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([2, 4])"
      ]
     },
     "execution_count": 35,
     "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: "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<h3 id=\"prod\">Product of Two Numpy Arrays</h3>"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Consider the following array <code>u</code>:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 36,
   "metadata": {
    "jupyter": {
     "outputs_hidden": false
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([1, 2])"
      ]
     },
     "execution_count": 36,
     "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>:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 37,
   "metadata": {
    "jupyter": {
     "outputs_hidden": false
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([3, 2])"
      ]
     },
     "execution_count": 37,
     "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:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 38,
   "metadata": {
    "jupyter": {
     "outputs_hidden": false
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([3, 4])"
      ]
     },
     "execution_count": 38,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Calculate the production of two numpy arrays\n",
    "\n",
    "z = u * v\n",
    "z"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<h3 id=\"dot\">Dot Product</h3>"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The dot product of the two numpy arrays <code>u</code> and <code>v</code> is given by:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 39,
   "metadata": {
    "jupyter": {
     "outputs_hidden": false
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "7"
      ]
     },
     "execution_count": 39,
     "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>"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Consider the following array: "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 40,
   "metadata": {
    "jupyter": {
     "outputs_hidden": false
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([ 1,  2,  3, -1])"
      ]
     },
     "execution_count": 40,
     "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:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 41,
   "metadata": {
    "jupyter": {
     "outputs_hidden": false
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([2, 3, 4, 0])"
      ]
     },
     "execution_count": 41,
     "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:"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<img src=\"https://s3-api.us-geo.objectstorage.softlayer.net/cf-courses-data/CognitiveClass/PY0101EN/Chapter%205/Images/NumOneAdd.gif\" width=\"500\" />"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<hr>"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<h2 id=\"math\">Mathematical Functions</h2>"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    " We can access the value of pie in numpy as follows :"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 42,
   "metadata": {
    "jupyter": {
     "outputs_hidden": false
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "3.141592653589793"
      ]
     },
     "execution_count": 42,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# The value of pie\n",
    "\n",
    "np.pi"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    " We can create the following numpy array in Radians:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 43,
   "metadata": {
    "jupyter": {
     "outputs_hidden": false
    }
   },
   "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:  "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 46,
   "metadata": {
    "jupyter": {
     "outputs_hidden": false
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([0.0000000e+00, 1.0000000e+00, 1.2246468e-16])"
      ]
     },
     "execution_count": 46,
     "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>"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<h2 id=\"lin\">Linspace</h2>"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    " A useful function for plotting mathematical functions is \"linespace\".   Linespace 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:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 47,
   "metadata": {
    "jupyter": {
     "outputs_hidden": false
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([-2., -1.,  0.,  1.,  2.])"
      ]
     },
     "execution_count": 47,
     "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: "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 48,
   "metadata": {
    "jupyter": {
     "outputs_hidden": false
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([-2. , -1.5, -1. , -0.5,  0. ,  0.5,  1. ,  1.5,  2. ])"
      ]
     },
     "execution_count": 48,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Makeup 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 line space to generate 100 evenly spaced samples from the interval 0 to 2π: "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 51,
   "metadata": {
    "jupyter": {
     "outputs_hidden": false
    }
   },
   "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": 51,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Makeup a numpy array within [0, 2π] and 100 elements \n",
    "\n",
    "x = np.linspace(0, 2*np.pi, num=100)\n",
    "x"
   ]
  },
  {
   "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>: "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 52,
   "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": 52,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Calculate the sine of x list\n",
    "\n",
    "y = np.sin(x)\n",
    "y"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 53,
   "metadata": {
    "jupyter": {
     "outputs_hidden": false
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[<matplotlib.lines.Line2D at 0x7f2c411205f8>]"
      ]
     },
     "execution_count": 53,
     "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>"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<h2 id=\"quiz\">Quiz on 1D Numpy Array</h2>"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Implement the following vector subtraction in numpy: u-v"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 54,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([ 1, -1])"
      ]
     },
     "execution_count": 54,
     "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",
    "x=u-v\n",
    "x"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Double-click __here__ for the solution.\n",
    "\n",
    "<!-- Your answer is below:\n",
    "u - v\n",
    "-->"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<hr>"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Multiply the numpy array z with -2:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 55,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([-4, -8])"
      ]
     },
     "execution_count": 55,
     "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"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Double-click __here__ for the solution.\n",
    "\n",
    "<!-- Your answer is below:\n",
    "-2 * z\n",
    "-->"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<hr>"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Consider the list <code>[1, 2, 3, 4, 5]</code> and <code>[1, 0, 1, 0, 1]</code>, and cast both lists to a numpy array then multiply them together:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 58,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([1, 0, 3, 0, 5])"
      ]
     },
     "execution_count": 58,
     "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",
    "l1*l2"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Double-click __here__ for the solution.\n",
    "\n",
    "<!-- Your answer is below:\n",
    "a = np.array([1, 2, 3, 4, 5])\n",
    "b = np.array([1, 0, 1, 0, 1])\n",
    "a * b\n",
    "-->"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<hr>"
   ]
  },
  {
   "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 the dot product:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 59,
   "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",
    "a=np.array([-1,1])\n",
    "b=np.array([1,-1])\n",
    "Plotvec2(a,b)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Double-click __here__ for the solution.\n",
    "\n",
    "<!-- Your answer is below:\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",
    "-->"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<hr>"
   ]
  },
  {
   "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 the dot product:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 60,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "-2"
      ]
     },
     "execution_count": 60,
     "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": [
    "# Write your code below and press Shift+Enter to execute\n",
    "a=np.array([1,0])\n",
    "b=np.array([1,-1])\n",
    "Plotvec2(a,b)\n",
    "mul=np.dot(a,b)\n",
    "mul"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Double-click __here__ for the solution.\n",
    "\n",
    "<!-- \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",
    " -->"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<hr>"
   ]
  },
  {
   "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 the dot product:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 62,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "1"
      ]
     },
     "execution_count": 62,
     "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": [
    "# Write your code below and press Shift+Enter to execute\n",
    "a=np.array([1,1])\n",
    "b=np.array([0,1])\n",
    "Plotvec2(a,b)\n",
    "mil=np.dot(a,b)\n",
    "mil"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Double-click __here__ for the solution.\n",
    "\n",
    "<!-- \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",
    "print(\"The dot product is\", np.dot(a, b))\n",
    " -->"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<hr>"
   ]
  },
  {
   "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>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "# Write your code below and press Shift+Enter to execute\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Double-click __here__ for the solution.\n",
    "\n",
    "<!-- \n",
    "The vectors used for question 4 and 5 are perpendicular. As a result, the dot product is zero. \n",
    "-->"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<h3>About the Authors:</h3>  \n",
    "<p><a href=\"https://www.linkedin.com/in/joseph-s-50398b136/\" target=\"_blank\">Joseph Santarcangelo</a> is a Data Scientist at IBM, and holds a PhD in Electrical Engineering. His research focused on using Machine Learning, Signal Processing, and Computer Vision to determine how videos impact human cognition. Joseph has been working for IBM since he completed his PhD.</p>"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Other contributors: <a href=\"www.linkedin.com/in/jiahui-mavis-zhou-a4537814a\">Mavis Zhou</a>"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<hr>"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<p>Copyright &copy; 2018 IBM Developer Skills Network. This notebook and its source code are released under the terms of the <a href=\"https://cognitiveclass.ai/mit-license/\">MIT License</a>.</p>"
   ]
  }
 ],
 "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",
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