explanations/linalg.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Math fundamentals\n",
    "\n",
    "To fully understand deep learning, you need to know some linear algebra and calculus.  We'll cover the basics in this lesson, including how to use NumPy to apply the math using Python.  This will give you a foundation for the rest of this course.\n",
    "\n",
    "Math terminology can be intimidating, but don't worry - everything will be explained using code, diagrams, and plain English."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Vectors\n",
    "\n",
    "At a very basic level, linear algebra is about manipulating vectors.  This manipulation happens by combining and scaling vectors.  A vector is a one-dimensional array of numbers.  Here's an example of a vector:\n",
    "\n",
    "$$v = [1,1]$$\n",
    "\n",
    "We can create this vector using NumPy, a Python library for working with arrays.  We'll use NumPy to create and manipulate vectors and matrices."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[1 1]\n"
     ]
    }
   ],
   "source": [
    "import numpy as np\n",
    "\n",
    "# Create a vector as a numpy array\n",
    "v = np.array([1,1])\n",
    "print(v)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "As you can see, our vector has two elements, `1` and `1`.  We call this one-dimensional since it only has elements in one direction.  Here's an example of a two-dimensional array, also called a matrix.  A matrix is a table of data with rows and columns. We can think of a vector as a single row or a single column from a matrix."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[[1 1]\n",
      " [1 0]]\n"
     ]
    }
   ],
   "source": [
    "# Create a matrix as a numpy array\n",
    "m = np.array([[1,1],[1,0]])\n",
    "print(m)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We'll discuss matrices more later, but for now, let's keep talking about vectors.  We can graph our vector to understand it better.  By convention, we always draw vectors starting at the origin (0,0).  Here's a diagram of our vector `v`:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<matplotlib.patches.FancyArrow at 0x1066cfa60>"
      ]
     },
     "execution_count": 3,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 640x480 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "import matplotlib.pyplot as plt\n",
    "\n",
    "plt.xlim(0, 3)  # set x-axis limits\n",
    "plt.ylim(0, 3)  # set y-axis limits\n",
    "\n",
    "# Plot the arrow\n",
    "plt.arrow(0, 0, v[0], v[1], head_width=0.1, head_length=0.2)  # draw arrow"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "As you can see, we draw a line from the coordinate `0,0` to the coordinate `1,1`.  The vector specifies the end of the line.  The first element in the vector is the x-coordinate of the end of the line, and the second element is the y-coordinate.  We draw the vector with an arrow to indicate the direction of the vector.  In this case, the vector is pointing up and to the right.\n",
    "\n",
    "The length of the vector, also called the norm, is the distance from the origin to the end of the line.  We can calculate the norm of a vector using this formula:\n",
    "\n",
    "$$||x||_{2} = \\sqrt{x_{1}^{2} + x_{2}^{2} + \\cdots + x_{n}^{2}}$$\n",
    "\n",
    "This is called the L2 norm, and is the most common norm in deep learning.  We're essentially taking the distance from the origin to the end of the vector.  We can calculate the L2 norm of our vector using NumPy:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "1.4142135623730951"
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "np.sqrt(np.sum(v**2))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We would draw a vector with 3 elements in 3-dimensional space.  Here's an example of a 3-dimensional vector:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[0 1 1]\n"
     ]
    }
   ],
   "source": [
    "# Create a vector as a numpy array\n",
    "v = np.array([0,1,1])\n",
    "print(v)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<mpl_toolkits.mplot3d.art3d.Line3DCollection at 0x11e435ba0>"
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 640x480 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "fig = plt.figure()\n",
    "ax = fig.add_subplot(111, projection='3d')\n",
    "\n",
    "ax.set_xlim([0, 3])\n",
    "ax.set_ylim([0, 3])\n",
    "ax.set_zlim([0, 3])\n",
    "\n",
    "# Plot vector\n",
    "ax.quiver(0,0,0, v[0], v[1], v[2], length=1)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The number of elements in a vector determines the number of dimensions of the *vector space* that the vector exists in.  A vector with 1 element exists in a 1-dimensional space, a vector with 2 elements exists in a 2-dimensional space, and a vector with 3 elements exists in a 3-dimensional space.  In deep learning, you'll often work with vectors that have thousands of elements - more than we can visualize.\n",
    "\n",
    "The dimensionality of a vector space is different from the dimensionality of an array.  A vector is a one-dimensional array, but it can exist in a multi-dimensional vector space.\n",
    "\n",
    "You may have noticed that we indexed `v` using the numbers `0`, `1`, and `2` when we drew our plot.  In NumPy, we index a vector with a single number (since it is a one-dimensional array).  The index corresponds to the position of the element we want."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Vector Operations\n",
    "\n",
    "We can scale vectors by scalars (single numbers).  For example, we can multiply our 2-dimensional vector `1,1` by `.5` to get `.5, .5`:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[0.5 0.5]\n"
     ]
    }
   ],
   "source": [
    "v = np.array([1,1]) * .5\n",
    "print(v)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<matplotlib.patches.FancyArrow at 0x11e4a63e0>"
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 640x480 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.xlim(0, 3)  # set x-axis limits\n",
    "plt.ylim(0, 3)  # set y-axis limits\n",
    "\n",
    "# Plot the arrow\n",
    "plt.arrow(0, 0, v[0], v[1], head_width=0.1, head_length=0.2)  # draw arrow"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "You can see that the vector now ends at `.5, .5`.  We can also add vectors together.  For example, we can add `1,1` and `1,0` to get `2,1`:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[2 1]\n"
     ]
    }
   ],
   "source": [
    "v = np.array([1,1]) + np.array([1,0])\n",
    "print(v)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<matplotlib.patches.FancyArrow at 0x11e4a4310>"
      ]
     },
     "execution_count": 10,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 640x480 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.xlim(0, 3)  # set x-axis limits\n",
    "plt.ylim(0, 3)  # set y-axis limits\n",
    "\n",
    "# Plot the arrow\n",
    "plt.arrow(0, 0, v[0], v[1], head_width=0.1, head_length=0.2)  # draw arrow"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "This vector now ends at `2,1`.\n",
    "\n",
    "We can use the basis vectors `0,1` and `1,0` along with our operations (multiply by a scalar and add vectors) to create any vector in the 2-d Euclidean space that we normally use in graphs.\n",
    "\n",
    "Here's a plot of the basis vectors:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<matplotlib.quiver.Quiver at 0x11e5a3640>"
      ]
     },
     "execution_count": 11,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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4cXGxSktLJf3t/pqsrKxQ/8cee0yffPKJfvazn+kvf/mLfvOb3+gPf/iD/vVf/9XJMtFKHT9+XJ988omkvwUdY0wzVwQAaGkcDToHDhzQ0KFDNXToUElSdna2hg4dqkWLFkmSysrKQqFHkvr06aM///nPys/P15AhQ7R06VK98cYbtX61HDe2b5/FOXHihP7nf/6nGasBALREjj5HJy0trc5/Zdf01OO0tDQdOnTIwapgi40bN4atv/vuu0pKSmqmagAALVGLukcHqK8vv/xSe/bsCWvjPh0AwHcRdNAqbdq0SdXV1WFtu3fv1rlz55qpIgBAS0TQQatU09mbqqoqbdmypRmqAQC0VAQdtDqXL1+uNdBw+QoA8G0EHbQ6O3fu1IULF2p8bfPmzTwlGQAQQtBBq/Pdb1t9G09JBgB8G0EHrcq3n4Zcm7qCEADgxkLQQavy7ach14b7dAAAVxF00KrUJ8R8/PHHKikpaYJqAAAtHUEHrcrVoDNhwgT9wz/8Q9hrTzzxhG6//fawfgCAGxtBB63Gl19+qc6dO2vv3r3atGmTBgwYEPb6Aw88oI8++ki//e1vOaMDAJDk8G9dAZHUuXPn656padu2rR566CFNmzatiaoCALRknNFBqxEVVf8/rg3pCwCwF58GAADAWgQdAABgLYIOAACwFkEHAABYi6ADAACsRdABAADWIugAAABrEXQAAIC1CDoAAMBaBB0AAGAtgg4AALAWQQcAAFiLoAMAAKxF0AEAANYi6AAAAGsRdAAAgLUIOgAAwFoEHQAAYC2CDgAAsBZBBwAAWIugAwAArEXQAQAA1mqSoLNs2TLdeuutiomJ0ahRo7R///5a++bl5cnlcoUtMTExTVEmAACwjONBZ82aNcrOztbixYt18OBBDRkyRBkZGTp79mytY2JjY1VWVhZaTp8+7XSZAADAQo4HnV/+8peaNWuWZsyYof79+2v58uXq0KGD3nzzzVrHuFwueb3e0BIXF+d0mQAAwEKOBp3Lly+rqKhI6enp//+GUVFKT09XYWFhreMuXryo3r17KzExUZMnT9axY8dq7VtZWalgMBi2AAAASA4HnS+++EJVVVXXnJGJi4uT3++vcUxSUpLefPNNbdiwQb///e9VXV2t1NRUffrppzX2z83NlcfjCS2JiYkRnwcAAGidWty3rnw+n7KyspSSkqKxY8fqnXfeUbdu3fT666/X2D8nJ0eBQCC0nDlzpokrBgAALVVbJzfetWtXtWnTRuXl5WHt5eXl8nq99dpGu3btNHToUJ04caLG191ut9xu9/euFQAA2MfRMzrR0dEaNmyYCgoKQm3V1dUqKCiQz+er1zaqqqp05MgRxcfHO1UmAACwlKNndCQpOztb06dP1/DhwzVy5Ei98sorqqio0IwZMyRJWVlZ6tGjh3JzcyVJTz31lEaPHq1+/frp/PnzevHFF3X69Gk98sgjTpcKAAAs43jQmTJlij7//HMtWrRIfr9fKSkp2rJlS+gG5dLSUkVF/f+Jpa+++kqzZs2S3+9X586dNWzYMO3Zs0f9+/d3ulQAAGAZlzHGNHcRkRQMBuXxeBQIBBQbG9vc5cBBCxcu1DPPPBNa379/v0aMGNGMFQEAGsupz+8W960rAACASCHoAAAAaxF0AACAtQg6AADAWgQdAABgLYIOAACwFkEHAABYi6ADAACsRdABAADWIugAAABrEXQAAIC1CDoAAMBaBB0AAGAtgg4AALAWQQcAAFiLoAMAAKxF0AEAANYi6AAAAGsRdAAAgLUIOgAAwFoEHQAAYC2CDgAAsBZBBwAAWIugAwAArEXQAQAA1iLoAAAAaxF0AACAtQg6AADAWgQdAABgLYIOAACwFkEHAABYi6ADAACsRdABAADWIugAcFwgEFBubq4OHz4sY0xzlwPgBkLQAeA4j8ej//7v/9aQIUN06623as6cOdq6dasqKyubuzQAlmuSoLNs2TLdeuutiomJ0ahRo7R///46+69du1bJycmKiYnRoEGDtGnTpqYoE4CDFi5cKJfLpdLSUi1btkz33XefbrnlFv3jP/6jVq5cqbNnzzZ3iQAs5HjQWbNmjbKzs7V48WIdPHhQQ4YMUUZGRq1/qe3Zs0dTp07VzJkzdejQIWVmZiozM1NHjx51ulQADhowYIB+9KMfhbVVVFRo3bp1+slPfiKv16vRo0fr2Wef5RIXgIhxGYf/Nhk1apRGjBihf//3f5ckVVdXKzExUf/yL/+iBQsWXNN/ypQpqqio0MaNG0Nto0ePVkpKipYvX37d9wsGg/J4PAoEAoqNjY3cRNDiLFy4UM8880xovU2bNnK5XM1YEa6nqqqq3gGmV69emjRpkiZNmqS0tDS53W6HqwPQnJz6/G4bsS3V4PLlyyoqKlJOTk6oLSoqSunp6SosLKxxTGFhobKzs8PaMjIytH79+hr7V1ZWhl3nDwaD379wtAoej0e9evXSF198oa+//lpVVVXNXRIi6OolrmXLlummm27S+PHjNWnSJE2ZMkUdOnRo7vIAtBKOXrr64osvVFVVpbi4uLD2uLg4+f3+Gsf4/f4G9c/NzZXH4wktiYmJkSkeLd5Pf/pTnT59Wn//93/f3KXAQe3atZPP51NaWprGjh1LyAHQII6e0WkKOTk5YWeAgsEgYecGM2DAAN1zzz3NXQbq4fTp0zp58uR1+91yyy26//77NWnSJGVkZHAZGkCjORp0unbtqjZt2qi8vDysvby8XF6vt8YxXq+3Qf3dbjfX7m9wTz75pJ588snmLgPXYYxRampqrUGnf//+oXtyRo8erTZt2jRxhQBs5Oilq+joaA0bNkwFBQWhturqahUUFMjn89U4xufzhfWXpPz8/Fr7A2gd3nvvPe3duze03q5dO6Wnp+vVV1/VyZMndezYMT333HO66667CDkAIsbxS1fZ2dmaPn26hg8frpEjR+qVV15RRUWFZsyYIUnKyspSjx49lJubK0maO3euxo4dq6VLl2rixIlavXq1Dhw4oBUrVjhdKgCHGGP0b//2b7rllls0ceJE/d3f/R2XpAA0CceDzpQpU/T5559r0aJF8vv9SklJ0ZYtW0I3HJeWlioq6v9PLKWmpmrVqlV68skn9Ytf/EK333671q9fr4EDBzpdKgCHXLhwQS+99BKXpAA0Ocefo9PUeI4OAACtj1Of3/zWFQAAsBZBBwAAWIugAwAArEXQAQAA1iLoAAAAaxF0AACAtQg6AADAWgQdAABgLYIOAACwFkEHAABYi6ADAACsRdABAADWIugAAABrEXQAAIC1CDoAAMBaBB0AAGAtgg4AALAWQQcAAFiLoAMAAKxF0AEAANYi6AAAAGsRdAAAgLUIOgAAwFoEHQAAYC2CDgAAsBZBBwAAWIugAwAArEXQAQAA1iLoAAAAaxF0AACAtQg6AADAWgQdAABgLYIOAACwFkEHAABYi6ADAACs5WjQOXfunKZNm6bY2Fh16tRJM2fO1MWLF+sck5aWJpfLFbY89thjTpYJAAAs1dbJjU+bNk1lZWXKz8/XlStXNGPGDD366KNatWpVneNmzZqlp556KrTeoUMHJ8sEAACWcizoHD9+XFu2bNGHH36o4cOHS5J+/etf6/7779dLL72khISEWsd26NBBXq/XqdIAAMANwrFLV4WFherUqVMo5EhSenq6oqKitG/fvjrHvvXWW+ratasGDhyonJwcff3117X2raysVDAYDFsAAAAkB8/o+P1+de/ePfzN2rZVly5d5Pf7ax334x//WL1791ZCQoIOHz6sn//85yopKdE777xTY//c3FwtWbIkorUDAAA7NDjoLFiwQM8//3ydfY4fP97ogh599NHQ/x80aJDi4+M1btw4nTx5Urfddts1/XNycpSdnR1aDwaDSkxMbPT7AwAAezQ46MyfP18PP/xwnX369u0rr9ers2fPhrX/9a9/1blz5xp0/82oUaMkSSdOnKgx6Ljdbrnd7npvDwAA3DgaHHS6deumbt26Xbefz+fT+fPnVVRUpGHDhkmStm3bpurq6lB4qY/i4mJJUnx8fENLBQAANzjHbka+4447dN9992nWrFnav3+/du/erTlz5ujBBx8MfePqs88+U3Jysvbv3y9JOnnypJ5++mkVFRXpf//3f/WnP/1JWVlZGjNmjAYPHuxUqQAAwFKOPjDwrbfeUnJyssaNG6f7779fP/zhD7VixYrQ61euXFFJSUnoW1XR0dF6//33NX78eCUnJ2v+/Pn6p3/6J7377rtOlgkAACzlMsaY5i4ikoLBoDwejwKBgGJjY5u7HAAAUA9OfX7zW1cAAMBaBB0AAGAtgg4AALAWQQcAAFiLoAMAAKxF0AEAANYi6AAAAGsRdAAAgLUIOgAAwFoEHQAAYC2CDgAAsBZBBwAAWIugAwAArEXQAQAA1iLoAAAAaxF0AACAtQg6AADAWgQdAABgLYIOAACwFkEHAABYi6ADAACsRdABAADWIugAAABrEXQAAIC1CDoAAMBaBB0AAGAtgg4AALAWQQcAAFiLoAMAAKxF0AEAANYi6AAAAGsRdAAAgLUIOgAAwFoEHQAAYC2CDgAAsBZBBwAAWMuxoPPss88qNTVVHTp0UKdOneo1xhijRYsWKT4+Xu3bt1d6ero+/vhjp0oEAACWcyzoXL58WQ888IAef/zxeo954YUX9Ktf/UrLly/Xvn37dNNNNykjI0OXLl1yqkwAAGAxlzHGOPkGeXl5mjdvns6fP19nP2OMEhISNH/+fP30pz+VJAUCAcXFxSkvL08PPvhgjeMqKytVWVkZWg8EAurVq5fOnDmj2NjYiM0DAAA4JxgMKjExUefPn5fH44nYdttGbEvf06lTp+T3+5Wenh5q83g8GjVqlAoLC2sNOrm5uVqyZMk17YmJiY7VCgAAnPHll1/aGXT8fr8kKS4uLqw9Li4u9FpNcnJylJ2dHVo/f/68evfurdLS0oj+h2rpribhG+1MFvNm3jcC5s28bwRXr8h06dIlotttUNBZsGCBnn/++Tr7HD9+XMnJyd+rqIZwu91yu93XtHs8nhvqD8hVsbGxzPsGwrxvLMz7xnKjzjsqKrK3Dzco6MyfP18PP/xwnX369u3bqEK8Xq8kqby8XPHx8aH28vJypaSkNGqbAADgxtagoNOtWzd169bNkUL69Okjr9ergoKCULAJBoPat29fg765BQAAcJVjXy8vLS1VcXGxSktLVVVVpeLiYhUXF+vixYuhPsnJyVq3bp0kyeVyad68eXrmmWf0pz/9SUeOHFFWVpYSEhKUmZlZ7/d1u91avHhxjZezbMa8mfeNgHkz7xsB847svB37evnDDz+s//iP/7im/YMPPlBaWtrf3tzl0sqVK0OXw4wxWrx4sVasWKHz58/rhz/8oX7zm9/oBz/4gRMlAgAAyzn+HB0AAIDmwm9dAQAAaxF0AACAtQg6AADAWgQdAABgLSuCzrPPPqvU1FR16NBBnTp1qtcYY4wWLVqk+Ph4tW/fXunp6fr444+dLTTCzp07p2nTpik2NladOnXSzJkzw76+X5O0tDS5XK6w5bHHHmuiihtn2bJluvXWWxUTE6NRo0Zp//79dfZfu3atkpOTFRMTo0GDBmnTpk1NVGlkNWTeeXl51+zXmJiYJqw2Mnbu3KlJkyYpISFBLpdL69evv+6Y7du3684775Tb7Va/fv2Ul5fneJ2R1tB5b9++/Zr97XK56vy5nJYmNzdXI0aMUMeOHdW9e3dlZmaqpKTkuuNa+/HdmHnbcHy/9tprGjx4cOhpzz6fT5s3b65zTKT2tRVB5/Lly3rggQca9GDBF154Qb/61a+0fPly7du3TzfddJMyMjJ06dIlByuNrGnTpunYsWPKz8/Xxo0btXPnTj366KPXHTdr1iyVlZWFlhdeeKEJqm2cNWvWKDs7W4sXL9bBgwc1ZMgQZWRk6OzZszX237Nnj6ZOnaqZM2fq0KFDyszMVGZmpo4ePdrElX8/DZ239LfHxX97v54+fboJK46MiooKDRkyRMuWLatX/1OnTmnixIm6++67VVxcrHnz5umRRx7R1q1bHa40sho676tKSkrC9nn37t0dqjDyduzYodmzZ2vv3r3Kz8/XlStXNH78eFVUVNQ6xobjuzHzllr/8d2zZ08999xzKioq0oEDB3TPPfdo8uTJOnbsWI39I7qvjUVWrlxpPB7PdftVV1cbr9drXnzxxVDb+fPnjdvtNv/5n//pYIWR89FHHxlJ5sMPPwy1bd682bhcLvPZZ5/VOm7s2LFm7ty5TVBhZIwcOdLMnj07tF5VVWUSEhJMbm5ujf1/9KMfmYkTJ4a1jRo1yvzzP/+zo3VGWkPnXd8/+62JJLNu3bo6+/zsZz8zAwYMCGubMmWKycjIcLAyZ9Vn3h988IGRZL766qsmqakpnD171kgyO3bsqLWPLcf3t9Vn3jYe38YY07lzZ/PGG2/U+Fok97UVZ3Qa6tSpU/L7/UpPTw+1eTwejRo1SoWFhc1YWf0VFhaqU6dOGj58eKgtPT1dUVFR2rdvX51j33rrLXXt2lUDBw5UTk6Ovv76a6fLbZTLly+rqKgobD9FRUUpPT291v1UWFgY1l+SMjIyWs1+lRo3b0m6ePGievfurcTExDr/pWQTG/b395GSkqL4+Hjde++92r17d3OX870EAgFJqvOXq23c3/WZt2TX8V1VVaXVq1eroqJCPp+vxj6R3NcN+q0rW1y9jh0XFxfWHhcX12qucfv9/mtOU7dt21ZdunSpcw4//vGP1bt3byUkJOjw4cP6+c9/rpKSEr3zzjtOl9xgX3zxhaqqqmrcT3/5y19qHOP3+1v1fpUaN++kpCS9+eabGjx4sAKBgF566SWlpqbq2LFj6tmzZ1OU3Sxq29/BYFDffPON2rdv30yVOSs+Pl7Lly/X8OHDVVlZqTfeeENpaWnat2+f7rzzzuYur8Gqq6s1b9483XXXXRo4cGCt/Ww4vr+tvvO25fg+cuSIfD6fLl26pJtvvlnr1q1T//79a+wbyX3dYoPOggUL9Pzzz9fZ5/jx40pOTm6iippGfefdWN++h2fQoEGKj4/XuHHjdPLkSd12222N3i6al8/nC/uXUWpqqu644w69/vrrevrpp5uxMjghKSlJSUlJofXU1FSdPHlSL7/8sn73u981Y2WNM3v2bB09elS7du1q7lKaVH3nbcvxnZSUpOLiYgUCAb399tuaPn26duzYUWvYiZQWG3Tmz58f+g2s2vTt27dR2/Z6vZKk8vJyxcfHh9rLy8tDv5zeXOo7b6/Xe82NqX/961917ty50PzqY9SoUZKkEydOtLig07VrV7Vp00bl5eVh7eXl5bXO0ev1Nqh/S9SYeX9Xu3btNHToUJ04ccKJEluM2vZ3bGystWdzajNy5MhWGRTmzJkT+jLF9c5O2HB8X9WQeX9Xaz2+o6Oj1a9fP0nSsGHD9OGHH+rVV1/V66+/fk3fSO7rFnuPTrdu3ZScnFznEh0d3aht9+nTR16vVwUFBaG2YDCoffv21Xq9sKnUd94+n0/nz59XUVFRaOy2bdtUXV0dCi/1UVxcLElhga+liI6O1rBhw8L2U3V1tQoKCmrdTz6fL6y/JOXn5zf7fm2Ixsz7u6qqqnTkyJEWuV8jyYb9HSnFxcWtan8bYzRnzhytW7dO27ZtU58+fa47xob93Zh5f5ctx3d1dbUqKytrfC2i+7oRN0q3OKdPnzaHDh0yS5YsMTfffLM5dOiQOXTokLlw4UKoT1JSknnnnXdC688995zp1KmT2bBhgzl8+LCZPHmy6dOnj/nmm2+aYwqNct9995mhQ4eaffv2mV27dpnbb7/dTJ06NfT6p59+apKSksy+ffuMMcacOHHCPPXUU+bAgQPm1KlTZsOGDaZv375mzJgxzTWF61q9erVxu90mLy/PfPTRR+bRRx81nTp1Mn6/3xhjzEMPPWQWLFgQ6r97927Ttm1b89JLL5njx4+bxYsXm3bt2pkjR4401xQapaHzXrJkidm6das5efKkKSoqMg8++KCJiYkxx44da64pNMqFCxdCx68k88tf/tIcOnTInD592hhjzIIFC8xDDz0U6v/JJ5+YDh06mCeeeMIcP37cLFu2zLRp08Zs2bKluabQKA2d98svv2zWr19vPv74Y3PkyBEzd+5cExUVZd5///3mmkKDPf7448bj8Zjt27ebsrKy0PL111+H+th4fDdm3jYc3wsWLDA7duwwp06dMocPHzYLFiwwLpfLvPfee8YYZ/e1FUFn+vTpRtI1ywcffBDqI8msXLkytF5dXW0WLlxo4uLijNvtNuPGjTMlJSVNX/z38OWXX5qpU6eam2++2cTGxpoZM2aEhbtTp06F/XcoLS01Y8aMMV26dDFut9v069fPPPHEEyYQCDTTDOrn17/+tenVq5eJjo42I0eONHv37g29NnbsWDN9+vSw/n/4wx/MD37wAxMdHW0GDBhg/vznPzdxxZHRkHnPmzcv1DcuLs7cf//95uDBg81Q9fdz9WvT312uznX69Olm7Nix14xJSUkx0dHRpm/fvmHHeWvR0Hk///zz5rbbbjMxMTGmS5cuJi0tzWzbtq15im+kmub73b+nbTy+GzNvG47vn/zkJ6Z3794mOjradOvWzYwbNy4Ucoxxdl+7jDGm4eeBAAAAWr4We48OAADA90XQAQAA1iLoAAAAaxF0AACAtQg6AADAWgQdAABgLYIOAACwFkEHAABYi6ADAACsRdABAADWIugAAABr/R9uyoG16rxvHQAAAABJRU5ErkJggg==\n",
      "text/plain": [
       "<Figure size 640x480 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "v1 = np.array([1,0])\n",
    "v2 = np.array([0,1])\n",
    "\n",
    "plt.xlim(-1, 3)  # set x-axis limits\n",
    "plt.ylim(-1, 3)  # set y-axis limits\n",
    "\n",
    "# Plot the arrows with plt.quiver\n",
    "plt.quiver([0,0], [0,0], [v1[0],v2[0]], [v1[1],v2[1]], angles='xy', scale_units='xy', scale=1)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "If we wanted to \"reach\" the point `.3,1`, we would do the following operations:\n",
    "\n",
    "1. Multiply `1,0` by `.3` to get `.3,0`\n",
    "2. Add `.3,0` to `0,1` to get `.3,1`\n",
    "\n",
    "We call the set of vectors that can be \"reached\" from a given set of vectors the *span* of the vectors.  In this case, the span of `0,1` and `1,0` is the entire 2-dimensional Euclidean space.\n",
    "\n",
    "One additional property that is important is that `0,1` and `1,0` are *orthogonal* to each other.  This means that they don't have any overlap in the directions that they point.  We can check this using NumPy:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0"
      ]
     },
     "execution_count": 12,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Find the dot product of v1 and v2\n",
    "np.dot(v1, v2)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The dot product tells you how much the direction of one vector overlaps with the direction of another vector.  Since the dot product is 0, we know that the vectors are orthogonal."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Basis Change\n",
    "\n",
    "We express coordinates in terms of a basis.  We know that the canonical basis vectors for the 2-dimensional Euclidean space are `0,1` and `1,0`.  A coordinate specifies how much we multiply each canonical basis vector to get the vector we want.  For example, the coordinate `2,1` specifies that we multiply `1,0` by `2` and `0,1` by `1` to get `2,0` and `0,1`, which we then add together to get `2,1`.\n",
    "\n",
    "We call this a *linear combination* of the basis vectors.\n",
    "\n",
    "In deep learning, we often change the basis that we use to express coordinates.  For example, we might want to express the coordinates `2,1` in terms of the basis vectors `1,1` and `1,-1`.  Showing you how to do a basis change is outside the scope of this lesson, but it's important to know that it's possible."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Matrices\n",
    "\n",
    "We can arrange vectors into matrices.  A matrix is a two-dimensional array of numbers.  By convention, we use lower-case letters to represent vectors, and upper-case letters to represent matrices. Here's an example of a 2x2 matrix with both of the vectors from above:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[[0 1 2]\n",
      " [1 0 2]\n",
      " [2 1 0]]\n"
     ]
    }
   ],
   "source": [
    "# Create a matrix as a numpy array\n",
    "M = np.array([[0,1,2], [1,0,2], [2,1,0]])\n",
    "print(M)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "A matrix has a certain number of rows and columns.  We can find the number of rows and columns in a matrix using the `shape` attribute:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(3, 3)"
      ]
     },
     "execution_count": 14,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "M.shape"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "To index a matrix, we need 2 numbers instead of 1.  The first number is the row index, and the second number is the column index.  For example, we can get the element at the first row and first column of `M` like this:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0"
      ]
     },
     "execution_count": 15,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "M[0,0]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We can also get entire rows and columns at once, or slices of them:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[0 1 2]\n",
      "[0 1 2]\n",
      "[[0 1]\n",
      " [1 0]]\n"
     ]
    }
   ],
   "source": [
    "# Select the entire first row\n",
    "print(M[0])\n",
    "# Select the entire first column\n",
    "print(M[:,0])\n",
    "# Select elements in the upper left corner\n",
    "print(M[:2, :2])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We can also use indexing for assignment:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {},
   "outputs": [],
   "source": [
    "# Set the first element to 10\n",
    "M[0,0] = 10"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "In deep learning, you will often work with arrays that are 3 or more dimensions.  To index these, you'll need to use as many index numbers as there are dimensions."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Linear Regression"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Now that we understand a bit about vectors and matrices, let's look at a concrete example.  We'll use linear regression to predict tomorrow's temperature.  We do this with the equation $\\hat{y}=wx + b$ where $\\hat{y}$ is the prediction, $w$ is the weight, $x$ is the input, and $b$ is the bias.\n",
    "\n",
    "We can first read in our data using pandas:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<div>\n",
       "<style scoped>\n",
       "    .dataframe tbody tr th:only-of-type {\n",
       "        vertical-align: middle;\n",
       "    }\n",
       "\n",
       "    .dataframe tbody tr th {\n",
       "        vertical-align: top;\n",
       "    }\n",
       "\n",
       "    .dataframe thead th {\n",
       "        text-align: right;\n",
       "    }\n",
       "</style>\n",
       "<table border=\"1\" class=\"dataframe\">\n",
       "  <thead>\n",
       "    <tr style=\"text-align: right;\">\n",
       "      <th></th>\n",
       "      <th>tmax</th>\n",
       "      <th>tmin</th>\n",
       "      <th>rain</th>\n",
       "      <th>tmax_tomorrow</th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>1970-01-01</th>\n",
       "      <td>60.0</td>\n",
       "      <td>35.0</td>\n",
       "      <td>0.0</td>\n",
       "      <td>52.0</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1970-01-02</th>\n",
       "      <td>52.0</td>\n",
       "      <td>39.0</td>\n",
       "      <td>0.0</td>\n",
       "      <td>52.0</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1970-01-03</th>\n",
       "      <td>52.0</td>\n",
       "      <td>35.0</td>\n",
       "      <td>0.0</td>\n",
       "      <td>53.0</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1970-01-04</th>\n",
       "      <td>53.0</td>\n",
       "      <td>36.0</td>\n",
       "      <td>0.0</td>\n",
       "      <td>52.0</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1970-01-05</th>\n",
       "      <td>52.0</td>\n",
       "      <td>35.0</td>\n",
       "      <td>0.0</td>\n",
       "      <td>50.0</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "</div>"
      ],
      "text/plain": [
       "            tmax  tmin  rain  tmax_tomorrow\n",
       "1970-01-01  60.0  35.0   0.0           52.0\n",
       "1970-01-02  52.0  39.0   0.0           52.0\n",
       "1970-01-03  52.0  35.0   0.0           53.0\n",
       "1970-01-04  53.0  36.0   0.0           52.0\n",
       "1970-01-05  52.0  35.0   0.0           50.0"
      ]
     },
     "execution_count": 18,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "import pandas as pd\n",
    "\n",
    "# Read in the data\n",
    "data = pd.read_csv(\"../data/clean_weather.csv\", index_col=0)\n",
    "# Fill in any missing values in the data with past values\n",
    "data = data.ffill()\n",
    "\n",
    "# Show the first 5 rows of the data\n",
    "data.head(5)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "In the table above, tomorrow's temperature is `tmax_tomorrow`, today's max temperature is `tmax`, today's min temperature is `tmin`, and how much it rained today is `rain`.  We're going to use `tmax`, `tmin`, and `rain` to predict `tmax_tomorrow`.\n",
    "\n",
    "Because we have more than one predictor, we can extend our equation to multiple predictors like this:\n",
    "\n",
    "$$\\hat{y} = w_{1}x_{1} + w_{2}x_{2} + w_{3}x_{3} + b$$\n",
    "\n",
    "So to get a prediction for tomorrow's temperature $\\hat{y}$, we can take a value called $w_{1}$, and multiply it by `tmax`, then take $w_{2}$ and multiply it by `tmin`, then take $w_{3}$ and multiply it by `rain`.  We'd then add in $b$.  Here's how that could look:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "62.5"
      ]
     },
     "execution_count": 19,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    ".7 * 60 + .3 * 35 + .1 * 0 + 10"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The above could be our prediction for the first row of the data, if our $w$ values are `.7`, `.3`, and `.1`, and our $b$ value is `10`.  We'll discuss how you can calculate the correct $w$ and $b$ values later, but for now, let's just use these values.\n",
    "\n",
    "Whenever we want to make a new prediction, we apply the same equation:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "58.099999999999994"
      ]
     },
     "execution_count": 20,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "w1 = .7\n",
    "w2 = .3\n",
    "w3 = .1\n",
    "b = 10\n",
    "\n",
    "w1 * 52 + w2 * 39 + w3 * 0 + b"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Vector multiplication\n",
    "\n",
    "You may notice that we're actually multiplying two vectors, then adding the results together.  We can think of $w$ and $x$ as vectors that we're multiplying.  We multiply each element in the $x$ vector by the corresponding element in the $w$ vector.  We then add up all the results.  This is called a *dot product*.\n",
    "\n",
    "Here's an example of a dot product:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "48.1"
      ]
     },
     "execution_count": 21,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "w = np.array([.7, .3, .1])\n",
    "x = np.array([52, 39, 0])\n",
    "\n",
    "# Multiply w and x elementwise\n",
    "np.dot(w, x)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Our result is `48.1`.  We then need to add the bias term `b` to get our prediction.  We can do this in NumPy like this:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "58.1"
      ]
     },
     "execution_count": 22,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "np.dot(w, x) + b"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Matrix Multiplication\n",
    "\n",
    "What if we want to make predictions for multiple rows at once?  It would be annoying to have to do the dot product for each row.  Luckily, we can use linear algebra to help us out.  We first need to convert our x data and our weights to a matrix.  We can then do a matrix multiplication to get our predictions.\n",
    "\n",
    "Matrix multiplication is a linear algebra operation defined like this:\n",
    "\n",
    "$$\n",
    "\\begin{equation}\n",
    "    A \\times B =\n",
    "    \\begin{bmatrix}\n",
    "      a_{11} & a_{12} \\\\\n",
    "      a_{21} & a_{22}\n",
    "    \\end{bmatrix}\n",
    "    \\times\n",
    "    \\begin{bmatrix}\n",
    "    b_{11} \\\\\n",
    "    b_{21}\n",
    "    \\end{bmatrix}\n",
    "    =\n",
    "    \\begin{bmatrix}\n",
    "      a_{11}b_{11} + a_{12}b_{21} \\\\\n",
    "      a_{21}b_{11} + a_{22}b_{21}\n",
    "    \\end{bmatrix}\n",
    "\\end{equation}\n",
    "$$\n",
    "\n",
    "We can visualize how it works with this gif:\n",
    "\n",
    "![Matrix mult](images/linalg/matrix_mult.gif)\n",
    "\n",
    "As you can see, we essentially take each row of matrix A and multiply it by each column of matrix B, then add everything together.  The number of columns in the first matrix has to equal the number of rows in the second matrix.\n",
    "\n",
    "This is very useful to us when we're multiplying weights by input numbers.  We can take our $x$ values, put them into a matrix, then multiply by the weights (also in a matrix):"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[60., 35.,  0.],\n",
       "       [52., 39.,  0.],\n",
       "       [52., 35.,  0.]])"
      ]
     },
     "execution_count": 23,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Convert the first 3 rows of data into a numpy matrix from a pandas dataframe\n",
    "# the first column of the matrix is tmax, second is tmin, third is rain\n",
    "\n",
    "X = data[[\"tmax\", \"tmin\", \"rain\"]].iloc[:3].to_numpy()\n",
    "X"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We can verify the shapes of our matrices to make sure they can be multiplied:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(3, 3)"
      ]
     },
     "execution_count": 24,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Print the shape (number of rows and columns) in X\n",
    "X.shape"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 25,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(3,)"
      ]
     },
     "execution_count": 25,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Print the shape of w\n",
    "w.shape"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "As we can see, `x` is a matrix with `3` rows and `3` columns.  `w` is actually a vector with `3` columns.\n",
    "\n",
    "`x` has length `3` in dimension `0` (rows) and length `1` columns.  `w` has length `3` in dimension `0`.\n",
    "\n",
    "We'll need to convert `w` from a vector into a matrix.  To do that, we can use the numpy `reshape` method.  We pass in our desired lengths in each dimension:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 26,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[0.7],\n",
       "       [0.3],\n",
       "       [0.1]])"
      ]
     },
     "execution_count": 26,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Reshape w into a 3 by 1 matrix\n",
    "# The new dimensions must match the number of elements in w\n",
    "W = w.reshape(3,1)\n",
    "W"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 27,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(3, 1)"
      ]
     },
     "execution_count": 27,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "W.shape"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We now have a `3x3` matrix, and a `3x1` matrix.  Since the number of columns in `x` matches the number of rows in `w`, we can multiply them:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 28,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[52.5],\n",
       "       [48.1],\n",
       "       [46.9]])"
      ]
     },
     "execution_count": 28,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "X @ W"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We can now add in our $b$ value, `10`.  $b$ is just a single number.  When we add it to `x @ w`, it will be broadcast across the matrix (added to each element):"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 29,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[62.5],\n",
       "       [58.1],\n",
       "       [56.9]])"
      ]
     },
     "execution_count": 29,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "b = np.array([10])\n",
    "X @ W + b"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "This gives us the same results as when we manually multiplied our `w` and `x` values.  This is the power of matrix multiplication - it enables us to store all of our parameters ($w$ and $b$ values) into arrays and then use them to modify our inputs.  This is a lot faster than keeping track of each individual variable!"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## The Normal Equation"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Earlier, we used values for $w$ and $b$ that were given to us.  We can also directly calculate the correct values for $w$ and $b$ using the *normal equation*.\n",
    "\n",
    "The normal equation is a mathematical equation that gives us the exact solution for $w$ and $b$ that minimizes the error of our model.  It's defined like this:\n",
    "\n",
    "$$W = (X^T X)^{-1} X^T Y$$\n",
    "\n",
    "You can think of this equation as finding the coefficients $W$ that project the coordinates $Y$ onto the basis $X$. Another way to say this is that we're multiplying the predictors $X$ by the weights $W$ and trying to get as close as possible to $Y$.\n",
    "\n",
    "Yet another way to say this (it's a complicated concept!) is that we're changing the basis vectors of $Y$ to the vectors in $X$.  The coefficients $W$ find a way to do this while minimizing error.\n",
    "\n",
    "We won't go into more detail on this here, but I recommend reading [this book](https://mml-book.github.io/) for more detail.\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Matrix Transposition\n",
    "\n",
    "You may have noticed a term in the equation that looks like $X^T$.  This is called *matrix transposition*.  It's a way to switch the rows and columns of a matrix.  For example, we can transpose our weights and $X$ values:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 30,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[62.5, 58.1, 56.9]])"
      ]
     },
     "execution_count": 30,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Use .T to get the transpose of a matrix\n",
    "W.T @ X.T + b"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Transposition takes the `3x1` weight matrix, and turns it into a `1x3` matrix.  It then transposes the `3x3` x matrix.  The dimensions of x will stay the same, but transposition will switch the rows and columns:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 31,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[0.7, 0.3, 0.1]])"
      ]
     },
     "execution_count": 31,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "W.T"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 32,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[60., 52., 52.],\n",
       "       [35., 39., 35.],\n",
       "       [ 0.,  0.,  0.]])"
      ]
     },
     "execution_count": 32,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "X.T"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "`W.T @ X.T` will multiply each column in the weight matrix by each row in the x matrix.  This is the exact same thing that will happen when we do `X @ W`.  It's just that `W.T @ X.T` returns a `1x3` matrix, and `X @ W` returns a `3x1` matrix.  So we need to do another transposition to correct this:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 33,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "True"
      ]
     },
     "execution_count": 33,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# The numpy allclose function tells us if two matrices are approximately equal\n",
    "np.allclose((W.T @ X.T).T, X @ W)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Above, we use `np.allclose` to ensure that both multiplications return the same result.  When comparing the results of matrix multiplication we usually use `np.allclose` instead of `==` because the way computers represent floating point numbers can cause the same number to appear slightly differently.  You can see an explanation for this [here](https://floating-point-gui.de/)."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Matrix Inversion\n",
    "\n",
    "You may have also noticed this term in the normal equation - $(X^T X)^{-1}$. This is called *matrix inversion*.  The inverse of a matrix is similar to the inverse of a number.  For example, the inverse of `4` is `1/4`, and `4 * 1/4` equals `1`.  $X \\cdot X^{-1}$ is equal to the identity matrix, $I$.  The identity matrix is a matrix with 1s on the diagonal, and 0s everywhere else.  It's a way to represent the number 1 in matrix form.\n",
    "\n",
    "Here's the `3x3` identity matrix:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 34,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[1., 0., 0.],\n",
       "       [0., 1., 0.],\n",
       "       [0., 0., 1.]])"
      ]
     },
     "execution_count": 34,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "np.eye(3)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Now we can invert our `X` matrix:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 35,
   "metadata": {},
   "outputs": [
    {
     "ename": "LinAlgError",
     "evalue": "Singular matrix",
     "output_type": "error",
     "traceback": [
      "\u001B[0;31m---------------------------------------------------------------------------\u001B[0m",
      "\u001B[0;31mLinAlgError\u001B[0m                               Traceback (most recent call last)",
      "Cell \u001B[0;32mIn[35], line 2\u001B[0m\n\u001B[1;32m      1\u001B[0m \u001B[38;5;66;03m# Use np.linalg.inv to get the inverse of a matrix\u001B[39;00m\n\u001B[0;32m----> 2\u001B[0m \u001B[43mnp\u001B[49m\u001B[38;5;241;43m.\u001B[39;49m\u001B[43mlinalg\u001B[49m\u001B[38;5;241;43m.\u001B[39;49m\u001B[43minv\u001B[49m\u001B[43m(\u001B[49m\u001B[43mX\u001B[49m\u001B[43m)\u001B[49m\n",
      "File \u001B[0;32m<__array_function__ internals>:200\u001B[0m, in \u001B[0;36minv\u001B[0;34m(*args, **kwargs)\u001B[0m\n",
      "File \u001B[0;32m~/.virtualenvs/nnets/lib/python3.10/site-packages/numpy/linalg/linalg.py:538\u001B[0m, in \u001B[0;36minv\u001B[0;34m(a)\u001B[0m\n\u001B[1;32m    536\u001B[0m signature \u001B[38;5;241m=\u001B[39m \u001B[38;5;124m'\u001B[39m\u001B[38;5;124mD->D\u001B[39m\u001B[38;5;124m'\u001B[39m \u001B[38;5;28;01mif\u001B[39;00m isComplexType(t) \u001B[38;5;28;01melse\u001B[39;00m \u001B[38;5;124m'\u001B[39m\u001B[38;5;124md->d\u001B[39m\u001B[38;5;124m'\u001B[39m\n\u001B[1;32m    537\u001B[0m extobj \u001B[38;5;241m=\u001B[39m get_linalg_error_extobj(_raise_linalgerror_singular)\n\u001B[0;32m--> 538\u001B[0m ainv \u001B[38;5;241m=\u001B[39m \u001B[43m_umath_linalg\u001B[49m\u001B[38;5;241;43m.\u001B[39;49m\u001B[43minv\u001B[49m\u001B[43m(\u001B[49m\u001B[43ma\u001B[49m\u001B[43m,\u001B[49m\u001B[43m \u001B[49m\u001B[43msignature\u001B[49m\u001B[38;5;241;43m=\u001B[39;49m\u001B[43msignature\u001B[49m\u001B[43m,\u001B[49m\u001B[43m \u001B[49m\u001B[43mextobj\u001B[49m\u001B[38;5;241;43m=\u001B[39;49m\u001B[43mextobj\u001B[49m\u001B[43m)\u001B[49m\n\u001B[1;32m    539\u001B[0m \u001B[38;5;28;01mreturn\u001B[39;00m wrap(ainv\u001B[38;5;241m.\u001B[39mastype(result_t, copy\u001B[38;5;241m=\u001B[39m\u001B[38;5;28;01mFalse\u001B[39;00m))\n",
      "File \u001B[0;32m~/.virtualenvs/nnets/lib/python3.10/site-packages/numpy/linalg/linalg.py:89\u001B[0m, in \u001B[0;36m_raise_linalgerror_singular\u001B[0;34m(err, flag)\u001B[0m\n\u001B[1;32m     88\u001B[0m \u001B[38;5;28;01mdef\u001B[39;00m \u001B[38;5;21m_raise_linalgerror_singular\u001B[39m(err, flag):\n\u001B[0;32m---> 89\u001B[0m     \u001B[38;5;28;01mraise\u001B[39;00m LinAlgError(\u001B[38;5;124m\"\u001B[39m\u001B[38;5;124mSingular matrix\u001B[39m\u001B[38;5;124m\"\u001B[39m)\n",
      "\u001B[0;31mLinAlgError\u001B[0m: Singular matrix"
     ]
    }
   ],
   "source": [
    "# Use np.linalg.inv to get the inverse of a matrix\n",
    "np.linalg.inv(X)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "If any rows and columns of a matrix are linear combinations of other rows and columns, then the matrix is singular.  This means that the matrix cannot be inverted.  Here's an example:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 36,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[1, 2],\n",
       "       [3, 6]])"
      ]
     },
     "execution_count": 36,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# This matrix is singular\n",
    "np.array([[1, 2], [2, 4]])\n",
    "\n",
    "# This matrix is not singular\n",
    "np.array([[1, 2], [3, 6]])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The reason for this is when a matrix is singular, the length of the matrix is 0 in one dimension.  This means that the area of the matrix is 0.  The formula for the inverse uses $1 / \\text{area}$ as a term, which is undefined.  This makes the inverse of a singular matrix also undefined.\n",
    "\n",
    "To correct for this, we can add a small \"ridge\" to the diagonal of the matrix by scaling the identity matrix.  This forces each row and column to be unique (not a linear combination of others).\n",
    "\n",
    "This is known as ridge regression:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 37,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[ 0.07378611, -0.06604895,  0.        ],\n",
       "       [-0.09812987,  0.11341549,  0.        ],\n",
       "       [-4.02332472, -5.34996509, 10.        ]])"
      ]
     },
     "execution_count": 37,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Add a ridge to the diagonal of the matrix\n",
    "np.linalg.inv(X + 0.1 * np.eye(3))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 38,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[ 1.00000000e+00,  5.55111512e-17,  0.00000000e+00],\n",
       "       [ 1.06185910e-16,  1.00000000e+00,  0.00000000e+00],\n",
       "       [ 1.66403976e-16, -2.96151798e-16,  1.00000000e+00]])"
      ]
     },
     "execution_count": 38,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# This returns the identity matrix\n",
    "ridge = X + 0.1 * np.eye(3)\n",
    "ridge @ np.linalg.inv(ridge)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Solving The Equation"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We can now use the normal equation to solve for $W$:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 39,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[0.38283829],\n",
       "       [0.86270249],\n",
       "       [0.        ]])"
      ]
     },
     "execution_count": 39,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Load in our Y values from the data\n",
    "Y = data[[\"tmax_tomorrow\"]].iloc[:3].to_numpy()\n",
    "\n",
    "# Use the normal equation to solve for W\n",
    "W = np.linalg.inv(X.T @ X + .1 * np.eye(x.shape[0])) @ X.T @ Y\n",
    "W"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 40,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[53.16488447],\n",
       "       [53.55298814],\n",
       "       [50.10217817]])"
      ]
     },
     "execution_count": 40,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Multiply X by W to get our predictions\n",
    "X @ W"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 41,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[52.],\n",
       "       [52.],\n",
       "       [53.]])"
      ]
     },
     "execution_count": 41,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Y"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "As you can see, our predictions are pretty close to the actual values.  We left out $b$ to make the calculation simpler, but you could also use this method to calculate it (but it's out of scope of this lesson!)."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Broadcasting\n",
    "\n",
    "A couple of sections back, we used broadcasting to add the bias `b` to the result of `X @ W`.  Broadcasting enables arrays with compatible shapes to be added or multiplied.\n",
    "\n",
    "If shapes are compatible, it means that:\n",
    "* The smaller array has all of its dimensions exactly matching the length dimensions of the larger array\n",
    "* Or the smaller array is length 1 in the non-matching dimensions.\n",
    "\n",
    "Here are some examples:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 42,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[2.],\n",
       "       [2.],\n",
       "       [2.],\n",
       "       [2.],\n",
       "       [2.]])"
      ]
     },
     "execution_count": 42,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# np.ones enables us to create an array filled with ones of a certain shape\n",
    "\n",
    "A = np.ones((5,1))\n",
    "\n",
    "# This works\n",
    "A + np.ones((1,1))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 43,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[2.],\n",
       "       [2.],\n",
       "       [2.],\n",
       "       [2.],\n",
       "       [2.]])"
      ]
     },
     "execution_count": 43,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# This works\n",
    "A + np.ones((1))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 44,
   "metadata": {},
   "outputs": [
    {
     "ename": "ValueError",
     "evalue": "operands could not be broadcast together with shapes (5,1) (2,1) ",
     "output_type": "error",
     "traceback": [
      "\u001B[0;31m---------------------------------------------------------------------------\u001B[0m",
      "\u001B[0;31mValueError\u001B[0m                                Traceback (most recent call last)",
      "Cell \u001B[0;32mIn[44], line 2\u001B[0m\n\u001B[1;32m      1\u001B[0m \u001B[38;5;66;03m# This does not work\u001B[39;00m\n\u001B[0;32m----> 2\u001B[0m \u001B[43mA\u001B[49m\u001B[43m \u001B[49m\u001B[38;5;241;43m+\u001B[39;49m\u001B[43m \u001B[49m\u001B[43mnp\u001B[49m\u001B[38;5;241;43m.\u001B[39;49m\u001B[43mones\u001B[49m\u001B[43m(\u001B[49m\u001B[43m(\u001B[49m\u001B[38;5;241;43m2\u001B[39;49m\u001B[43m,\u001B[49m\u001B[38;5;241;43m1\u001B[39;49m\u001B[43m)\u001B[49m\u001B[43m)\u001B[49m\n",
      "\u001B[0;31mValueError\u001B[0m: operands could not be broadcast together with shapes (5,1) (2,1) "
     ]
    }
   ],
   "source": [
    "# This does not work\n",
    "A + np.ones((2,1))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We can also use broadcasting to multiply individual elements:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 45,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[2.],\n",
       "       [2.],\n",
       "       [2.],\n",
       "       [2.],\n",
       "       [2.]])"
      ]
     },
     "execution_count": 45,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# np.full initializes an array filled with a given value\n",
    "# Using the * means that each element in array b is multiplied by the elements in array b\n",
    "A * np.full((1,1), 2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 46,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[0.60888367],\n",
       "       [0.27445549],\n",
       "       [0.41672986],\n",
       "       [0.18109412],\n",
       "       [0.89991161]])"
      ]
     },
     "execution_count": 46,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# np.random.rand generates random numbers between 0 and 1 of a given shape\n",
    "A * np.random.rand(5,1)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Derivatives\n",
    "\n",
    "The last math concept we'll cover is derivatives, a key part of calculus.  We'll cover derivatives in a lot more depth later, so this will just be a high-level overview.\n",
    "\n",
    "Derivatives tell us the rate of change of a function.  For example, the derivative of $x^2$ is $2x$.  Let's look at how this works, by first plotting $f(x) = x^2$:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 47,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[<matplotlib.lines.Line2D at 0x14cf0be80>]"
      ]
     },
     "execution_count": 47,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 640x480 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "import matplotlib.pyplot as plt\n",
    "\n",
    "# Create a list of numbers from -10 to 10, stepping by .1\n",
    "xs = np.arange(-10, 10, .1)\n",
    "# Square all of our x values\n",
    "ys = xs ** 2\n",
    "\n",
    "# Plot the result\n",
    "plt.plot(xs, ys)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The function graph has a curved shape.  This is because the difference between y values is increasing as x gets further from `0`.   For example, if x is positive, $(x+2) ^ 2 - (x+1) ^ 2$ is greater than $(x+1) ^ 2 - x ^ 2$.  If x is negative, $(x-2) ^ 2 - (x-1) ^ 2$ is greater than $(x-1) ^ 2 - x ^ 2$.  We can also say that the slope of the function is increasing as we get further from `0`.\n",
    "\n",
    "As you can see below, the derivative is the slope between 2 points on the function curve:\n",
    "\n",
    "![slope](images/linalg/slope.svg)\n",
    "\n",
    "We can calculate the slope by the equation $\\frac{y_2 - y_1}{x_2 - x_1}$, where $x_2$ and $x_1$ are the x values of 2 points, and $y_2$ and $y_1$ are the y values of the same points."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 48,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "10.0"
      ]
     },
     "execution_count": 48,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Find 2 x points close to 5\n",
    "x1 = 5 - 1e-8\n",
    "x2 = 5 + 1e-8\n",
    "\n",
    "# Calculate the corresponding y values\n",
    "y1 = x1 ** 2\n",
    "y2 = x2 ** 2\n",
    "\n",
    "# Calculate the slope (or derivative)\n",
    "# This will equal 2 times the midpoint of the x values (5)\n",
    "(y2 - y1) / (x2 - x1)\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Derivatives are at a single point, so we would say that $f'(5) = 10$.  This means that the rate of change of $x^2$ at $x=5$ is $10$.\n",
    "\n",
    "We can now plot the derivative, $f'(x) = 2x$:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 49,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[<matplotlib.lines.Line2D at 0x14d1896f0>]"
      ]
     },
     "execution_count": 49,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 640x480 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "ys = xs * 2\n",
    "\n",
    "# Plot the derivative\n",
    "plt.plot(xs, ys)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "As you can see, the derivative is an upward-sloping line.  This tells us that the rate of change of the function increases as x increases.\n",
    "\n",
    "Derivatives are important for training neural networks.  They allow us to determine how much to increase or decrease our network parameters.  We'll learn more about this in the next lesson.\n",
    "\n",
    "You usually don't need to know how to find the equation of the derivative yourself, since you'll be given it beforehand.  In a later lesson, we'll discuss how to find the derivative if you need to."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Wrap-up\n",
    "\n",
    "In the next lesson, we'll build on what we learned here to automatically calculate the correct $w$ and $b$ values for linear regression using gradient descent.  Gradient descent is a foundational concept of neural networks.  I hope to see you there!"
   ]
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "Python 3 (ipykernel)",
   "language": "python",
   "name": "python3"
  },
  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
    "version": 3
   },
   "file_extension": ".py",
   "mimetype": "text/x-python",
   "name": "python",
   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython3",
   "version": "3.10.8"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 1
}