0X_vector.ipynb

{
  "cells": [
    {
      "cell_type": "markdown",
      "metadata": {
        "execution": {},
        "id": "hSnyF6nM_EG5"
      },
      "source": [
        "# Tutorial 0: Vectors"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "execution": {},
        "id": "Rufq8bWJ_EG-"
      },
      "source": [
        "---\n",
        "# Setup\n",
        "\n",
        "Execute the following cells to set up the notebook environment"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": 1,
      "metadata": {
        "execution": {},
        "id": "lJgocvPS_EHA"
      },
      "outputs": [],
      "source": [
        "# Imports\n",
        "import numpy as np\n",
        "import matplotlib.pyplot as plt"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": 2,
      "metadata": {
        "execution": {},
        "id": "A1JEHLeW_EHA"
      },
      "outputs": [],
      "source": [
        "#@title Figure settings\n",
        "import ipywidgets as widgets  # interactive display\n",
        "from ipywidgets import fixed\n",
        "%config InlineBackend.figure_format = 'retina'\n",
        "plt.style.use(\"https://raw.githubusercontent.com/NeuromatchAcademy/course-content/master/nma.mplstyle\")"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": 3,
      "metadata": {
        "execution": {},
        "id": "vZjiw4Ae_EHB",
        "cellView": "form"
      },
      "outputs": [],
      "source": [
        "# @title Plotting functions\n",
        "from matplotlib.patches import FancyArrowPatch\n",
        "from mpl_toolkits.mplot3d import proj3d\n",
        "\n",
        "\n",
        "def visualize_vectors(v, v_unit):\n",
        "  \"\"\" Plots a 2D vector and the corresponding unit vector\n",
        "\n",
        "  Args:\n",
        "    v (ndarray): array of size (2,) with the vector coordinates\n",
        "    v_unit (ndarray): array of size (2, ) with the unit vector coordinates\n",
        "\n",
        "  \"\"\"\n",
        "  fig, ax = plt.subplots()\n",
        "\n",
        "  # Set up plot aesthetics\n",
        "  ax.spines['top'].set_color('none')\n",
        "  ax.spines['bottom'].set_position('zero')\n",
        "  ax.spines['left'].set_position('zero')\n",
        "  ax.spines['right'].set_color('none')\n",
        "  ax.set(xlim = [-6, 6], ylim = [-6, 6])\n",
        "  ax.grid(True, alpha=.4, linewidth=1, zorder=0)\n",
        "\n",
        "  # Plot vectors\n",
        "  v_arr = ax.arrow(0, 0, v[0], v[1], width=0.08, color='#648FFF',\n",
        "                   length_includes_head=True, zorder=2);\n",
        "  v_unit_arr = ax.arrow(0, 0, v_unit[0], v_unit[1], width=0.08,\n",
        "                        color='#DC267F',\n",
        "                        length_includes_head=True, zorder=3);\n",
        "  ax.set(xlim = [-4, 4], ylim = [-4, 4]);\n",
        "\n",
        "  # Add legend\n",
        "  leg = ax.legend([v_arr, v_unit_arr],\n",
        "                  [r\"Vector $\\mathbf{v}$\",\n",
        "                   r\"Normalized vector $\\tilde{\\mathbf{v}}$\"],\n",
        "                  handlelength=0, fontsize=20, loc='upper left')\n",
        "  for handle, label in zip(leg.legendHandles, leg.texts):\n",
        "    label.set_color(handle.get_facecolor())\n",
        "    handle.set_visible(False)\n",
        "\n",
        "class Arrow3D(FancyArrowPatch):\n",
        "    def __init__(self, xs, ys, zs, *args, **kwargs):\n",
        "      FancyArrowPatch.__init__(self, (0,0), (0,0), *args, **kwargs)\n",
        "      self._verts3d = xs, ys, zs\n",
        "\n",
        "    def draw(self, renderer):\n",
        "      xs3d, ys3d, zs3d = self._verts3d\n",
        "      xs, ys, zs = proj3d.proj_transform(xs3d, ys3d, zs3d, renderer.M)\n",
        "      self.set_positions((xs[0],ys[0]),(xs[1],ys[1]))\n",
        "      return FancyArrowPatch.draw(self, renderer)\n",
        "\n",
        "    def do_3d_projection(self, renderer=None):\n",
        "      xs3d, ys3d, zs3d = self._verts3d\n",
        "      xs, ys, zs = proj3d.proj_transform(xs3d, ys3d, zs3d, self.axes.M)\n",
        "      self.set_positions((xs[0], ys[0]), (xs[1], ys[1]))"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "execution": {},
        "id": "9l27aaZj_EHD"
      },
      "source": [
        "---\n",
        "\n",
        "# Section 1: Intro to vectors"
      ]
    },
    {
      "cell_type": "markdown",
      "source": [
        "## Section 1.1: What is a vector?\n",
        "\n",
        "A vector $\\mathbf{x}$ can be considered from at least two perspectives: as an ordered list of numbers or as an arrow with the base at the origin of a coordinate system. These are two ways of looking at a single thing: in the arrow case, the tip of the arrow is defined by a coordinate (which can be represented by the ordered list).\n",
        "<br/><br/>\n",
        "\n",
        "The **dimensionality of a vector** is determined by the number of components in the ordered list (or the dimensionality of the space in which the arrow exists). For example, $\\mathbf{x} = \\begin{bmatrix} 4 \\\\ 2 \\\\ 1  \\end{bmatrix}$ is a 3 dimensional vector. We can refer to components by $\\mathbf{x}_i$ where i indicates the position of the component. In this vector, $\\mathbf{x_1} = 4$, $\\mathbf{x_2} = 2$, and $\\mathbf{x_3} = 1$.\n",
        "\n",
        "<br/><br/>\n",
        "One defining property of a vector is its length: $||\\mathbf{x}||$. This is the length of the arrow and can be computed as the square root of the sum of all components squared:\n",
        "\n",
        "\\begin{equation}\n",
        "||\\mathbf{x}|| = \\sqrt{\\sum_{i=1}^N \\mathbf{x}_i^2}\n",
        "\\end{equation}\n",
        "\n",
        "<br/><br/>\n",
        "We have two special types of vectors with specific names. **Zero vectors** have length 0 (and all components equal 0). **Unit vectors** have length 1. You can normalize a vector and create a unit vector, $\\tilde{\\mathbf{x}}$, by dividing by its length,$||\\mathbf{x}||$ :\n",
        "\n",
        "\\begin{equation}\n",
        "\\tilde{\\mathbf{x}} = \\frac{\\mathbf{x}}{||\\mathbf{x}||}\n",
        "\\end{equation}"
      ],
      "metadata": {
        "id": "gAKTcpWo0Pgy"
      }
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "execution": {},
        "id": "MKtovFAM_EHE"
      },
      "source": [
        "### Coding Exercise 1.1: Normalizing vectors\n",
        "\n",
        "In this exercise, you will first create the below vector, $\\mathbf{v}$, using a numpy array. You will then implement a function, `normalize_vector`, that outputs a normalized version of the input vector (by dividing each component by the vector length).  You will then visualize both the original vector and the unit vector.\n",
        "\n",
        "First think: how do you think the normalized unit vector will compare to the original vector in terms of direction and length?\n",
        "\n",
        "\n",
        "\\begin{equation}\n",
        "\\mathbf{v} =\n",
        "\\begin{bmatrix}\n",
        "4 \\\\\n",
        "1\n",
        "\\end{bmatrix}\n",
        "\\end{equation}"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": 4,
      "metadata": {
        "execution": {},
        "id": "FQbMfdkc_EHE"
      },
      "outputs": [],
      "source": [
        "def normalize_vector(input_vector):\n",
        "  \"\"\" Normalizes a vector to length 1\n",
        "\n",
        "  Inputs:\n",
        "    input_vector (ndarray): array of shape (n_dim, ) where n_dim is the\n",
        "        dimensionality of the vector\n",
        "\n",
        "  Outputs:\n",
        "    (ndarray): normalized vector of length 1\n",
        "  \"\"\"\n",
        "  # raise NotImplementedError(\"Student exercise: complete normalize_vector function\")\n",
        "\n",
        "\n",
        "  # Compute vector length (use np.linalg.norm)\n",
        "  # SOLUTION\n",
        "  vector_length = np.linalg.norm(input_vector)\n",
        "\n",
        "  # SOLUTION\n",
        "  # Create normalized vector\n",
        "  normalized_vector = input_vector/vector_length\n",
        "\n",
        "  return normalized_vector"
      ]
    },
    {
      "cell_type": "code",
      "source": [
        "# Create vector v from above (use np.array)\n",
        "# SOLUTION\n",
        "v = np.array([4,1])\n",
        "\n",
        "# Call the function\n",
        "v_unit = normalize_vector(v)\n",
        "\n",
        "# Visualize the vectors\n",
        "visualize_vectors(v, v_unit)"
      ],
      "metadata": {
        "id": "30Ukk5Ry8Q6d",
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 647
        },
        "outputId": "737961a3-20df-43dd-c5c6-816ede1dc20d"
      },
      "execution_count": 5,
      "outputs": [
        {
          "output_type": "stream",
          "name": "stderr",
          "text": [
            "<ipython-input-3-25232a2bc73d>:37: MatplotlibDeprecationWarning: The legendHandles attribute was deprecated in Matplotlib 3.7 and will be removed two minor releases later. Use legend_handles instead.\n",
            "  for handle, label in zip(leg.legendHandles, leg.texts):\n"
          ]
        },
        {
          "output_type": "display_data",
          "data": {
            "text/plain": [
              "<Figure size 800x600 with 1 Axes>"
            ],
            "image/png": "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\n"
          },
          "metadata": {
            "image/png": {
              "width": 775,
              "height": 575
            }
          }
        }
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "execution": {},
        "id": "gJVEy0az_EHE"
      },
      "source": [
        "*Example output:*\n",
        "\n",
        "<img alt='Solution hint' align='left' width=775.0 height=575.0 src=https://raw.githubusercontent.com/NeuromatchAcademy/precourse/main/tutorials/W0D3_LinearAlgebra/static/W0D3_Tutorial1_Solution_121dde78_352.png>\n",
        "\n"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "execution": {},
        "id": "a6N3SuQ4_EHE"
      },
      "source": [
        "\n",
        "\n",
        "Note that the unit vector still points in the same direction as the original (it overlays it). Dividing each component by a constant (the length) does not change the direction. The only change is the length - the normalized vector has a length of 1."
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "execution": {},
        "id": "Z517eH90_EHF"
      },
      "source": [
        "## Section 1.2: Vector operations\n"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "execution": {},
        "id": "WZCSUDhJ_EHF"
      },
      "source": [
        "\n",
        "We have several mathematical operations we perform on or with vectors. **Scalar multiplication** of a vector changes its length but not its direction (except for sometimes reversing by 180 degrees as we'll see in the next demo). When we multiply a vector, $\\mathbf{x}$,  by a scalar, $a$, we multiply each individual component by the scalar:\n",
        "\n",
        "$$ a\\mathbf{x} = \\begin{bmatrix}\n",
        "    a\\mathbf{x}_1 \\\\ a\\mathbf{x}_2 \\\\ \\vdots \\\\ a\\mathbf{x}_N\n",
        "\\end{bmatrix}$$\n",
        "\n",
        "You can visualize **vector addition** as the stacking of the two arrows head to tail. In essence, you are moving the second vector so its base is at the tip of the first, without changing its direction. The tip of this stacked second vector is the new added vector's tip. In order words, if you visualize two vectors as sides of a parallelogram, adding them will create a vector with the 4th vertex of that parallelogram as its tip. Numerically, this means that we add the corresponding components of each vector:\n",
        "\n",
        "\n",
        " $$\\mathbf{x} + \\mathbf{y} = \\begin{bmatrix}\n",
        "           \\mathbf{x}_{1} + \\mathbf{y}_1 \\\\ \\mathbf{x}_{2} + \\mathbf{y}_2\\\\ \\vdots \\\\ \\mathbf{x}_{N} + \\mathbf{y}_N\n",
        "\\end{bmatrix}$$\n",
        "\n",
        "\n",
        "We call a group of 2 or more vectors a **set of vectors**. A **linear combination** of a set of vectors is a combination of the set using scalar multiplication and vector addition. Essentially, we are multiplying each vector in the set by a scalar and then adding all the scalar multiplied vectors together: the output of this (another vector) is a linear combination of the set. More formally defined, a vector, $\\mathbf{u}$, is a linear combination of a set of vectors $\\mathbf{v}^1, \\mathbf{v}^2, ..., \\mathbf{v}^N$ with (scalar) weights $c_1, c_2, ...c_N$ if:\n",
        "\n",
        "$$\\mathbf{u} = c_1\\mathbf{v}^1 + c_2\\mathbf{v}^2 + ... + c_n\\mathbf{v}^N $$.\n",
        "</details>\n"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "execution": {},
        "id": "trqBsIrE_EHF"
      },
      "source": [
        "### Interactive Demo: Linear combination of vectors\n",
        "In the following demo, we will demonstrate a linear combination of two vectors, $\\mathbf{x}$ and $\\mathbf{y}$.\n",
        "\n",
        "\\begin{equation}\n",
        "\\mathbf{z} = a\\mathbf{x} + b\\mathbf{y}\n",
        "\\end{equation}\n",
        "\n",
        "where $\\mathbf{x} = \\begin{bmatrix}3 \\\\ 1 \\end{bmatrix}$ and $\\mathbf{y} = \\begin{bmatrix}-1 \\\\ 2 \\end{bmatrix}$.\n",
        "\n",
        "You will play with the scalar multiples $a$ and $b$ to visualize both scalar multiplication and vector addition. Think and talk through the following questions:\n",
        "\n",
        "1. How does $a\\mathbf{x}$ compare to $\\mathbf{x}$ when $a$ is negative?\n",
        "2. How does $a\\mathbf{x}$ compare to $\\mathbf{x}$ when $a$ is a fraction?\n",
        "3. Can you get $\\mathbf{z}$ to point to anywhere in the 2D space with combinations of $a$ and $b$?\n",
        "4. Would this be the case no matter what $\\mathbf{x}$ and $\\mathbf{y}$ are, as long as they are both 2D vectors?"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {
        "cellView": "form",
        "execution": {},
        "id": "AkLZIg-q_EHF"
      },
      "outputs": [],
      "source": [
        " #@markdown Make sure you execute this cell to enable the widget! Move the sliders for “a” and “b”. After releasing the slider, be patient for a couple of seconds to see the desired change.\n",
        "\n",
        "\n",
        "def plot_arrows(x, y, a_times_x, b_times_y):\n",
        "  fig, ax = plt.subplots(figsize=(10, 7))\n",
        "\n",
        "  ax.spines['top'].set_color('none')\n",
        "  ax.spines['bottom'].set_position('zero')\n",
        "  ax.spines['left'].set_position('zero')\n",
        "  ax.spines['right'].set_color('none')\n",
        "\n",
        "  ax.set_aspect('equal', adjustable='box')\n",
        "  ax.set(xlim = [-10, 10], ylim = [-10, 10], xticks = np.arange(-10, 10), yticks = np.arange(-10, 10), xticklabels = [], yticklabels = [])\n",
        "\n",
        "  ax.grid(alpha=.4)\n",
        "\n",
        "  z = a_times_x + b_times_y\n",
        "  z_arr = ax.arrow(0, 0, z[0], z[1], width=.08, color='k', length_includes_head = True);\n",
        "\n",
        "  x_orig, = ax.plot([0, x[0]], [0, x[1]], '--', color='#648FFF')\n",
        "  y_orig, = ax.plot([0, y[0]], [0, y[1]], '--',  color='#DC267F')\n",
        "\n",
        "  ax_arr = ax.arrow(0, 0, a_times_x[0], a_times_x[1], width=.08, color='#648FFF', length_includes_head = True);\n",
        "  by_arr = ax.arrow(0, 0, b_times_y[0], b_times_y[1], width=.08, color='#DC267F', length_includes_head = True);\n",
        "\n",
        "  ax.plot([a_times_x[0], z[0]], [a_times_x[1], z[1]], '--k')\n",
        "  ax.plot([b_times_y[0], z[0]], [b_times_y[1], z[1]], '--k')\n",
        "\n",
        "\n",
        "  leg = ax.legend([x_orig, y_orig, ax_arr, by_arr, z_arr], [r\"$\\mathbf{x}$\", r\"$\\mathbf{y}$\", r\"$a\\mathbf{x}$\", r\"$b\\mathbf{y}$\", r\"$\\mathbf{z} = a\\mathbf{x} + b\\mathbf{y}$\"], handlelength = 2, fontsize = 25, loc = 'center left', bbox_to_anchor=(1.05, .5))\n",
        "  for handle, label in zip(leg.legendHandles, leg.texts):\n",
        "      try:\n",
        "          label.set_color(handle.get_facecolor())\n",
        "      except:\n",
        "          label.set_color(handle.get_color())\n",
        "      #handle.set_visible(False)\n",
        "\n",
        "@widgets.interact(a = widgets.FloatSlider(value=1.0, min=-2, max=2, step=0.1), b = widgets.FloatSlider(value=1.0, min=-2, max=2, step=0.1))\n",
        "def plot_linear_combination(a, b):\n",
        "  x = np.array([3, 1])\n",
        "  y = np.array([-1, 2])\n",
        "\n",
        "  plot_arrows(x, y, a*x, b*y)"
      ]
    },
    {
      "cell_type": "code",
      "source": [],
      "metadata": {
        "id": "pbLY-zEO7IH5"
      },
      "execution_count": null,
      "outputs": []
    }
  ],
  "metadata": {
    "colab": {
      "provenance": []
    },
    "kernel": {
      "display_name": "Python 3",
      "language": "python",
      "name": "python3"
    },
    "kernelspec": {
      "display_name": "Python 3",
      "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": "3.9.17"
    }
  },
  "nbformat": 4,
  "nbformat_minor": 0
}