linear_algebra/notebook/transform.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Basis Vectors Transformation\n",
    "In this example, we show how to perform transformation on basis vectors. We will show that when we change the basis vectors, the coordinates of points change as well.\n",
    "\n",
    "\n",
    "Let us start with the canonical/standard basis vectors:\n",
    "\n",
    "\\begin{equation*}\n",
    "B = \n",
    "\\begin{pmatrix}\n",
    "\\begin{bmatrix}\n",
    "1\\\\\n",
    "0\\\\\n",
    "\\end{bmatrix},\n",
    "\\begin{bmatrix}\n",
    "0\\\\\n",
    "1\\\\\n",
    "\\end{bmatrix}\n",
    "\\end{pmatrix}\n",
    "\\end{equation*}\n",
    "\n",
    "and generate grid of points from $y=-2$ to $y=1$ and from $x=-2$ to $x=1$ with respect to the canonical basis vectors."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {},
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "import matplotlib.pyplot as plt"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "metadata": {},
   "outputs": [],
   "source": [
    "x = np.arange(-2, 2, 1)\n",
    "y = np.arange(-2, 2, 1)\n",
    "xx, yy = np.meshgrid(x, y, sparse=False)\n",
    "xx = np.reshape(xx,(-1,1))\n",
    "yy = np.reshape(yy,(-1,1))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "(16, 1) (16, 1)\n"
     ]
    }
   ],
   "source": [
    "a = np.array(list(zip(xx,yy)))\n",
    "x = a[:,0]\n",
    "y = a[:,1]\n",
    "print(x.shape, y.shape)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(16, 1)"
      ]
     },
     "execution_count": 23,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "index = np.reshape(np.array([i for i in range(a.shape[0])]), (-1,1))\n",
    "index.shape"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.scatter(x, y, c=index, cmap='viridis')\n",
    "plt.savefig(\"canonical.png\")\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Rotate Transform\n",
    "Let us rotate the basis vectors by $45^{\\circ}$ or $\\frac{\\pi}{4}$ counter clockwise:\n",
    "\n",
    "\\begin{equation*}\n",
    "T = \n",
    "\\begin{bmatrix}\n",
    "\\cos\\frac{\\pi}{4} & -\\sin\\frac{\\pi}{4} \\\\\n",
    "\\sin\\frac{\\pi}{4} & \\cos\\frac{\\pi}{4}\\\\\n",
    "\\end{bmatrix}\n",
    "\\end{equation*}"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 25,
   "metadata": {},
   "outputs": [],
   "source": [
    "import math\n",
    "T = np.array([[math.cos(math.pi/4), -math.sin(math.pi/4)],[math.sin(math.pi/4), math.cos(math.pi/4)]])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 26,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[[ 0.70710678 -0.70710678]\n",
      " [ 0.70710678  0.70710678]]\n"
     ]
    }
   ],
   "source": [
    "print(T)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 27,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(2, 2)"
      ]
     },
     "execution_count": 27,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "T.shape"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 29,
   "metadata": {},
   "outputs": [],
   "source": [
    "a = np.reshape(a, (16,2))\n",
    "b = T.dot(a.T)\n",
    "b = b.T"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The last point in the canonical basis vectors has coordinates:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 30,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([1, 1])"
      ]
     },
     "execution_count": 30,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "a.T[:,-1]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "This is transformed to $\\begin{bmatrix}0.0 \\\\ 1.4\\end{bmatrix}$:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 31,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([1.11022302e-16, 1.41421356e+00])"
      ]
     },
     "execution_count": 31,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "b[-1]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 32,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "x = b[:,0]\n",
    "y = b[:,1]\n",
    "x = np.reshape(x, (-1,1))\n",
    "y = np.reshape(y, (-1,1))\n",
    "plt.scatter(x, y, c=index, cmap='viridis')\n",
    "plt.savefig(\"rotation.png\")\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We can see from the plot that the points are correspondingly rotated. For example, in the canonical basis vectors, the *yellow* point:\n",
    "\n",
    "\\begin{equation*}\n",
    "\\mathbf{p} = \n",
    "\\begin{bmatrix}\n",
    "1 \\\\\n",
    "1\n",
    "\\end{bmatrix}\n",
    "\\end{equation*}\n",
    "\n",
    "has a new set of coordinates:\n",
    "\n",
    "\\begin{equation*}\n",
    "\\mathbf{p} = \n",
    "\\begin{bmatrix}\n",
    "0.0 \\\\\n",
    "1.4\n",
    "\\end{bmatrix}\n",
    "\\end{equation*}"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Similarity Transform\n",
    "\n",
    "Let us apply similarity transform of the form:\n",
    "\n",
    "\\begin{equation*}\n",
    "T = \n",
    "\\begin{bmatrix}\n",
    "\\lambda\\cos\\theta & -\\lambda\\sin\\theta \\\\\n",
    "\\lambda\\sin\\theta & \\lambda\\cos\\theta\\\\\n",
    "\\end{bmatrix}\n",
    "\\end{equation*}\n",
    "\n",
    "Where $0<\\lambda<1.0$ and $0<\\theta<\\pi$: "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "theta: 0.9319398238849279\n",
      "lambda: 0.7479030644047094\n"
     ]
    }
   ],
   "source": [
    "lamb = np.random.rand(1)[0]\n",
    "theta = math.pi*np.random.rand(1)[0]\n",
    "T = lamb*np.array([[math.cos(theta), -math.sin(theta)],[math.sin(theta), math.cos(theta)]])\n",
    "print(\"theta:\", theta)\n",
    "print(\"lambda:\", lamb)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[[ 0.30888437 -2.09271651]\n",
      " [ 0.7548424  -1.49231629]\n",
      " [ 1.20080044 -0.89191607]\n",
      " [ 1.64675848 -0.29151585]\n",
      " [-0.29151585 -1.64675848]\n",
      " [ 0.15444218 -1.04635826]\n",
      " [ 0.60040022 -0.44595804]\n",
      " [ 1.04635826  0.15444218]\n",
      " [-0.89191607 -1.20080044]\n",
      " [-0.44595804 -0.60040022]\n",
      " [ 0.          0.        ]\n",
      " [ 0.44595804  0.60040022]\n",
      " [-1.49231629 -0.7548424 ]\n",
      " [-1.04635826 -0.15444218]\n",
      " [-0.60040022  0.44595804]\n",
      " [-0.15444218  1.04635826]]\n"
     ]
    }
   ],
   "source": [
    "a = np.reshape(a, (16,2))\n",
    "c = T.dot(a.T)\n",
    "c = c.T\n",
    "print(c)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "x = c[:,0]\n",
    "y = c[:,1]\n",
    "x = np.reshape(x, (-1,1))\n",
    "y = np.reshape(y, (-1,1))\n",
    "plt.scatter(x, y, c=index, cmap='viridis')\n",
    "plt.savefig(\"similarity.png\")\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Affine Transform\n",
    "Let us apply similarity transform of the form:\n",
    "\n",
    "\\begin{equation*}\n",
    "T = \n",
    "\\begin{bmatrix}\n",
    "\\lambda_{11} & \\lambda_{12} \\\\\n",
    "\\lambda_{21} & \\lambda_{22}\\\\\n",
    "\\end{bmatrix}\n",
    "\\end{equation*}\n",
    "\n",
    "Where $0<\\lambda<1.0$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {},
   "outputs": [],
   "source": [
    "T = np.random.rand(2,2)\n",
    "a = np.reshape(a, (16,2))\n",
    "d = T.dot(a.T)\n",
    "d = d.T"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "x = d[:,0]\n",
    "y = d[:,1]\n",
    "x = np.reshape(x, (-1,1))\n",
    "y = np.reshape(y, (-1,1))\n",
    "plt.scatter(x, y, c=index, cmap='viridis')\n",
    "plt.savefig(\"affine.png\")\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "Python 3.9.12 64-bit",
   "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": "3.9.12"
  },
  "vscode": {
   "interpreter": {
    "hash": "aee8b7b246df8f9039afb4144a1f6fd8d2ca17a180786b69acc140d282b71a49"
   }
  }
 },
 "nbformat": 4,
 "nbformat_minor": 2
}