Ch5_Linear_Transform_Ex5.1.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "id": "3b91c43e",
   "metadata": {},
   "source": [
    "<b>Question 1</b>\n",
    "\n",
    "Consider the transformation $T: \\mathbb{R}^2 \\to \\mathbb{R}^2$ given by $T(u) = Au$ where $A = \\begin{pmatrix}\n",
    "0&1\\\\\n",
    "1&0\\\\\n",
    "\\end{pmatrix}$. Determine the vector $T(u)$ where $u$ is given by:\n",
    "\n",
    "(a) $u = \\begin{pmatrix}\n",
    "1\\\\\n",
    "3\\\\\n",
    "\\end{pmatrix}\\qquad$ (b) $u = \\begin{pmatrix}\n",
    "-1\\\\\n",
    "5\\\\\n",
    "\\end{pmatrix}\\qquad$ (c) $u = \\begin{pmatrix}\n",
    "\\sqrt{2}\\\\\n",
    "1\\\\\n",
    "\\end{pmatrix}\\qquad$ (d) $u = \\begin{pmatrix}\n",
    "-2\\\\\n",
    "-3\\\\\n",
    "\\end{pmatrix}$\n",
    "\n",
    "For each of these vectos $u$ and $T(u)$ plot them on the same place $\\mathbb{R}^2$"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "8e459db8",
   "metadata": {},
   "source": [
    "<b> Solution </b> \n",
    "\n",
    "(a) $T(u) = \\begin{pmatrix}\n",
    "0&1\\\\\n",
    "1&0\\\\\n",
    "\\end{pmatrix}\\begin{pmatrix}\n",
    "1\\\\\n",
    "3\\\\\n",
    "\\end{pmatrix} = \\begin{pmatrix}\n",
    "0(1) + 1(3)\\\\\n",
    "1(1) + 0(3)\\\\\n",
    "\\end{pmatrix}$\n",
    "\n",
    "$T(u) = \\begin{pmatrix}\n",
    "3\\\\\n",
    "1\\\\\n",
    "\\end{pmatrix}$\n",
    "\n",
    "(b) $T(u) = \\begin{pmatrix}\n",
    "0&1\\\\\n",
    "1&0\\\\\n",
    "\\end{pmatrix}\\begin{pmatrix}\n",
    "-1\\\\\n",
    "5\\\\\n",
    "\\end{pmatrix} = \\begin{pmatrix}\n",
    "0(-1) + 1(5)\\\\\n",
    "1(-1) + 0(5)\\\\\n",
    "\\end{pmatrix}$\n",
    "\n",
    "$T(u) = \\begin{pmatrix}\n",
    "5\\\\\n",
    "-1\\\\\n",
    "\\end{pmatrix}$\n",
    "\n",
    "\n",
    "(c) $T(u) = \\begin{pmatrix}\n",
    "0&1\\\\\n",
    "1&0\\\\\n",
    "\\end{pmatrix}\\begin{pmatrix}\n",
    "\\sqrt{2}\\\\\n",
    "1\\\\\n",
    "\\end{pmatrix} = \\begin{pmatrix}\n",
    "0(\\sqrt{2}) + 1(1)\\\\\n",
    "1(\\sqrt{2}) + 0(1)\\\\\n",
    "\\end{pmatrix}$\n",
    "\n",
    "$T(u) = \\begin{pmatrix}\n",
    "1\\\\\n",
    "\\sqrt{2}\\\\\n",
    "\\end{pmatrix}$\n",
    "\n",
    "\n",
    "(d) $T(u) = \\begin{pmatrix}\n",
    "0&1\\\\\n",
    "1&0\\\\\n",
    "\\end{pmatrix}\\begin{pmatrix}\n",
    "-2\\\\\n",
    "-3\\\\\n",
    "\\end{pmatrix} = \\begin{pmatrix}\n",
    "0(-2) + 1(-3)\\\\\n",
    "1(-2) + 0(-3)\\\\\n",
    "\\end{pmatrix}$\n",
    "\n",
    "$T(u) = \\begin{pmatrix}\n",
    "-3\\\\\n",
    "-2\\\\\n",
    "\\end{pmatrix}$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "id": "827e8b4d",
   "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": [
    "import matplotlib.pyplot as plt\n",
    "import numpy as np\n",
    "\n",
    "def plot_vector(vector, color, label):\n",
    "    plt.quiver(0, 0, vector[0], vector[1], angles='xy', scale_units='xy', scale=1, color=color, label=label)\n",
    "\n",
    "# Define the vectors u and T(u)\n",
    "u = np.array([1, 3])\n",
    "Tu = np.array([3, 1])\n",
    "\n",
    "# Create a figure and axis\n",
    "fig, ax = plt.subplots()\n",
    "\n",
    "# Plot the vectors u and T(u)\n",
    "plot_vector(u, 'b', 'u')\n",
    "plot_vector(Tu, 'r', 'T(u)')\n",
    "\n",
    "# Set the axis limits to match the vectors\n",
    "ax.set_xlim(0, max(u[0], Tu[0]))\n",
    "ax.set_ylim(0, max(u[1], Tu[1]))\n",
    "\n",
    "ax.set_xlabel('X-axis')\n",
    "ax.set_ylabel('Y-axis')\n",
    "ax.legend()\n",
    "\n",
    "\n",
    "# Show the plot\n",
    "plt.grid()\n",
    "plt.axhline(0, color='black', linewidth=1)\n",
    "plt.axvline(0, color='black', linewidth=1)\n",
    "plt.gca().set_aspect('equal', adjustable='box')\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "id": "2afcad5c",
   "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": [
    "def plot_vector(vector, color, label):\n",
    "    plt.quiver(0, 0, vector[0], vector[1], angles='xy', scale_units='xy', scale=1, color=color, label=label)\n",
    "\n",
    "# Define the vectors u and T(u)\n",
    "u = np.array([1, np.sqrt(2)])\n",
    "Tu = np.array([np.sqrt(2), 1])\n",
    "\n",
    "# Create a figure and axis\n",
    "fig, ax = plt.subplots()\n",
    "\n",
    "# Plot the vectors u and T(u)\n",
    "plot_vector(u, 'b', 'u')\n",
    "plot_vector(Tu, 'r', 'T(u)')\n",
    "\n",
    "# Set the axis limits to match the vectors\n",
    "ax.set_xlim(0, max(u[0], Tu[0]))\n",
    "ax.set_ylim(0, max(u[1], Tu[1]))\n",
    "ax.legend()\n",
    "\n",
    "\n",
    "# Show the plot\n",
    "plt.grid()\n",
    "plt.axhline(0, color='black', linewidth=1)\n",
    "plt.axvline(0, color='black', linewidth=1)\n",
    "plt.gca().set_aspect('equal', adjustable='box')\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "id": "8dee8333",
   "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": [
    "def plot_vector(vector, color, label):\n",
    "    plt.quiver(0, 0, vector[0], vector[1], angles='xy', scale_units='xy', scale=1, color=color, label=label)\n",
    "\n",
    "# Define the vectors u and T(u)\n",
    "u = np.array([-1, 5])\n",
    "Tu = np.array([5, -1])\n",
    "\n",
    "# Create a figure and axis\n",
    "fig, ax = plt.subplots()\n",
    "\n",
    "# Plot the vectors u and T(u)\n",
    "plot_vector(u, 'b', 'u')\n",
    "plot_vector(Tu, 'r', 'T(u)')\n",
    "\n",
    "# Set the axis limits to match the vectors\n",
    "ax.set_xlim(-6, max(u[0], Tu[0]))\n",
    "ax.set_ylim(-2, max(u[1], Tu[1]))\n",
    "ax.legend()\n",
    "\n",
    "# Show the plot\n",
    "plt.grid()\n",
    "plt.axhline(0, color='black', linewidth=1)\n",
    "plt.axvline(0, color='black', linewidth=1)\n",
    "plt.gca().set_aspect('equal', adjustable='box')\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "22d170b3",
   "metadata": {},
   "source": [
    "-------------------------"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "e6eeecc1",
   "metadata": {},
   "source": [
    "<b> Question 2 </b>\n",
    "\n",
    "Consider the transformation $T: \\mathbb{R}^3 \\to \\mathbb{R}^2$ given by $\\begin{pmatrix}\n",
    "\\begin{bmatrix}\n",
    "x\\\\\n",
    "y\\\\\n",
    "z\\\\\n",
    "\\end{bmatrix}\n",
    "\\end{pmatrix} = \\begin{pmatrix}\n",
    "xy\\\\\n",
    "xz\\\\\n",
    "\\end{pmatrix}\n",
    "$. Determine the vector $T(u)$ where $u$ is given by:\n",
    "\n",
    "(a)$u = \\begin{pmatrix}\n",
    "1\\\\\n",
    "3\\\\\n",
    "5\\\\\n",
    "\\end{pmatrix}\\qquad$ (b)$u = \\begin{pmatrix}\n",
    "-1\\\\\n",
    "-4\\\\\n",
    "4\\\\\n",
    "\\end{pmatrix}\\qquad$ (c)$u = \\begin{pmatrix}\n",
    "\\sqrt{2}\\\\\n",
    "\\sqrt{8}\\\\\n",
    "\\sqrt{18}\\\\\n",
    "\\end{pmatrix}\\qquad$ (d) $u = \\begin{pmatrix}\n",
    "-2\\\\\n",
    "-3\\\\\n",
    "-4\\\\\n",
    "\\end{pmatrix}$"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "3532d8fe",
   "metadata": {},
   "source": [
    "<b> Solution </b> \n",
    "\n",
    "(a) Here, $x = 1, y = 3$ and $z = 5$\n",
    "\n",
    "Therefore, with $T: \\mathbb{R}^3 \\to \\mathbb{R}^2$ given by $\\begin{pmatrix}\n",
    "\\begin{bmatrix}\n",
    "x\\\\\n",
    "y\\\\\n",
    "z\\\\\n",
    "\\end{bmatrix}\n",
    "\\end{pmatrix} = \\begin{pmatrix}\n",
    "xy\\\\\n",
    "xz\\\\\n",
    "\\end{pmatrix}$\n",
    "\n",
    "we have $T(u) = \\begin{pmatrix}\n",
    "1\\times 3\\\\\n",
    "1 \\times 5\\\\\n",
    "\\end{pmatrix} = \\begin{pmatrix}\n",
    "3\\\\\n",
    "5\\\\\n",
    "\\end{pmatrix}$\n",
    "\n",
    "\n",
    "\n",
    "(b) Here, $x = -1, y = -4$ and $z = 4$\n",
    "\n",
    "Therefore, with $T: \\mathbb{R}^3 \\to \\mathbb{R}^2$ given by $\\begin{pmatrix}\n",
    "\\begin{bmatrix}\n",
    "x\\\\\n",
    "y\\\\\n",
    "z\\\\\n",
    "\\end{bmatrix}\n",
    "\\end{pmatrix} = \\begin{pmatrix}\n",
    "xy\\\\\n",
    "xz\\\\\n",
    "\\end{pmatrix}$\n",
    "\n",
    "we have $T(u) = \\begin{pmatrix}\n",
    "-1\\times (-4)\\\\\n",
    "-1 \\times 4\\\\\n",
    "\\end{pmatrix} = \\begin{pmatrix}\n",
    "4\\\\\n",
    "-4\\\\\n",
    "\\end{pmatrix}$\n",
    "\n",
    "\n",
    "(c) Here, $x = \\sqrt{2}, y = \\sqrt{8}$ and $z = \\sqrt{18}$\n",
    "\n",
    "Therefore, with $T: \\mathbb{R}^3 \\to \\mathbb{R}^2$ given by $\\begin{pmatrix}\n",
    "\\begin{bmatrix}\n",
    "x\\\\\n",
    "y\\\\\n",
    "z\\\\\n",
    "\\end{bmatrix}\n",
    "\\end{pmatrix} = \\begin{pmatrix}\n",
    "xy\\\\\n",
    "xz\\\\\n",
    "\\end{pmatrix}$\n",
    "\n",
    "we have $T(u) = \\begin{pmatrix}\n",
    "\\sqrt{2}\\times \\sqrt{8}\\\\\n",
    "\\sqrt{2} \\times \\sqrt{18}\\\\\n",
    "\\end{pmatrix} = \\begin{pmatrix}\n",
    "\\sqrt{16}\\\\\n",
    "\\sqrt{36}\\\\\n",
    "\\end{pmatrix} = \\begin{pmatrix}\n",
    "4\\\\\n",
    "6\\\\\n",
    "\\end{pmatrix}$\n",
    "\n",
    "\n",
    "(d) Here, $x = -2, y = -3$ and $z = -4$\n",
    "\n",
    "Therefore, with $T: \\mathbb{R}^3 \\to \\mathbb{R}^2$ given by $\\begin{pmatrix}\n",
    "\\begin{bmatrix}\n",
    "x\\\\\n",
    "y\\\\\n",
    "z\\\\\n",
    "\\end{bmatrix}\n",
    "\\end{pmatrix} = \\begin{pmatrix}\n",
    "xy\\\\\n",
    "xz\\\\\n",
    "\\end{pmatrix}$\n",
    "\n",
    "we have $T(u) = \\begin{pmatrix}\n",
    "-2\\times (-3)\\\\\n",
    "-2\\times (-4)\\\\\n",
    "\\end{pmatrix} = \\begin{pmatrix}\n",
    "6\\\\\n",
    "8\\\\\n",
    "\\end{pmatrix}$"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "3914af4e",
   "metadata": {},
   "source": [
    "---------------------------------------------------------"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "f2de07a2",
   "metadata": {},
   "source": [
    "<b>Question 3 </b>\n",
    "\n",
    "Let $T: \\mathbb{R}^3 \\to \\mathbb{R}^3$ be given by $\\begin{pmatrix}\n",
    "\\begin{bmatrix}\n",
    "x\\\\\n",
    "y\\\\\n",
    "z\\\\\n",
    "\\end{bmatrix}\n",
    "\\end{pmatrix} = \\begin{pmatrix}\n",
    "x+y\\\\\n",
    "y+z\\\\\n",
    "z+x\\\\\n",
    "\\end{pmatrix}$. Find $T(u)$ where\n",
    "\n",
    "(a) $u = \\begin{pmatrix}\n",
    "2\\\\\n",
    "4\\\\\n",
    "7\\\\\n",
    "\\end{pmatrix}\\qquad$ (b) $u = \\begin{pmatrix}\n",
    "-3\\\\\n",
    "8\\\\\n",
    "-6\\\\\n",
    "\\end{pmatrix}\\qquad$ (c) $u = \\begin{pmatrix}\n",
    "\\pi \\\\\n",
    "2 \\pi \\\\\n",
    "5 \\pi \\\\\n",
    "\\end{pmatrix}\\qquad$ (d) $u = \\begin{pmatrix}\n",
    "1/2 \\\\\n",
    "2/3 \\\\\n",
    "3/4 \\\\\n",
    "\\end{pmatrix}$"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "a412a355",
   "metadata": {},
   "source": [
    "<b> Solution </b>\n",
    "\n",
    "(a) Here, $x = 2, y = 4$ and $z = 7$\n",
    "\n",
    "Therefore, with $T: \\mathbb{R}^3 \\to \\mathbb{R}^2$ given by $\\begin{pmatrix}\n",
    "\\begin{bmatrix}\n",
    "x\\\\\n",
    "y\\\\\n",
    "z\\\\\n",
    "\\end{bmatrix}\n",
    "\\end{pmatrix} = \\begin{pmatrix}\n",
    "x+y\\\\\n",
    "y+z\\\\\n",
    "z + x\\\\\n",
    "\\end{pmatrix}$\n",
    "\n",
    "we have $T(u) = \\begin{pmatrix}\n",
    "2 + 4\\\\\n",
    "4 + 7\\\\\n",
    "7 + 2\\\\\n",
    "\\end{pmatrix} = \\begin{pmatrix}\n",
    "6\\\\\n",
    "11\\\\\n",
    "9\\\\\n",
    "\\end{pmatrix}$\n",
    "\n",
    "\n",
    "(b) Here, $x = -3, y = 8$ and $z = -6$\n",
    "\n",
    "Therefore, with $T: \\mathbb{R}^3 \\to \\mathbb{R}^2$ given by $\\begin{pmatrix}\n",
    "\\begin{bmatrix}\n",
    "x\\\\\n",
    "y\\\\\n",
    "z\\\\\n",
    "\\end{bmatrix}\n",
    "\\end{pmatrix} = \\begin{pmatrix}\n",
    "x + y\\\\\n",
    "y + z\\\\\n",
    "z + x\\\\\n",
    "\\end{pmatrix}$\n",
    "\n",
    "we have $T(u) = \\begin{pmatrix}\n",
    "-3 + 8\\\\\n",
    "8 + (-6)\\\\\n",
    "-6 + (-3)\\\\\n",
    "\\end{pmatrix} = \\begin{pmatrix}\n",
    "5\\\\\n",
    "2\\\\\n",
    "-9\\\\\n",
    "\\end{pmatrix}$\n",
    "\n",
    "\n",
    "(c) Here, $x = \\pi, y = 2\\pi$ and $z = 5\\pi$\n",
    "\n",
    "Therefore, with $T: \\mathbb{R}^3 \\to \\mathbb{R}^2$ given by $\\begin{pmatrix}\n",
    "\\begin{bmatrix}\n",
    "x\\\\\n",
    "y\\\\\n",
    "z\\\\\n",
    "\\end{bmatrix}\n",
    "\\end{pmatrix} = \\begin{pmatrix}\n",
    "x + y\\\\\n",
    "y + z\\\\\n",
    "z + x\\\\\n",
    "\\end{pmatrix}$\n",
    "\n",
    "we have $T(u) = \\begin{pmatrix}\n",
    "\\pi + 2\\pi \\\\\n",
    "2\\pi + 5\\pi \\\\\n",
    "5\\pi + \\pi \\\\\n",
    "\\end{pmatrix} = \\begin{pmatrix}\n",
    "3\\pi \\\\\n",
    "7\\pi \\\\\n",
    "6\\pi \\\\\n",
    "\\end{pmatrix}$\n",
    "\n",
    "\n",
    "(d) Here, $x = 1/2, y = 2/3$ and $z = 3/4$\n",
    "\n",
    "Therefore, with $T: \\mathbb{R}^3 \\to \\mathbb{R}^2$ given by $\\begin{pmatrix}\n",
    "\\begin{bmatrix}\n",
    "x\\\\\n",
    "y\\\\\n",
    "z\\\\\n",
    "\\end{bmatrix}\n",
    "\\end{pmatrix} = \\begin{pmatrix}\n",
    "x + y\\\\\n",
    "y + z\\\\\n",
    "z + x\\\\\n",
    "\\end{pmatrix}$\n",
    "\n",
    "we have $T(u) = \\begin{pmatrix}\n",
    "1/2 + 2/3 \\\\\n",
    "2/3 + 3/4 \\\\\n",
    "3/4 + 1/2 \\\\\n",
    "\\end{pmatrix} = \\begin{pmatrix}\n",
    "7/6 \\\\\n",
    "17/12 \\\\\n",
    "5/4 \\\\\n",
    "\\end{pmatrix}$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "d37f6f62",
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "markdown",
   "id": "a5df2af0",
   "metadata": {},
   "source": [
    "-------------------------------------"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "dbc085d7",
   "metadata": {},
   "source": [
    "<b> Question 4 </b>\n",
    "\n",
    "Explain why $T: U \\to V$ given by $T(u) = \\pm \\sqrt{u}$ is not a transformation."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "8b378b9e",
   "metadata": {},
   "source": [
    "<b> Solution </b> \n",
    "\n",
    "For $T$ to be a transformation, it has to have unique output.\n",
    "\n",
    "Given $T(u) = \\pm \\sqrt{u}$ will take vector in two different direction. Therefore, $T(u)$ is not a transformation."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "76591988",
   "metadata": {},
   "source": [
    "-------------------------------------"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "2fe539b5",
   "metadata": {},
   "source": [
    "<b> Question 5 </b> \n",
    "\n",
    "Determine whether the following transformations (mapping) are linear:\n",
    "\n",
    "(a) $T: \\mathbb{R}^2 \\to \\mathbb{R}^2$ given by$\\qquad$ (i)$T\\begin{pmatrix}\\begin{bmatrix}\n",
    "x\\\\\n",
    "y\\\\\n",
    "\\end{bmatrix}\n",
    "\\end{pmatrix} = \\begin{pmatrix}\n",
    "y\\\\\n",
    "x\\\\\n",
    "\\end{pmatrix}\\qquad$ (ii)$T\\begin{pmatrix}\\begin{bmatrix}\n",
    "x\\\\\n",
    "y\\\\\n",
    "\\end{bmatrix}\n",
    "\\end{pmatrix} = \\begin{pmatrix}\n",
    "x^2\\\\\n",
    "y^2\\\\\n",
    "\\end{pmatrix}\\qquad$\n",
    "\n",
    "\n",
    "(b) $T: \\mathbb{R}^3 \\to \\mathbb{R}^2$ given by$\\qquad T\\begin{pmatrix}\\begin{bmatrix}\n",
    "x\\\\\n",
    "y\\\\\n",
    "z\\\\\n",
    "\\end{bmatrix}\n",
    "\\end{pmatrix} = \\begin{pmatrix}\n",
    "xy\\\\\n",
    "xz\\\\\n",
    "\\end{pmatrix}\\qquad$\n",
    "\n",
    "\n",
    "(c) $T: \\mathbb{R}^3 \\to \\mathbb{R}^3$ given by$\\qquad$ (i)$T\\begin{pmatrix}\\begin{bmatrix}\n",
    "x\\\\\n",
    "y\\\\\n",
    "z\\\\\n",
    "\\end{bmatrix}\n",
    "\\end{pmatrix} = \\begin{pmatrix}\n",
    "x+y\\\\\n",
    "y+z\\\\\n",
    "z+x\\\\\n",
    "\\end{pmatrix}\\qquad$ (ii)$T\\begin{pmatrix}\\begin{bmatrix}\n",
    "x\\\\\n",
    "y\\\\\n",
    "z\\\\\n",
    "\\end{bmatrix}\n",
    "\\end{pmatrix} = \\begin{pmatrix}\n",
    "\\sqrt{x}\\\\\n",
    "\\sqrt{y}\\\\\n",
    "\\sqrt{z}\\\\\n",
    "\\end{pmatrix}\\qquad$"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "a3a0ec5d",
   "metadata": {},
   "source": [
    "<b> Solution </b> \n",
    "\n",
    "We know that, <br>\n",
    "A transformation $T: V \\to W$ is called a linear transformation $\\iff$ for all vector $u$ and $v$ in the vector space $V$ and for any scalar $k$, we have <br>\n",
    "\n",
    "(a)$T(u+v) = T(u) + T(v)$<br>\n",
    "(b)$T(ku) = kT(u)$\n",
    "\n",
    "\n",
    "(a)(i)<br> <b>First Condition</b> <br>\n",
    "\n",
    "Let $u = \\begin{pmatrix}\n",
    "x\\\\\n",
    "y\\\\\n",
    "\\end{pmatrix},\\quad v = \\begin{pmatrix}\n",
    "s\\\\\n",
    "t\\\\\n",
    "\\end{pmatrix}$\n",
    "\n",
    "Then, $T(u + v) = T\\begin{pmatrix}\n",
    "x +s\\\\\n",
    "y + t\\\\\n",
    "\\end{pmatrix}$\n",
    "\n",
    "And, as transformation is $T\\begin{pmatrix}\\begin{bmatrix}\n",
    "x\\\\\n",
    "y\\\\\n",
    "\\end{bmatrix}\\end{pmatrix} = \\begin{pmatrix}\n",
    "y\\\\\n",
    "x\\\\\n",
    "\\end{pmatrix}$\n",
    "\n",
    "$T(u + v) = \\begin{pmatrix}\n",
    "y + t\\\\\n",
    "x + s\\\\\n",
    "\\end{pmatrix}$\n",
    "\n",
    "Then, $T(u) + T(v) = T\\begin{pmatrix}\n",
    "x\\\\\n",
    "y\\\\\n",
    "\\end{pmatrix} + T\\begin{pmatrix}\n",
    "s\\\\\n",
    "t\\\\\n",
    "\\end{pmatrix}$\n",
    "\n",
    "After, applying transformation, <br>\n",
    "\n",
    "$T(u) + T(v) = \\begin{pmatrix}\n",
    "y\\\\\n",
    "x\\\\\n",
    "\\end{pmatrix} + \\begin{pmatrix}\n",
    "t\\\\\n",
    "s\\\\\n",
    "\\end{pmatrix}$\n",
    "\n",
    "$T(u) + T(v) = \\begin{pmatrix}\n",
    "y + t\\\\\n",
    "x + s\\\\\n",
    "\\end{pmatrix}$\n",
    "\n",
    "Hence, first condition of $T(u + v) = T(u) + T(v)$ satisfied\n",
    "\n",
    "(b)<br>\n",
    "\n",
    "<b>Second Condition</b>\n",
    "\n",
    "$T(ku) = T\\begin{pmatrix}\n",
    "kx\\\\\n",
    "ky\\\\\n",
    "\\end{pmatrix}$\n",
    "\n",
    "And, after applying transformation, $T(ku) = \\begin{pmatrix}\n",
    "ky\\\\\n",
    "kx\\\\\n",
    "\\end{pmatrix}$\n",
    "\n",
    "$\\Rightarrow T(ku) = k\\begin{pmatrix}\n",
    "y\\\\\n",
    "x\\\\\n",
    "\\end{pmatrix}$\n",
    "\n",
    "$\\Rightarrow T(ku) = kT(u)$\n",
    "\n",
    "hence, both condition satisfied for this transformation. Therefore, it is a <b>Linear Transformation</b>\n",
    "\n",
    "\n",
    "(a)(ii)<br>\n",
    "\n",
    "Let $u = \\begin{pmatrix}\n",
    "x\\\\\n",
    "y\\\\\n",
    "\\end{pmatrix}, v = \\begin{pmatrix}\n",
    "s\\\\\n",
    "t\\\\\n",
    "\\end{pmatrix}$\n",
    "\n",
    "Then, $T(u + v) = T\\begin{pmatrix}\n",
    "x+s\\\\\n",
    "y+t\\\\\n",
    "\\end{pmatrix}$\n",
    "\n",
    "$\\Rightarrow T(u + v) = \\begin{pmatrix}\n",
    "(x+s)^2\\\\\n",
    "(y+t)^2\\\\\n",
    "\\end{pmatrix} = \\begin{pmatrix}\n",
    "x^2 + 2xs + s^2\\\\\n",
    "y^2 + 2yt + t^2\\\\\n",
    "\\end{pmatrix}$\n",
    "\n",
    "On the other hand,<br> $T(u) + T(v)  = T\\begin{pmatrix}\n",
    "x\\\\\n",
    "y\\\\\n",
    "\\end{pmatrix} + T\\begin{pmatrix}\n",
    "s\\\\\n",
    "t\\\\\n",
    "\\end{pmatrix}$\n",
    "\n",
    "$\\Rightarrow \\begin{pmatrix}\n",
    "x^2\\\\\n",
    "y^2\\\\\n",
    "\\end{pmatrix} + \\begin{pmatrix}\n",
    "s^2\\\\\n",
    "t^2\\\\\n",
    "\\end{pmatrix} = \\begin{pmatrix}\n",
    "x^2 + s^2\\\\\n",
    "y^2 + t^2\\\\\n",
    "\\end{pmatrix}$\n",
    "\n",
    "$\\Rightarrow T(u + v) \\ne T(u) + T(v)$\n",
    "\n",
    "Hence, <b>not a linear transformation</b>\n",
    "\n",
    "\n",
    "<b>Solution (b)</b>\n",
    "\n",
    "Let $u = \\begin{pmatrix}\n",
    "x\\\\\n",
    "y\\\\\n",
    "z\\\\\n",
    "\\end{pmatrix}, v = \\begin{pmatrix}\n",
    "r\\\\\n",
    "s\\\\\n",
    "t\\\\\n",
    "\\end{pmatrix}$\n",
    "\n",
    "Then, $T(u + v) = T\\begin{pmatrix}\n",
    "x+r\\\\\n",
    "y+s\\\\\n",
    "z+t\\\\\n",
    "\\end{pmatrix}$ \n",
    "\n",
    "And, after applying transformation, it is \n",
    "\n",
    "$T(u+v) = \\begin{pmatrix}\n",
    "(x+r)(y+s)\\\\\n",
    "(x+r)(z+t)\\\\\n",
    "\\end{pmatrix} = \\begin{pmatrix}\n",
    "xy + xs + ry + rs\\\\\n",
    "xz + xt + rz + rt\\\\\n",
    "\\end{pmatrix}$\n",
    "\n",
    "Next, <br>\n",
    "\n",
    "$T(u) + T(v) = T\\begin{pmatrix}\n",
    "x\\\\\n",
    "y\\\\\n",
    "z\\\\\n",
    "\\end{pmatrix} + T\\begin{pmatrix}\n",
    "r\\\\\n",
    "s\\\\\n",
    "t\\\\\n",
    "\\end{pmatrix}$\n",
    "\n",
    "\n",
    "$T(u) + T(v) = \\begin{pmatrix}\n",
    "xy\\\\\n",
    "xz\\\\\n",
    "\\end{pmatrix} + \\begin{pmatrix}\n",
    "rs\\\\\n",
    "rt\\\\\n",
    "\\end{pmatrix} = \\begin{pmatrix}\n",
    "xy + rs\\\\\n",
    "xz + rt\\\\\n",
    "\\end{pmatrix}$\n",
    "\n",
    "We get $T(u + v) \\ne T(u) + T(v)$<br>\n",
    "Hence, <b>not a linear transformation</b>\n",
    "\n",
    "<b>Solution (c)</b>\n",
    "\n",
    "Let $u = \\begin{pmatrix}\n",
    "x\\\\\n",
    "y\\\\\n",
    "z\\\\\n",
    "\\end{pmatrix}, v = \\begin{pmatrix}\n",
    "r\\\\\n",
    "s\\\\\n",
    "t\\\\\n",
    "\\end{pmatrix}$\n",
    "\n",
    "$T(u+v) = T\\begin{pmatrix}\n",
    "x + r\\\\\n",
    "y + s\\\\\n",
    "z + t\\\\\n",
    "\\end{pmatrix}$\n",
    "\n",
    "and, after applying transformation $T(u+v) = \\begin{pmatrix}\n",
    "x + r + y + s\\\\\n",
    "y + s + z + t\\\\\n",
    "z + t + x + r\\\\\n",
    "\\end{pmatrix}$\n",
    "\n",
    "Next, <br>\n",
    "\n",
    "$T(u) + T(v) = T\\begin{pmatrix}\n",
    "x\\\\\n",
    "y\\\\\n",
    "z\\\\\n",
    "\\end{pmatrix} + T\\begin{pmatrix}\n",
    "r\\\\\n",
    "s\\\\\n",
    "t\\\\\n",
    "\\end{pmatrix}$ <br>And, after applying transformation\n",
    "\n",
    "$T(u) + T(v) = \\begin{pmatrix}\n",
    "x+y\\\\\n",
    "y+z\\\\\n",
    "z+x\\\\\n",
    "\\end{pmatrix} + \\begin{pmatrix}\n",
    "r+s\\\\\n",
    "s+t\\\\\n",
    "t+r\\\\\n",
    "\\end{pmatrix} = \\begin{pmatrix}\n",
    "x+ y+ r+ s\\\\\n",
    "y+z+s+t\\\\\n",
    "z+x+t+r\\\\\n",
    "\\end{pmatrix}$\n",
    "\n",
    "$\\Rightarrow T(u+v) = T(u) + T(v)$\n",
    "\n",
    "\n",
    "Next, <br>\n",
    "Second condition: <br>\n",
    "\n",
    "$T(ku) = T\\begin{pmatrix}\n",
    "kx\\\\\n",
    "ky\\\\\n",
    "kz\\\\\n",
    "\\end{pmatrix}$ \n",
    "\n",
    "$\\Rightarrow T(ku) = \\begin{pmatrix}\n",
    "kx + ky\\\\\n",
    "ky + kz\\\\\n",
    "kz + kx\\\\\n",
    "\\end{pmatrix} = \\begin{pmatrix}\n",
    "k(x+y)\\\\\n",
    "k(y+z)\\\\\n",
    "k(z+x)\\\\\n",
    "\\end{pmatrix} = k \\begin{pmatrix}\n",
    "x+y\\\\\n",
    "y+z\\\\\n",
    "z+x\\\\\n",
    "\\end{pmatrix} = kT(u)$\n",
    "\n",
    "Thus, both condition satisfied, and <b>it is a linear transformation</b>\n",
    "\n",
    "<b>Solution (c)(ii)</b>\n",
    "\n",
    "Let $u = \\begin{pmatrix}\n",
    "x\\\\\n",
    "y\\\\\n",
    "z\\\\\n",
    "\\end{pmatrix}, v = \\begin{pmatrix}\n",
    "r\\\\\n",
    "s\\\\\n",
    "t\\\\\n",
    "\\end{pmatrix}$\n",
    "\n",
    "Then, $T(u + v) = T\\begin{pmatrix}\n",
    "x+r\\\\\n",
    "y+s\\\\\n",
    "z+t\\\\\n",
    "\\end{pmatrix}$\n",
    "And, after applying transformation, <br>\n",
    "\n",
    "$T(u+v) = \\begin{pmatrix}\n",
    "\\sqrt{x+r}\\\\\n",
    "\\sqrt{y+s}\\\\\n",
    "\\sqrt{z+t}\\\\\n",
    "\\end{pmatrix}$\n",
    "\n",
    "Next, <br>\n",
    "\n",
    "$T(u) + T(v) = T\\begin{pmatrix}\n",
    "x\\\\\n",
    "y\\\\\n",
    "z\\\\\n",
    "\\end{pmatrix} + T\\begin{pmatrix}\n",
    "r\\\\\n",
    "s\\\\\n",
    "t\\\\\n",
    "\\end{pmatrix}$\n",
    "\n",
    "After applying transformation, <br>\n",
    "$T(u) + T(v) = \\begin{pmatrix}\n",
    "\\sqrt{x}\\\\\n",
    "\\sqrt{y}\\\\\n",
    "\\sqrt{z}\\\\\n",
    "\\end{pmatrix} + \\begin{pmatrix}\n",
    "\\sqrt{r}\\\\\n",
    "\\sqrt{s}\\\\\n",
    "\\sqrt{t}\\\\\n",
    "\\end{pmatrix}$\n",
    "\n",
    "$T(u) + T(v) = \\begin{pmatrix}\n",
    "\\sqrt{x} + \\sqrt{r}\\\\\n",
    "\\sqrt{y} + \\sqrt{s}\\\\\n",
    "\\sqrt{z} + \\sqrt{t}\\\\\n",
    "\\end{pmatrix}$\n",
    "\n",
    "$\\Rightarrow T(u+v) \\ne T(u) + T(v)$\n",
    "\n",
    "Hence, <b>not a linear transformation</b>"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "800a5c42",
   "metadata": {},
   "source": [
    "------------------------------------"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "9d7acd26",
   "metadata": {},
   "source": [
    "<b> Question 6 </b> \n",
    "\n",
    "Let $P_2$ be the vector space of polynomials of degree 2 or less. Decide whether the following transformations are linear:\n",
    "\n",
    "(a)$T: P_2 \\to P_2$ given by $T(c_2x^2 + c_1x + c_0) = c_0x^2 + c_1x + c_2$<br>\n",
    "\n",
    "(b)$T: P_2 \\to P_2$ given by $T(c_2x^2 + c_1x + c_0) = c_2^{2}x^2 + c_1^{2}x + c_0^{2}$"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "43056a89",
   "metadata": {},
   "source": [
    "<b> Solution (a) </b> \n",
    "\n",
    "Let $u = a_2x^2 = a_1x + a_0$<br>\n",
    "and, $v = b_2x^2 + b_1x + b_0$\n",
    "\n",
    "Then, <br>\n",
    "\n",
    "$T(u+v) = T(a_2x^2 + a_1x + a_0 + b_2x^2 + b_1x + b_0)$\n",
    "\n",
    "$\\Rightarrow T(u+v) = T\\{ (a_2 + b_2)x^2 + (a_1 + b_1)x + (a_0 + b_0)\\}$\n",
    "\n",
    "And, after applying transformation, <br>\n",
    "\n",
    "$T(u+v) = (a_0 + b_0)x^2 + (a_1 + b_1)x + (a_2 + b_2)$\n",
    "\n",
    "Next, <br>\n",
    "\n",
    "$T(u) + T(v) = T(a_2x^2 + a_1x + a_0) + T(b_2x^2 + b_1x + b_0)$\n",
    "\n",
    "After applying transformation, we have<br>\n",
    "\n",
    "$T(u) + T(v) = a_0x^2 + a_1x + a_2 + b_0x^2 + b_1x + b_2$\n",
    "\n",
    "$\\Rightarrow T(u) + T(v) = (a_0 + b_0)x^2 + (a_1 + b_1)x + (a_2 + b_2)$\n",
    "\n",
    "Hence, first condition satisfied i.e,  $T(u + v) = T(u) + T(v)$\n",
    "\n",
    "Second, <br>\n",
    "\n",
    "$T(ku) = T(k(a_2x^2 + a_1x + a_0))$<br>\n",
    "$\\quad = T(ka_2x^2 + ka_1x + ka_0)$<br>\n",
    "$\\quad = ka_0x^2 + ka_1x + ka_2\\qquad$ (after transformation)\n",
    "\n",
    "$\\quad = k(a_0x^2 + a_1x + a_2) = kT(u)$\n",
    "\n",
    "Now since both condition satisfied it is a linear transformation.\n",
    "\n",
    "\n",
    "<b> Solution (b)</b>\n",
    "\n",
    "Checking $T(Ku) = kT(u)$ first, \n",
    "\n",
    "If we have $u = a_2x^2 + a_1x + a_0$, then<br>\n",
    "\n",
    "$T(ku) = T(k(a_2x^2 + a_1x + a_0))$<br>\n",
    "$\\Rightarrow T(ku) = T(ka_2x^2 + ka_1x + ka_0)$\n",
    "\n",
    "WE have $T: P_2 \\to P_2$ given by $T(c_2x^2 + c_1x + c_0) = c_2^{2}x^2 + c_1^{2}x + c_0^{2}$\n",
    "\n",
    "Then, after transformation, <br>\n",
    "\n",
    "$T(ku) = (ka_2)^2x^2 + (ka_1)x + (ka_0)^2$\n",
    "\n",
    "$\\Rightarrow T(ku) = k^2a_2^2x^2 + k^2a_1^2x + k^2a_0^2$\n",
    "\n",
    "Next, <br>\n",
    "$kT(u) = kT(a_2x^2 + a_1x + a_0)$<br>\n",
    "\n",
    "After transformation, \n",
    "\n",
    "$\\Rightarrow kT(u) = k(a_2^2x^2 + a_1^2x + a_0) = ka_2x^2 + ka_1^2x + ka_0^2$\n",
    "\n",
    "Now, we have $T(ku) \\ne kT(u)$\n",
    "\n",
    "Therefore, not a linear transformation"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "d273c968",
   "metadata": {},
   "source": [
    "---------------------------------"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "348dd2ed",
   "metadata": {},
   "source": [
    "<b> Question 7 </b> \n",
    "\n",
    "Let $M_{nn}$ be the vector space of size $n$ by $n$ matrices. Decide whether the following transformations (mapping) are linear:\n",
    "\n",
    "(a) $T: M_{22} \\to M_{22}$ given by $T(A) = A^T$ where $A^T$ is the transpose of the matrix $A$. <br>\n",
    "\n",
    "(b) $T: M_{22} \\to \\mathbb{R}$ given by $T(A) = tr(A)$ is the trace of the matrix $A$\n",
    "\n",
    "(c) $T: M_{nn} \\to \\mathbb{R}$ given by $T(A) = a_{11}a_{22}a_{33}\\cdots a_{nn}$ where $a_{jj}$ are the entries along the leading diagonal of a $n$ by $n$ matrix."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "fcd301dc",
   "metadata": {},
   "source": [
    "We know that, <br>\n",
    "A transformation $T: V \\to W$ is called a linear transformation $\\iff$ for all vector $u$ and $v$ in the vector space $V$ and for any scalar $k$, we have <br>\n",
    "\n",
    "(a)$T(u+v) = T(u) + T(v)$<br>\n",
    "(b)$T(ku) = kT(u)$\n",
    "\n",
    "<b> Solution (a)</b> \n",
    "\n",
    "Let $u = \\begin{pmatrix}\n",
    "a_{11}&a_{12}\\\\\n",
    "a_{21}&a_{22}\\\\\n",
    "\\end{pmatrix}$ and $v = \\begin{pmatrix}\n",
    "b_{11}&b_{12}\\\\\n",
    "b_{21}&b_{22}\\\\\n",
    "\\end{pmatrix}$\n",
    "\n",
    "$T(u + v) = T \\begin{pmatrix}\n",
    "\\begin{pmatrix}\n",
    "a_{11}&a_{12}\\\\\n",
    "a_{21}&a_{22}\\\\\n",
    "\\end{pmatrix} + \\begin{pmatrix}\n",
    "b_{11}&b_{12}\\\\\n",
    "b_{21}&b_{22}\\\\\n",
    "\\end{pmatrix}\\end{pmatrix}$\n",
    "\n",
    "$\\quad = T\\begin{pmatrix}\n",
    "a_{11} + b_{11} & a_{12} + b_{12}\\\\\n",
    "a_{21} + b_{21} & a_{22} + b_{22}\\\\\n",
    "\\end{pmatrix}$\n",
    "\n",
    "After, applying transpose as transformation, we have\n",
    "\n",
    "$\\Rightarrow T(u + v) = \\begin{pmatrix}\n",
    "a_{11} + b_{11} & a_{21} + b_{21}\\\\\n",
    "a_{12} + b_{12} & a_{22} + b_{22}\\\\\n",
    "\\end{pmatrix}$\n",
    "\n",
    "\n",
    "Next, <br>\n",
    "\n",
    "$T(u) + T(v) = T\\begin{pmatrix}\n",
    "a_{11}&a_{12}\\\\\n",
    "a_{21}&a_{22}\\\\\n",
    "\\end{pmatrix} + T\\begin{pmatrix}\n",
    "b_{11}&b_{12}\\\\\n",
    "b_{21}&b_{22}\\\\\n",
    "\\end{pmatrix}$\n",
    "\n",
    "After transpose as transformation, we have <br>\n",
    "$\\Rightarrow T(u) + T(v) = \\begin{pmatrix}\n",
    "a_{11} & a_{21}\\\\\n",
    "a_{12} & a_{22}\\\\\n",
    "\\end{pmatrix} + \\begin{pmatrix}\n",
    "b_{11} & b_{21}\\\\\n",
    "b_{12} & b_{22}\\\\\n",
    "\\end{pmatrix}$\n",
    "\n",
    "$T(u) + T(v) = \\begin{pmatrix}\n",
    "a_{11}+b_{11}&a_{21}+b_{21}\\\\\n",
    "a_{12}+b_{12} & a_{22} + b_{22}\\\\\n",
    "\\end{pmatrix}$\n",
    "\n",
    "<b> Solution (c)</b>\n",
    "\n",
    "Checking second condition first, \n",
    "\n",
    "Let $A = \\begin{pmatrix}\n",
    "a_{11} \\cdots a_{1n}\\\\\n",
    "\\vdots \\ddots \\vdots\\\\\n",
    "a_{n1} \\cdots a_{nn}\\\\\n",
    "\\end{pmatrix}$\n",
    "\n",
    "$\\Rightarrow T(K(A)) = T\\begin{pmatrix} k \\begin{pmatrix}\n",
    "a_{11} \\cdots a_{1n}\\\\\n",
    "\\vdots \\ddots \\vdots\\\\\n",
    "a_{n1} \\cdots a_{nn}\\\\\n",
    "\\end{pmatrix}\\end{pmatrix}$\n",
    "\n",
    "$\\quad = T\\begin{pmatrix}\n",
    "ka_{11} \\cdots ka_{1n}\\\\\n",
    "\\vdots \\ddots \\vdots\\\\\n",
    "ka_{n1} \\cdots ka_{nn}\\\\\n",
    "\\end{pmatrix}$\n",
    "\n",
    "After transformation, we get:<br>\n",
    "\n",
    "$T(k(A)) = ka_{11}ka_{22}\\cdots ka_{nn}$<br>\n",
    "$\\qquad = k^n (a_{11}a_{22}\\cdots a_{nn}) = k^nT(A)$<br>\n",
    "This is not equal to $k(T(A))$\n",
    "\n",
    "Therefore, not a linear transform"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "534ffa47",
   "metadata": {},
   "source": [
    "---------------------------------"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "0f369995",
   "metadata": {},
   "source": [
    "<b> Question 8 </b>\n",
    "\n",
    "Show that the transpose of any square matrix is a linear transformation "
   ]
  },
  {
   "cell_type": "markdown",
   "id": "c0dc60da",
   "metadata": {},
   "source": [
    "<b> Solution </b>\n",
    "\n",
    "Let $A$ and $B$ be $n$ by $n$ matrices, and $T: M_{22} \\to M_{22}$ is transformation such that $T(A) = A^T$\n",
    "\n",
    "Now, checking first condition<br>\n",
    "\n",
    "Here, $T(A + B))$ after aplying transformation:\n",
    "\n",
    "$\\quad = (A + B)^T$<br>\n",
    "$\\quad = A^T + B^T$<br>\n",
    "\n",
    "$\\quad = T(A) + T(B)$ \n",
    "\n",
    "Thus, first condition $T(A+ B) = T(A) + T(B)$ satisfied.\n",
    "\n",
    "now, second condition<br>\n",
    "$T(kA)  = (kA)^T$ (after applying transformation)\n",
    "$\\quad = kA^T$\n",
    "\n",
    "$\\quad = kT(A)$. \n",
    "\n",
    "Thus, both condition satisfied, and we can say the transpose of a square matrix is a linear transform."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "bf4a32ba",
   "metadata": {},
   "source": [
    "-------------------------------"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "11ce56ba",
   "metadata": {},
   "source": [
    "<b> Question 9 </b> \n",
    "\n",
    "Let $T: C[0, 1] \\to \\mathbb{R}$ be a transformation (mapping) given by $T(f) = \\int_0^1 f(x) dx$ where $f$ is continuous function is defined in the interval $[0, 1]$. Show that $T$ is a linear transformation "
   ]
  },
  {
   "cell_type": "markdown",
   "id": "63f46023",
   "metadata": {},
   "source": [
    "<b> Solution </b> \n",
    "\n",
    "Let $f$ and $g$ be the function of $C[0,1]$\n",
    "\n",
    "$T(f+ g) = \\int_0^1 [f(x) + g(x)] dx$\n",
    "\n",
    "$\\quad = \\int_0^1 f(x)dx + \\int_0^1 g(x) dx$\n",
    "\n",
    "$\\quad = T(f) + T(g)$\n",
    "\n",
    "Thus, first condition, $T(f + g) = T(f) + T(g)$ satisfied.\n",
    "\n",
    "Next, <br>\n",
    "\n",
    "$T(kf) = \\int_0^1[kf(x)] dx$<br>\n",
    "$\\qquad = k\\int_0^1 f(x) dx$<br>\n",
    "\n",
    "$T(kf) = kT(f)$\n",
    "\n",
    "Now since both condition satisfied, this integration is linear "
   ]
  },
  {
   "cell_type": "markdown",
   "id": "1574e850",
   "metadata": {},
   "source": [
    "---------------------------"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "24ba69b0",
   "metadata": {},
   "source": [
    "<b> Question 10 </b> \n",
    "\n",
    "Let $T: V \\to W$ be a transform such $T(O) \\ne O$. Prove that $T$ is not a linear transform."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "77077c50",
   "metadata": {},
   "source": [
    "<b> Solution </b>\n",
    "\n",
    "Let $u$ be a vector and $T: V \\to W$ is the transform such that $T(O) \\ne O$\n",
    "\n",
    "If $T$ is a linear transform, then<br>\n",
    "$T(u) = T(u + O)$ which can be expressed as:\n",
    "\n",
    "$T(u) = T(u) + T(O)$ (as per the first condition)\n",
    "\n",
    "But $T(O) \\ne O$, therefore, <br>\n",
    "\n",
    "$T(u) \\ne T(u)$.\n",
    "\n",
    "This is a contradiction that $T(u) \\ne T(u)$ therfore, $T$ is not a linear transformation."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "d5bdf47e",
   "metadata": {},
   "source": [
    "-----------------------"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "87825597",
   "metadata": {},
   "source": [
    "<b> Question 11 </b> \n",
    "\n",
    "Let $T: V \\to W$ be the zero transform, that is $T(v) = O$ for all vectors $v \\in V$. Show that $T$ is a linear transformation"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "3ba07100",
   "metadata": {},
   "source": [
    "<b> Solution </b>\n",
    "\n",
    "As $T: V \\to W$ is a zero transform, we have\n",
    "\n",
    "$T(u + v) = O$. <br>\n",
    "\n",
    "Similarly, $T(u) + T(v) = O + O = O $\n",
    "\n",
    "Thus, first condition satisfied.\n",
    "\n",
    "Now, second<br>\n",
    "Again with zero transform, we get\n",
    "\n",
    "$T(ku) = O$, <br>\n",
    "and, $kT(u) = O$\n",
    "\n",
    "$\\Rightarrow T(ku) = kT(u)$\n",
    "\n",
    "Thus, both conditions satisfied, therefore, $T(v) = O$ is a linear transform"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "2f8e3f4a",
   "metadata": {},
   "source": [
    "--------------------------"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "bdd3a7bb",
   "metadata": {},
   "source": [
    "<b> Question 12 </b> \n",
    "\n",
    "Let $T: V \\to W$ be a linear transformation of an $n-$dimensional vector space into a vector space $W$. Let $\\{v_1, v_2, \\cdots v_n \\}$ be a basis for $V$. Prove that if $u$ is any vector in $V$ then we can write $T(u)$ as a linear combination of $\\{T(v_1), T(v_2), T(v_3), \\cdots T(v_4)\\}$"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "1ab69d7d",
   "metadata": {},
   "source": [
    "<b> Solution </b> \n",
    "\n",
    "Since $\\{ v_1, v_2, \\cdots, v_n\\}$ is a basis for $V$, <br>\n",
    "and, $u$ is in $V$\n",
    "\n",
    "Therefore, $u$ can be written as: $u = k_1v_1, k_2v_2, \\cdots k_nv_n$\n",
    "\n",
    "Since, $T$ is a linear transformation\n",
    "\n",
    "$T(u) = T(k_1v_1, k_2v_2, \\cdots k_nv_n)$\n",
    "\n",
    "$\\quad = k_1T(v_1), k_2T(v_2), \\cdots, k_nT(v_n)$\n",
    "\n",
    "Thus, we can stated that $T(u)$ is a linear combination of $\\{T(v_1), T(v_2), T(v_3), \\cdots T(v_4)\\}$"
   ]
  }
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