{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## CS 132 Homework 01 \n",
    "\n",
    "### Due Wednesday July 14th at Midnight (1 minute after 11:59pm) in Gradescope (with grace period of 6 hours)\n",
    "### Homeworks may be submitted up to 24 hours late with a 10% penalty (same grace period)\n",
    "\n",
    "In this first homework, you will become familiar with how to use Jupyter Notebooks to write text and display the results of Python code. In addition, we will do some basic analytical exercises based on\n",
    "the lectures this week. All homeworks in the course will be written in notebooks and submitted in Gradescope. \n",
    "\n",
    "For additional help, check out the tutorials in my tutoral directory linked from the class web page. \n",
    "\n",
    "My YouTube channel contains video presentations of the Markdown Tutorial and the basic Python for CS 237/132 Tutorial. \n",
    "\n",
    "Please read through the entire notebook, reading the expository material and then do the problems, following the instuctions to try various features; among the exposition of various features of Python there\n",
    "are three problems which will be graded.  \n",
    "\n",
    "All homework problems are worth the same, 10 points. \n",
    "\n",
    "You will need to complete this notebook and then convert to a PDF file in order to submit to Gradescope. Instructions are given here:\n",
    "\n",
    "https://www.cs.bu.edu/fac/snyder/cs132/HWSubmissionInstructions.html\n",
    "\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [],
   "source": [
    "# Here are some imports which will be used in the code in the rest of the lab  \n",
    "\n",
    "# Imports used for the code in CS 237\n",
    "\n",
    "import numpy as np                # arrays and functions which operate on arrays\n",
    "\n",
    "import matplotlib.pyplot as plt   # normal plotting\n",
    "\n",
    "\n",
    "# NOTE: You may not use any other libraries than those listed here without explicit permission."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Basic Jupyter Notebook Concepts\n",
    "\n",
    "A Jupyter notebook contains two types of cells: Code cells and Markdown cells, the first for writing and\n",
    "running Python code, and the second for typesetting text and mathematics. \n",
    "\n",
    "This cell is a Markdown cell, and contains text. \n",
    "\n",
    "A code cell contains Python code and will appear as follows:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "This is a code cell!\n"
     ]
    },
    {
     "data": {
      "text/plain": [
       "9"
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "print(\"This is a code cell!\")\n",
    "\n",
    "x = 4\n",
    "x + 5"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "You can add and delete cells, and change their type using the menu at the top of the window,\n",
    "and there are useful shortcuts for most commands: for example, to run a cell, hold down Control and\n",
    "then hit Return; to change a cell from Markdown to Code, select the cell and type \"y\" and to turn\n",
    "a Code cell to Markdown, select it and type \"m\".  "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Markdown: Typesetting in Notebooks\n",
    "\n",
    "Markdown is the language which is used to format text in cells which are indicated to be of Markdown type. \n",
    "\n",
    "**(A)**  If you simply type text, it will be formatted with line breaks according to the width of the cell (as with HTML in a web page--if you change the width of this window, you will see the effect). \n",
    "\n",
    "To get a line break, put a slash at the end of the line.  \n",
    "\n",
    "So\n",
    "\n",
    "      Hello\\\n",
    "      World\n",
    "      \n",
    "would be rendered as\n",
    "\n",
    "Hello\\\n",
    "World\n",
    "\n",
    "\n",
    "The same effect occurs if you put two blank spaces at the end,\n",
    "but the slash is better because you can see it. \n",
    "\n",
    "To separate text into paragraphs, with a blank line between, \n",
    "simply insert a blank line.\n",
    "\n",
    "There is a blank line above this one. \n",
    "\n",
    "This is the second paragraph, which is going to be followed by a third.\n",
    "\n",
    "Yup, here is the third. \n",
    "\n",
    "\n",
    "This *usually* works fine, but if in doubt, leave two blank lines!\n",
    "\n",
    "**(B)**  The other thing you can do, if you don't know anything better, is to **indent** the\n",
    "text -- Markdown will assume that this is \"preformatted text\" as with the HTML `<pre>` command. \n",
    "\n",
    "    Here I have simply hit a tab before the line, and\n",
    "    so it will simply give me the characters I type with no\n",
    "    attempt to format them.  So I can type <b> this </b> and\n",
    "    it will not be formatted. \n",
    "    \n",
    "**(C)**  So the simplest thing is to write text as normal, \n",
    "\n",
    "    and then indented, so you can just use \"ASCII Art\" or \"ASCII Math\" to\n",
    "    write your answers:\n",
    "    \n",
    "    The probability of getting a Head for the first time after K tosses\n",
    "    is P( 2^k ), and the probability of getting a head in the first 4 tosses\n",
    "    is hence:\n",
    "    \n",
    "             P( Heads within 4 tosses ) = 1/2 + 1/4 + 1/8 + 1/16 = 15/16. \n",
    "             \n",
    "In the next problem, we will explore how to do this with Latex, so you can get better-looking\n",
    "math:\n",
    "\n",
    "$$P(\\,\\text{Heads within $4$ tosses}\\,) = \\frac{1}{2^1} + \\frac{1}{2^2} + \\frac{1}{2^3} + \\frac{1}{2^4}  = \\frac{1}{2} + \\frac{1}{4} + \\frac{1}{8} + \\frac{1}{16}  = \\frac{15}{16}.$$\n",
    "\n",
    "\n",
    "\n",
    "**(D)**  If for any reason, you need a line break, you can use the HTML command for a break, as shown here:\n",
    "\n",
    "    This is a line with a <br> break in the middle, and now, <br> another one. \n",
    "\n",
    "This is a line with a <br> break in the middle, and now, <br> another one. "
   ]
  },
  {
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"
    }
   },
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Problem 1\n",
    "\n",
    "Create a cell below this one by selecting this cell (click to the left of the cell), and then typing \"b\"; then\n",
    "convert it from a code cell (the default) to a Markdown cell, and write your favorite Haiku (Google for examples\n",
    "if you don't have a favorite). \n",
    "\n",
    "Here is my favorite, by an American author:\n",
    "\n",
    "![Screen%20Shot%202021-05-24%20at%2011.27.31%20PM.png](attachment:Screen%20Shot%202021-05-24%20at%2011.27.31%20PM.png)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Solution:**\n",
    "\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Basic Latex\n",
    "\n",
    "Latex allows you to typeset math expressions, using dollar signs `$` to indicate\n",
    "the beginning and ending of \"math mode.\"  \n",
    "\n",
    "\n",
    "There are two modes in Latex, essentially inline and display math:\n",
    "\n",
    "**(A)** Inline Mode (surround a math expression with $):\n",
    "\n",
    "    If you enclose a math expression in single dollar signs, it will be rendered\n",
    "    in-line: $2^0 + 2^1 + 2^2 + 2^3$.\n",
    "\n",
    "If you enclose a math expression in single dollar signs, it will be rendered in-line: $2^0 + 2^1 + 2^2 + 2^3$.\n",
    "\n",
    "This works fine for simple mathematical statements:\n",
    "\n",
    "    $P(H) = 0.5$\n",
    "\n",
    "$P(H) = 0.5$\n",
    "\n",
    "   \n",
    "**(B)**  Display Mode (surround a math expression with $$):\n",
    "\n",
    "    If you enclose a math expression in double dollar signs, separated by blank lines, \n",
    "    it will be centered in its own paragraph:\n",
    "    \n",
    "    $$2^0 + 2^1 + 2^2 + 2^3$$.\n",
    "\n",
    "If you enclose a math expression in double dollar signs, separated by blank lines, \n",
    "it will be centered in its own paragraph:\n",
    "    \n",
    "$$2^0 + 2^1 + 2^2 + 2^3$$.\n",
    "\n",
    "**(C)**  Making superscripts and subscripts is easy, using `^` (hat) and `_` (underscore), but make sure to put the whole sub- or superscripted\n",
    "expression in curly braces, that is, make sure you do this:\n",
    "\n",
    "    $2^{2k+1}$\n",
    "\n",
    "$2^{2k+1}$\n",
    "\n",
    "instead of this, which is surely not correct:\n",
    "\n",
    "    $2^2k+1$\n",
    "\n",
    "$2^2k+1$\n",
    "\n",
    "**(D)** \n",
    "\n",
    "The Latex `\\frac{...}{...}` command is very easy to use, as is the equivalent\n",
    "expression `{ ... \\over ... }`; here are some simple examples:\n",
    "\n",
    "\n",
    "| Expression   | Latex  | \n",
    "|----------|--------|\n",
    "| $\\frac{1}{2}$      | `\\frac{1}{2}`    | \n",
    "| ${x+1 \\over y - 1}$      | `{x+1 \\over y - 1}`    | \n",
    "| $\\frac{\\frac{x}{y} + \\frac{e^x}{\\pi}}{2z \\over 5}$      | `frac{\\frac{x}{y} + \\frac{e^x}{\\pi}}{2z \\over 5}`    | \n",
    "\n",
    "**(E)** \n",
    "\n",
    "To write matrices in Latex, you can use a couple of different environments, which require\n",
    "using `begin` and `end` as shown here:\n",
    "\n",
    "`matrix` just presents the matrix without brackets:\n",
    "\n",
    "        $\\begin{matrix}\n",
    "        1 & 2 & 3 \\\\\n",
    "        4 & 5 & 6\n",
    "     \\end{matrix}$\n",
    "\n",
    "$\\begin{matrix}\n",
    "1 & 2 & 3 \\\\\n",
    "4 & 5 & 6\n",
    "\\end{matrix}$\n",
    "\n",
    "`pmatrix` adds \"parenthesis brackets\":\n",
    "\n",
    "        $\\begin{pmatrix}\n",
    "        1 & 2 & 3 & 4\n",
    "        \\end{pmatrix}$\n",
    "\n",
    "$\\begin{pmatrix}\n",
    "1 & 2 & 3 & 4\n",
    "\\end{pmatrix}$\n",
    "\n",
    "Of course, you can simply write a tuple with comma very simply:\n",
    "\n",
    "        $(1,2,3,4)$\n",
    "\n",
    "$(1,2,3,4)$\n",
    "\n",
    "`bmatrix` adds square brackets, which are standard in linear algebra texts:\n",
    "\n",
    "        $\\begin{bmatrix}\n",
    "        1 & 2 & 3 \\\\\n",
    "        4 & 5 & 6\n",
    "        \\end{bmatrix}$\n",
    "\n",
    "$\\begin{bmatrix}\n",
    "1 & 2 & 3 \\\\\n",
    "4 & 5 & 6\n",
    "\\end{bmatrix}$\n",
    "\n",
    "\n",
    "## Basic Latex\n",
    "\n",
    "Latex allows you to typeset math expressions, using dollar signs `$` to indicate\n",
    "the beginning and ending of \"math mode.\"  \n",
    "\n",
    "\n",
    "There are two modes in Latex, essentially inline and display math:\n",
    "\n",
    "**(A)** Inline Mode (surround a math expression with $):\n",
    "\n",
    "    If you enclose a math expression in single dollar signs, it will be rendered\n",
    "    in-line: $2^0 + 2^1 + 2^2 + 2^3$.\n",
    "\n",
    "If you enclose a math expression in single dollar signs, it will be rendered in-line: $2^0 + 2^1 + 2^2 + 2^3$.\n",
    "\n",
    "This works fine for simple mathematical statements:\n",
    "\n",
    "    $P(H) = 0.5$\n",
    "\n",
    "$P(H) = 0.5$\n",
    "\n",
    "   \n",
    "**(B)**  Display Mode (surround a math expression with $$):\n",
    "\n",
    "    If you enclose a math expression in double dollar signs, separated by blank lines, \n",
    "    it will be centered in its own paragraph:\n",
    "    \n",
    "    $$2^0 + 2^1 + 2^2 + 2^3$$.\n",
    "\n",
    "If you enclose a math expression in double dollar signs, separated by blank lines, \n",
    "it will be centered in its own paragraph:\n",
    "    \n",
    "$$2^0 + 2^1 + 2^2 + 2^3$$.\n",
    "\n",
    "**(C)**  Making superscripts and subscripts is easy, using `^` (hat) and `_` (underscore), but make sure to put the whole sub- or superscripted\n",
    "expression in curly braces, that is, make sure you do this:\n",
    "\n",
    "    $2^{2k+1}$\n",
    "\n",
    "$2^{2k+1}$\n",
    "\n",
    "instead of this, which is surely not correct:\n",
    "\n",
    "    $2^2k+1$\n",
    "\n",
    "$2^2k+1$\n",
    "\n",
    "**(D)** \n",
    "\n",
    "The Latex `\\frac{...}{...}` command is very easy to use, as is the equivalent\n",
    "expression `{ ... \\over ... }`; here are some simple examples:\n",
    "\n",
    "\n",
    "| Expression   | Latex  | \n",
    "|----------|--------|\n",
    "| $\\frac{1}{2}$      | `\\frac{1}{2}`    | \n",
    "| ${x+1 \\over y - 1}$      | `{x+1 \\over y - 1}`    | \n",
    "| $\\frac{\\frac{x}{y} + \\frac{e^x}{\\pi}}{2z \\over 5}$      | `frac{\\frac{x}{y} + \\frac{e^x}{\\pi}}{2z \\over 5}`    | \n",
    "\n",
    "**(E)** \n",
    "\n",
    "To write matrices in Latex, you can use a couple of different environments, which require\n",
    "using `begin` and `end` as shown here:\n",
    "\n",
    "`matrix` just presents the matrix without brackets:\n",
    "\n",
    "        $\\begin{matrix}\n",
    "        1 & 2 & 3 \\\\\n",
    "        4 & 5 & 6\n",
    "     \\end{matrix}$\n",
    "\n",
    "$\\begin{matrix}\n",
    "1 & 2 & 3 \\\\\n",
    "4 & 5 & 6\n",
    "\\end{matrix}$\n",
    "\n",
    "`pmatrix` adds \"parenthesis brackets\":\n",
    "\n",
    "        $\\begin{pmatrix}\n",
    "        1 & 2 & 3 & 4\n",
    "        \\end{pmatrix}$\n",
    "\n",
    "$\\begin{pmatrix}\n",
    "1 & 2 & 3 & 4\n",
    "\\end{pmatrix}$\n",
    "\n",
    "Of course, you can simply write a tuple with comma very simply:\n",
    "\n",
    "        $(1,2,3,4)$\n",
    "\n",
    "$(1,2,3,4)$\n",
    "\n",
    "`bmatrix` adds square brackets, which are standard in linear algebra texts:\n",
    "\n",
    "        $\\begin{bmatrix}\n",
    "        1 & 2 & 3 \\\\\n",
    "        4 & 5 & 6\n",
    "        \\end{bmatrix}$\n",
    "\n",
    "$\\begin{bmatrix}\n",
    "1 & 2 & 3 \\\\\n",
    "4 & 5 & 6\n",
    "\\end{bmatrix}$\n",
    "\n",
    "\n",
    "  "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Problem 2 \n",
    "\n",
    "Consider the following system of linear equations:\n",
    "$$\\begin{aligned}\n",
    "        2 x_1 + 4x_2 &= 4           \\\\       \n",
    "        5 x_1 + 7x_2 &= 14           \\\\  \n",
    "  \\end{aligned}$$\n",
    "\n",
    "### Part A\n",
    "\n",
    "In the following cell, give  the augmented matrix corresponding to this system (Hint: just cut and paste\n",
    "and modify from the previous cell). Give the numbers in floating-point form. "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Solution A:**\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    " "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Part B\n",
    "\n",
    "Solve this system using elementary row operations on the augmented matrix and give the solution\n",
    "as a tuple of two floating-point numbers, using Latex.  Round to 4 decimal places.\n",
    "\n",
    "      "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Solution B:**\n",
    "\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Numpy Arrays\n",
    "\n",
    "Numpy provides an (multi-dimensional) array type, <code>ndarray</code>, which is exactly what we\n",
    "need for linear algebra computations. \n",
    "\n",
    "\n",
    "### Basic One-Dimensional Arrays\n",
    "\n",
    "The numpy library functions return arrays instead of normal Python lists. Since\n",
    "numpy automatically converts normal lists into numpy arrays, we can use lists\n",
    "without any problems.  \n",
    "\n",
    "However, the reverse is not true:  normal Python functions which expect lists will NOT\n",
    "work with numpy arrays, and you'll have to make adjustments. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[12  3  5]\n"
     ]
    }
   ],
   "source": [
    "# Creating a numpy array from a list\n",
    "\n",
    "ex1 = np.array( [ 12,3,5 ] )\n",
    "print(ex1)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Multidimensional Ndarrays\n",
    "\n",
    "Numpy is designed to provide high-speed access to multi-dimensional\n",
    "arrays, for linear algebra and data science.  These will be used extensively in CS 132.  "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[[ 1  2  3  4]\n",
      " [ 5  6  7  8]\n",
      " [ 9 10 11 12]]\n"
     ]
    },
    {
     "data": {
      "text/plain": [
       "array([[ 1,  2,  3,  4],\n",
       "       [ 5,  6,  7,  8],\n",
       "       [ 9, 10, 11, 12]])"
      ]
     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# 2D array\n",
    "\n",
    "X = np.array([[ 1, 2, 3, 4],\n",
    "              [ 5, 6, 7, 8],\n",
    "              [ 9, 10, 11, 12]]   )\n",
    "\n",
    "print(X)\n",
    "X"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The size of the array along each dimension can be found using the `shape` function:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(3, 4)"
      ]
     },
     "execution_count": 10,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# X has 3 rows and 4 columns\n",
    "np.shape(X)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Numpy array types. \n",
    "\n",
    "Numpy assigns a type to each matrix – for example, float or int. If you construct a matrix\n",
    "using the constructor `array`, and all of the entries are integers, then numpy will auto-detect this as an integer matrix. If even one of the values is float, then the whole matrix will be float. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[1, 2, 3],\n",
       "       [5, 6, 7]])"
      ]
     },
     "execution_count": 11,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "np.array( [ [ 1, 2, 3], [5, 6, 7]])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[1., 2., 3.],\n",
       "       [5., 6., 7.]])"
      ]
     },
     "execution_count": 12,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "np.array( [ [ 1, 2, 3], [5, 6, 7.0]])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "When you assign values to an integer matrix they will be rounded to the nearest integer. This is not what you\n",
    "want. So you do not want to work with integer matrices in general.\n",
    "\n",
    "So it is a good idea to make sure that the inputs your your functions are floating point matrices. To convert an\n",
    "integer matrix to a floating point matrix you can convert it using the function `astype`:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[ 1.,  2.,  3.,  4.],\n",
       "       [ 5.,  6.,  7.,  8.],\n",
       "       [ 9., 10., 11., 12.]])"
      ]
     },
     "execution_count": 13,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "X1 = X.astype(float)\n",
    "X1"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Problem 3\n",
    "\n",
    "### Part A\n",
    "\n",
    "Create a variable Y which holds the matrix from the previous problem:\n",
    "\n",
    "$$\\begin{bmatrix}\n",
    "        2.0 & 4.0 & 4.0 \\\\\n",
    "        5.0 & 7.0 & 14.0\n",
    "\\end{bmatrix}$$\n",
    "\n",
    "Be sure to create an array of floats, not integers!\n",
    "Simply print out the value of Y as your answer. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
   "outputs": [],
   "source": [
    "# Solution A:\n",
    "    \n",
    "    "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Part B\n",
    "\n",
    "Now calculate and print out an array Y1 in which you have multiplied the first row of Y by 2, and added\n",
    "it to the second row:\n",
    "\n",
    "$$\\begin{bmatrix}\n",
    "        2.0 & 4.0 & 4.0 \\\\\n",
    "        9.0 & 15.0 & 22.0\n",
    "\\end{bmatrix}$$\n",
    "\n",
    "You must calculate Y1 from Y, and not simply use `np.array`. Hint: Use `np.copy(...)` to make a fresh copy of\n",
    "the array Y, then make the changes to the new copy Y1. \n",
    "\n",
    "Hint: You can do this by changing each individual entry, but numpy allows you to apply arithmetic operations to\n",
    "entire rows, which can simplify the computation. In fact, after making the copy, you only need one line!"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {},
   "outputs": [],
   "source": [
    "# SolutionB:\n",
    "\n"
   ]
  },
  {
   "attachments": {
    "Screen%20Shot%202021-07-08%20at%2010.51.22%20AM.png": {
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"
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"
    }
   },
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Problem 4\n",
    "\n",
    "This problem concerns the  *parametric* forms of solutions, as presented at the end of section 1.1 in the online textbook. \n",
    "\n",
    "For parts A -- C, state (i) what shape (point, line, plane) the solution set has, (ii) the *Degrees of Freedom* in the solution set, and (iii) whether the given point is in the solution set. \n",
    "\n",
    "(Optional: on your own, try to imagine what the solution sets looks like graphically.)\n",
    "\n",
    "(A)  Solution:  $(x,y) = (a,0)$ for any $a\\in \\mathbb{R}$. Point: $(0,0)$.\n",
    "\n",
    "(B)  Solution:  $(x,y) = (a,-2a)$ for any $a\\in \\mathbb{R}$. Point: $(1,0)$.\n",
    "\n",
    "(C)  Solution:  $(x,y,z) = (a,b+1,a+b)$ for any $a,b\\in \\mathbb{R}$. Point: $(1,10,10)$.\n",
    "\n",
    "For parts D and E, for each of the graphs, give (i) the equation for the line in the form $ax+by = c$, and\n",
    "(ii) supposing this is a graph of the solution to some set of equations, give the parametric form of the solution\n",
    "in the same form as in parts A -- C. \n",
    "\n",
    "(D)  \n",
    "\n",
    "![Screen%20Shot%202021-07-08%20at%2010.53.59%20AM.png](attachment:Screen%20Shot%202021-07-08%20at%2010.53.59%20AM.png)\n",
    "\n",
    "(E)  \n",
    "\n",
    "![Screen%20Shot%202021-07-08%20at%2010.51.22%20AM.png](attachment:Screen%20Shot%202021-07-08%20at%2010.51.22%20AM.png)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Solution:**\n",
    "\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Problem 5\n",
    "\n",
    "For each of the following, state whether the given matrix is in \n",
    "\n",
    "     (E) Row Echelon Form, \n",
    "     (R) Reduced Row Echelon Form, or\n",
    "     (N) Neither.\n",
    "\n",
    "\n",
    "(A)  $\\begin{bmatrix}\n",
    "        1 & 1 & 0 & 1\\\\\n",
    "        0 & 0 & 1 & 1\\\\\n",
    "        0 & 0 & 0 & 0\\\\\n",
    "     \\end{bmatrix}$\n",
    "\n",
    "(B)  $\\begin{bmatrix}\n",
    "        1 & 1 & 0 & 0\\\\\n",
    "        0 & 1 & 1 & 0\\\\\n",
    "        0 & 0 & 1 & 1\\\\\n",
    "     \\end{bmatrix}$\n",
    "\n",
    "(C)  $\\begin{bmatrix}\n",
    "        1 & 0 & 0 & 0\\\\\n",
    "        1 & 1 & 0 & 0\\\\\n",
    "        0 & 1 & 3 & 7\\\\\n",
    "        0 & 0 & 1 & 1\\\\\n",
    "     \\end{bmatrix}$\n",
    "     \n",
    "(D)  $\\begin{bmatrix}\n",
    "        0 & 1 & 1 & 1 & 1\\\\\n",
    "        0 & 0 & 2 & 2 & 2\\\\\n",
    "        0 & 0 & 0 & 0 & 3\\\\\n",
    "        0 & 0 & 0 & 0 & 0\\\\\n",
    "     \\end{bmatrix}$\n",
    "     \n",
    "(E) $\\begin{bmatrix}\n",
    "        1 & 1 & 2 & 1\\\\\n",
    "        0 & 0 & 0 & 0\\\\\n",
    "        0 & 0 & 0 & 0\\\\\n",
    "     \\end{bmatrix}$\n",
    "\n",
    "\n",
    "\n",
    "\n",
    "(F)   $\\begin{bmatrix}\n",
    "        0 & 0 & 0 & 2\\\\\n",
    "     \\end{bmatrix}$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Solution:**\n",
    "\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Note\n",
    "\n",
    "For the following three problems, you will need to trace the evolution of the matrix as you perform Gaussian Elimination.  You do not need to format these in Latex, you may simply choose to indent your entire answer, or you may inclose it in the HTML tags `<pre>` and `</pre>` to get preformatted text, like this:\n",
    "\n",
    "    1  3  5  7 \n",
    "    3  5  7  9 \n",
    "    5  7  9  1\n",
    "\n",
    "    R2 = R2 - 3*R1\n",
    "\n",
    "    1  3  5  7 \n",
    "    0 -4 -8 -12 \n",
    "    5  7  9  1 \n",
    " \n",
    "etc. OR:\n",
    "\n",
    "\n",
    "<pre> \n",
    " 1  3  5  7 \n",
    " 3  5  7  9 \n",
    " 5  7  9  1\n",
    " \n",
    "R2 = R2 - 3*R1\n",
    "\n",
    " 1  3  5  7 \n",
    " 0 -4 -8 -12 \n",
    " 5  7  9  1 \n",
    " \n",
    "</pre>\n",
    "\n",
    "Hint:  We've designed these problems so you get integers throughout the computation -- not the usual situation, but easier for hand computation!"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Problem 6\n",
    "\n",
    "Consider the following matrix:\n",
    "\n",
    "$$\\begin{bmatrix}\n",
    "        1 & 3 & 5 & 7\\\\\n",
    "        3 & 5 & 7 & 9\\\\\n",
    "        5 & 7 & 9 & 1\\\\\n",
    "     \\end{bmatrix}$$\n",
    "\n",
    "(A) Use the elimination algorithm from lecture to convert this matrix to reduced echelon form. Show all work (including the matrices at each step) and put a box around the pivot positions in the final matrix,\n",
    "\n",
    "(B) List the pivot columns, and\n",
    "\n",
    "(C) State whether this system is consistent or inconsistent and why. \n",
    "\n",
    "\n",
    "\n",
    "Hint:  To enclose a number in Latex in a box, use the Latex command `\\boxed{...}` as shown here:\n",
    "\n",
    "            $$\\begin{bmatrix}\n",
    "             \\boxed{1} & 0 \\\\\n",
    "             0 & \\boxed{1} \\\\\n",
    "          \\end{bmatrix}$$\n",
    "           \n",
    "$$\\begin{bmatrix}\n",
    "        \\boxed{1} & 0 \\\\\n",
    "        0 & \\boxed{1} \\\\\n",
    "     \\end{bmatrix}$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Solution**\n",
    "\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Problem 7\n",
    "\n",
    "Find the *unique solution* to the following set of equations by using the Gaussian Elimination algorithm by hand (the *solution* is\n",
    "the listing of the values of each of the variables, not just the reduced echelon form).   Show all work, including each matrix that results after the row operations. \n",
    "\n",
    "$$\\begin{aligned}\n",
    "        3x_1 - 3x_2 + 9x_3 &= 24             \\\\       \n",
    "        2x_1 - 2x_2 + 7x_3         &= 17    \\\\\n",
    "         -x_1 +2x_2 - 4x_3        &= -11               \\\\ \n",
    "  \\end{aligned}$$\n",
    "\n",
    "\n",
    "\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Solution:**\n",
    "\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Problem 8\n",
    "\n",
    " \n",
    "  \n",
    "(B)  Find the *parametric form* for the solution to the following set of equations by using the Gaussian Elimination algorithm by hand (the *general solution* is\n",
    "the listing of the values of each of the variables, in terms of the free variables, which are then parameters for an infinite set of solutions).  \n",
    "\n",
    "Show all work, including the matrices produced after each row operation, and give the parametric form solution (hint: the variable $x_3$ will be free and hence a  parameter). \n",
    "\n",
    "Again, you should have integers at each step, just to make the calculation easier by hand.  \n",
    "\n",
    "\n",
    "$$\\begin{aligned}\n",
    "        3x_1 - 3x_2 + 3x_3 &= 9             \\\\       \n",
    "        2x_1 - x_2  + 4x_3  &= 7   \\\\\n",
    "        3x_1 - 5x_2  - x_3 &= 7              \\\\ \n",
    "  \\end{aligned}$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Solution:**\n",
    "\n"
   ]
  },
  {
   "attachments": {
    "Screen%20Shot%202021-05-26%20at%2011.54.00%20PM.png": {
     "image/png": 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"
    }
   },
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Problem 9\n",
    "\n",
    "In this problem we are going to apply the elimination algorithm to solve a problem in \"curve fitting,\" that is,\n",
    "finding a quadratic equation of the form\n",
    "\n",
    "$$f(x) = a + bx + cx^2$$\n",
    "\n",
    "for some constants $a$, $b$, and $c$, which goes through the points $(1,-1)$, $(2,3)$, and $(3,13)$: \n",
    "\n",
    "![Screen%20Shot%202021-05-26%20at%2011.54.00%20PM.png](attachment:Screen%20Shot%202021-05-26%20at%2011.54.00%20PM.png)\n",
    "\n",
    "We can accomplish this useful task by\n",
    "construing the problem in terms of finding the unknowns $a$, $b$, and $c$ (hence, these are the variables, which\n",
    "we normally call $x_1$, $x_2$, and $x_3$). This technique, which involves treating the unknown\n",
    "constants as variables, is a fairly typical application of Gaussian Elimination. \n",
    "\n",
    "We will set up the problem using one equation for each of the points which the curve must fit, for example,\n",
    "the point $(2,3)$ must satisfy\n",
    "\n",
    "$$a + b\\cdot 2 + c\\cdot 2^2  = 3$$\n",
    "\n",
    "which, phrased as an equation in the three variables $a$, $b$, and $c$, is\n",
    "\n",
    "$$a + 2b + 4c = 3$$\n",
    "\n",
    "(A)  Give the set of three equations in the variables $a$, $b$, and $c$ representing the problem to be solved. \n",
    "\n",
    "(B) Solve the problem by using Gaussian Elimination by hand, showing the matrix at each step, which will arrive\n",
    "at a unique solution. \n",
    "\n",
    "(C)  Give the unique solution as a formula for $f(x)$ which fits the points. \n",
    "\n",
    "(D) Insert your values for $a$, $b$, and $c$ into the code fragment below to display the points and the curve. \n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Solution:**\n",
    "\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Solution:** \n",
    "\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "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": [
    "# Solution for (D):  insert your values for a, b, and c here:\n",
    "\n",
    "a = 1       # replace with your values found above \n",
    "b = 1\n",
    "c = 1\n",
    "\n",
    "def f(t):\n",
    "    return a + b*t + c*t*t\n",
    "\n",
    "\n",
    "plt.title('Curve Fitting Example')\n",
    "plt.xlabel(\"X\")\n",
    "plt.ylabel(\"Y\")\n",
    "X = np.linspace(0,3.5,100)           \n",
    "Y = [f(x) for x in X]\n",
    "plt.plot(X,Y)\n",
    "plt.scatter([1,2,3], [-1,3,13])\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Problem 10\n",
    "\n",
    "In this problem, you must find the circle that runs through the points (5,5), (4,6), and (6,2). Write your equation in the form\n",
    "\n",
    "$$a+bx+cy+x^2+y^2 = 0.$$ \n",
    "\n",
    "(A) Give the original set of equations encoding this problem, the initial matrix, and the reduced row echelon form\n",
    "after running GE (do by hand, it s not that hard).  \n",
    "\n",
    "(B) Calculate the center and radius of this circle;\n",
    "\n",
    "(C) Complete the code template below to display the circle, verifying that it runs through the points given. \n",
    "\n",
    "Hints: \n",
    "\n",
    "- The points are of the form (x,y) - plug each point into the equation to generate your system of linear equations with variables a, b, and c.  \n",
    "\n",
    "- To find the center $(x_c,y_c)$ and radius $r$ of the resulting equation, rearrange it into the form \n",
    "\n",
    "$$(x-x_c)^2+(y- y_c)^2 = r^2.$$\n",
    "\n",
    "Here is a further explanation of how to do this rearrangement. \n",
    "\n",
    "\n",
    "First, find values for a, b, and c.   Then, you need to find values $x_c, y_c, r$ so you can transform\n",
    "\n",
    "the first equation into the second:\n",
    "\n",
    "\n",
    "\n",
    "\n",
    "$$a+bx+cy+x^2+y^2 = 0$$\n",
    "\n",
    "\n",
    "\n",
    "\n",
    "$$x^2 + bx + y^2 + cy = -a$$\n",
    "\n",
    "\n",
    "\n",
    "\n",
    "$$(x^2 + bx + x_c^2) + (y^2 + cy + y_c^2) = x_c^2 + y_c^2 - a$$\n",
    "\n",
    "\n",
    "\n",
    "\n",
    "$$(x - x_c)^2 + (y - y_c)^2 = r^2$$\n",
    "\n",
    "\n",
    "\n",
    "\n",
    "The last step consists in \"completing the squares\" to get the final equation (this process is discussed in\n",
    "the  Math Review, or you can Google it!). \n",
    "\n",
    "Once you get the a, b, and c, this may be more obvious.  \n",
    "\n",
    "\n"
   ]
  },
  {
   "attachments": {},
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Solution:**\n",
    "\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 720x720 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "# Solution for (C): \n",
    "\n",
    "x_c = 1              # put your solutions here\n",
    "y_c = 1\n",
    "r = 1\n",
    "\n",
    "# You shouldn't have to touch any of this code\n",
    "\n",
    "plt.figure(figsize=(10,10))\n",
    "plt.title('Solution (C)')\n",
    "plt.grid()\n",
    "plt.xlabel(\"X\")\n",
    "plt.ylabel(\"Y\")\n",
    "plt.xlim([-8,8])\n",
    "plt.ylim([-8,8])\n",
    "T = np.linspace(0,2*np.pi,100)      #    (cos(theta), sin(theta))         \n",
    "X = [r*np.cos(theta)+x_c for theta in T]\n",
    "Y = [r*np.sin(theta)+y_c for theta in T]\n",
    "plt.plot(X,Y)\n",
    "plt.plot([-8,8,0,0,0],[0,0,0,8,-8],color='k')\n",
    "plt.scatter([5,4,6], [5,6,2])\n",
    "plt.scatter([x_c], [y_c],marker='x')\n",
    "plt.show()"
   ]
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "Python 3",
   "language": "python",
   "name": "python3"
  },
  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
    "version": 3
   },
   "file_extension": ".py",
   "mimetype": "text/x-python",
   "name": "python",
   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython3",
   "version": "3.7.6"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 2
}