08-02-linear-algebra.ipynb

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   "source": [
    "# Linear algebra\n",
    "Linear algebra is the branch of mathematics that deals with **vector spaces**. "
   ]
  },
  {
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   "source": [
    "import re, math, random # regexes, math functions, random numbers\n",
    "import matplotlib.pyplot as plt # pyplot\n",
    "from collections import defaultdict, Counter\n",
    "from functools import partial, reduce"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
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     "slide_type": "slide"
    }
   },
   "source": [
    "## Vectors\n",
    "\n",
    "Vectors are points in some finite-dimensional space. "
   ]
  },
  {
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   "execution_count": 19,
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     "slide_type": "subslide"
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   "outputs": [],
   "source": [
    "v = [1, 2]\n",
    "w = [2, 1]\n",
    "vectors = [v, w]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
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    },
    "slideshow": {
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    }
   },
   "outputs": [],
   "source": [
    "def vector_add(v, w):\n",
    "    \"\"\"adds two vectors componentwise\"\"\"\n",
    "    return [v_i + w_i for v_i, w_i in zip(v,w)]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
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     "end_time": "2018-09-12T02:00:08.880719Z",
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    "slideshow": {
     "slide_type": "fragment"
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   "outputs": [
    {
     "data": {
      "text/plain": [
       "[3, 3]"
      ]
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     "execution_count": 3,
     "metadata": {},
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   ],
   "source": [
    "vector_add(v, w)"
   ]
  },
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   "execution_count": 5,
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    "slideshow": {
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   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[-1, 1]"
      ]
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "def vector_subtract(v, w):\n",
    "    \"\"\"subtracts two vectors componentwise\"\"\"\n",
    "    return [v_i - w_i for v_i, w_i in zip(v,w)]\n",
    "\n",
    "vector_subtract(v, w)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {
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     "end_time": "2018-09-12T02:01:22.749575Z",
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   },
   "outputs": [],
   "source": [
    "def vector_sum(vectors):\n",
    "    return reduce(vector_add, vectors)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {
    "ExecuteTime": {
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   "outputs": [
    {
     "data": {
      "text/plain": [
       "[3, 3]"
      ]
     },
     "execution_count": 20,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "vector_sum(vectors)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2018-09-12T02:56:37.552577Z",
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     "slide_type": "subslide"
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   },
   "outputs": [],
   "source": [
    "def scalar_multiply(c, v):\n",
    "    # c is a number, v is a vector\n",
    "    return [c * v_i for v_i in v]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2018-09-12T02:02:59.938350Z",
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   "outputs": [
    {
     "data": {
      "text/plain": [
       "[2.5, 5.0]"
      ]
     },
     "execution_count": 13,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "scalar_multiply(2.5, v)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2018-09-12T02:03:15.086122Z",
     "start_time": "2018-09-12T02:03:15.082201Z"
    },
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "outputs": [],
   "source": [
    "def vector_mean(vectors):\n",
    "    \"\"\"compute the vector whose i-th element is the mean of the\n",
    "    i-th elements of the input vectors\"\"\"\n",
    "    n = len(vectors)\n",
    "    return scalar_multiply(1/n, vector_sum(vectors))\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "metadata": {
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   "outputs": [
    {
     "data": {
      "text/plain": [
       "[1.5, 1.5]"
      ]
     },
     "execution_count": 21,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "vector_mean(vectors)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2018-09-12T02:56:56.631003Z",
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    "slideshow": {
     "slide_type": "subslide"
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   },
   "outputs": [],
   "source": [
    "def dot(v, w):\n",
    "    \"\"\"v_1 * w_1 + ... + v_n * w_n\"\"\"\n",
    "    return sum(v_i * w_i for v_i, w_i in zip(v, w))\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {
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   "outputs": [
    {
     "data": {
      "text/plain": [
       "4"
      ]
     },
     "execution_count": 18,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "dot(v, w)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "The dot product measures how far the vector v extends in the w direction. \n",
    "- For example, if w = [1, 0] then dot(v, w) is just the first component of v. \n",
    "\n",
    "The dot product measures the length of the vector you’d get if you projected v onto w."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2018-09-12T02:07:35.950195Z",
     "start_time": "2018-09-12T02:07:35.946818Z"
    },
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "outputs": [],
   "source": [
    "def sum_of_squares(v):\n",
    "    \"\"\"v_1 * v_1 + ... + v_n * v_n\"\"\"\n",
    "    return dot(v, v)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2018-09-12T02:07:36.983243Z",
     "start_time": "2018-09-12T02:07:36.978652Z"
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   "outputs": [
    {
     "data": {
      "text/plain": [
       "5"
      ]
     },
     "execution_count": 23,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sum_of_squares(v)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2018-09-12T02:07:57.941211Z",
     "start_time": "2018-09-12T02:07:57.937912Z"
    },
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "outputs": [],
   "source": [
    "def magnitude(v):\n",
    "    return math.sqrt(sum_of_squares(v))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 25,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2018-09-12T02:08:06.030797Z",
     "start_time": "2018-09-12T02:08:06.026317Z"
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   "outputs": [
    {
     "data": {
      "text/plain": [
       "2.23606797749979"
      ]
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     "execution_count": 25,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "magnitude(v)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 26,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2018-09-12T02:08:55.689307Z",
     "start_time": "2018-09-12T02:08:55.686062Z"
    },
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "outputs": [],
   "source": [
    "def squared_distance(v, w):\n",
    "    return sum_of_squares(vector_subtract(v, w))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 27,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2018-09-12T02:09:02.988465Z",
     "start_time": "2018-09-12T02:09:02.983903Z"
    },
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     "slide_type": "fragment"
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   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "2"
      ]
     },
     "execution_count": 27,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "squared_distance(v, w)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 28,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2018-09-12T02:09:26.432641Z",
     "start_time": "2018-09-12T02:09:26.429550Z"
    },
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "outputs": [],
   "source": [
    "def distance(v, w):\n",
    "    return math.sqrt(squared_distance(v, w))\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 29,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2018-09-12T02:09:38.557639Z",
     "start_time": "2018-09-12T02:09:38.553786Z"
    },
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     "slide_type": "fragment"
    }
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   "outputs": [
    {
     "data": {
      "text/plain": [
       "1.4142135623730951"
      ]
     },
     "execution_count": 29,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "distance(v, w)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "Using lists as vectors \n",
    "- is great for exposition \n",
    "- but terrible for performance.\n",
    "    - to use the NumPy library."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Matrices"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "A matrix is a two-dimensional collection of numbers. \n",
    "- We will represent matrices as lists of lists\n",
    "    - If A is a matrix, then A[i][j] is the element in the ith row and the jth column."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 30,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2018-09-12T02:15:43.067895Z",
     "start_time": "2018-09-12T02:15:43.063749Z"
    },
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "outputs": [],
   "source": [
    "A = [[1, 2, 3],\n",
    "     [4, 5, 6]]\n",
    "\n",
    "B = [[1, 2],\n",
    "     [3, 4],\n",
    "     [5, 6]]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 31,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2018-09-12T02:16:12.577964Z",
     "start_time": "2018-09-12T02:16:12.574091Z"
    },
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "outputs": [],
   "source": [
    "def shape(A):\n",
    "    num_rows = len(A)\n",
    "    num_cols = len(A[0]) if A else 0\n",
    "    return num_rows, num_cols"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 32,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2018-09-12T02:16:20.887496Z",
     "start_time": "2018-09-12T02:16:20.883170Z"
    },
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(2, 3)"
      ]
     },
     "execution_count": 32,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "shape(A)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 34,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2018-09-12T02:16:44.013585Z",
     "start_time": "2018-09-12T02:16:44.010635Z"
    },
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "outputs": [],
   "source": [
    "def get_row(A, i):\n",
    "    return A[i]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 35,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2018-09-12T02:16:45.185135Z",
     "start_time": "2018-09-12T02:16:45.180832Z"
    },
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[4, 5, 6]"
      ]
     },
     "execution_count": 35,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "get_row(A, 1)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 37,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2018-09-12T02:17:11.445393Z",
     "start_time": "2018-09-12T02:17:11.442110Z"
    },
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "outputs": [],
   "source": [
    "def get_column(A, j):\n",
    "    return [A_i[j] for A_i in A]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 38,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2018-09-12T02:17:12.220804Z",
     "start_time": "2018-09-12T02:17:12.216292Z"
    },
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[3, 6]"
      ]
     },
     "execution_count": 38,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "get_column(A, 2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 50,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2018-09-12T02:21:41.402495Z",
     "start_time": "2018-09-12T02:21:41.396419Z"
    },
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "outputs": [],
   "source": [
    "def make_matrix(num_rows, num_cols, entry_fn):\n",
    "    \"\"\"returns a num_rows x num_cols matrix\n",
    "    whose (i,j)-th entry is entry_fn(i, j),\n",
    "    entry_fn is a function for generating matrix elements.\"\"\"\n",
    "    return [[entry_fn(i, j) \n",
    "             for j in range(num_cols)]\n",
    "                for i in range(num_rows)]\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 71,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2018-09-12T02:37:16.809334Z",
     "start_time": "2018-09-12T02:37:16.802710Z"
    },
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[[0, 1, 2, 3, 4],\n",
       " [1, 2, 3, 4, 5],\n",
       " [2, 3, 4, 5, 6],\n",
       " [3, 4, 5, 6, 7],\n",
       " [4, 5, 6, 7, 8]]"
      ]
     },
     "execution_count": 71,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "def entry_add(i, j):\n",
    "    \"\"\"a function for generating matrix elements. \"\"\"\n",
    "    return i+j\n",
    "\n",
    "make_matrix(5, 5, entry_add)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 53,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2018-09-12T02:23:07.805069Z",
     "start_time": "2018-09-12T02:23:07.797716Z"
    },
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[[1, 0, 0, 0, 0],\n",
       " [0, 1, 0, 0, 0],\n",
       " [0, 0, 1, 0, 0],\n",
       " [0, 0, 0, 1, 0],\n",
       " [0, 0, 0, 0, 1]]"
      ]
     },
     "execution_count": 53,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "def is_diagonal(i, j):\n",
    "    \"\"\"1's on the 'diagonal', \n",
    "       0's everywhere else\"\"\"\n",
    "    return 1 if i == j else 0\n",
    "\n",
    "identity_matrix = make_matrix(5, 5, is_diagonal)\n",
    "identity_matrix"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "ExecuteTime": {
     "end_time": "2018-09-12T02:25:53.668226Z",
     "start_time": "2018-09-12T02:25:53.663461Z"
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    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "### Matrices will be important.\n",
    "- using a matrix to represent a dataset\n",
    "- using an n × k matrix to represent a linear function that maps k-dimensional vectors to n-dimensional vectors. \n",
    "- using matrix to represent binary relationships. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 58,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2018-09-12T02:28:11.642703Z",
     "start_time": "2018-09-12T02:28:11.636503Z"
    },
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "outputs": [],
   "source": [
    " friendships = [(0, 1), \n",
    "                (0, 2), \n",
    "                (1, 2), \n",
    "                (1, 3), \n",
    "                (2, 3), \n",
    "                (3, 4),\n",
    "                (4, 5), \n",
    "                (5, 6), \n",
    "                (5, 7), \n",
    "                (6, 8), \n",
    "                (7, 8), \n",
    "                (8, 9)]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 60,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2018-09-12T02:29:26.728950Z",
     "start_time": "2018-09-12T02:29:26.716314Z"
    },
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "outputs": [],
   "source": [
    "friendships = [[0, 1, 1, 0, 0, 0, 0, 0, 0, 0], # user 0\n",
    "               [1, 0, 1, 1, 0, 0, 0, 0, 0, 0], # user 1\n",
    "               [1, 1, 0, 1, 0, 0, 0, 0, 0, 0], # user 2\n",
    "               [0, 1, 1, 0, 1, 0, 0, 0, 0, 0], # user 3\n",
    "               [0, 0, 0, 1, 0, 1, 0, 0, 0, 0], # user 4\n",
    "               [0, 0, 0, 0, 1, 0, 1, 1, 0, 0], # user 5\n",
    "               [0, 0, 0, 0, 0, 1, 0, 0, 1, 0], # user 6\n",
    "               [0, 0, 0, 0, 0, 1, 0, 0, 1, 0], # user 7\n",
    "               [0, 0, 0, 0, 0, 0, 1, 1, 0, 1], # user 8\n",
    "               [0, 0, 0, 0, 0, 0, 0, 0, 1, 0]] # user 9"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 62,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2018-09-12T02:29:36.913500Z",
     "start_time": "2018-09-12T02:29:36.908769Z"
    },
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "True"
      ]
     },
     "execution_count": 62,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "friendships[0][2] == 1 # True, 0 and 2 are friends "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 74,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2018-09-12T02:37:50.949363Z",
     "start_time": "2018-09-12T02:37:50.942135Z"
    },
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "outputs": [],
   "source": [
    "def matrix_add(A, B):\n",
    "    if shape(A) != shape(B):\n",
    "        raise ArithmeticError(\"cannot add matrices with different shapes\")\n",
    "\n",
    "    num_rows, num_cols = shape(A)\n",
    "    def entry_fn(i, j): return A[i][j] + B[i][j]\n",
    "\n",
    "    return make_matrix(num_rows, num_cols, entry_fn)\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 76,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2018-09-12T02:37:56.803580Z",
     "start_time": "2018-09-12T02:37:56.796449Z"
    },
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[[1, 1, 2, 3, 4],\n",
       " [1, 3, 3, 4, 5],\n",
       " [2, 3, 5, 5, 6],\n",
       " [3, 4, 5, 7, 7],\n",
       " [4, 5, 6, 7, 9]]"
      ]
     },
     "execution_count": 76,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "A = make_matrix(5, 5, is_diagonal)\n",
    "B = make_matrix(5, 5, entry_add)\n",
    "\n",
    "matrix_add(A, B)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 104,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2018-09-12T03:26:10.982125Z",
     "start_time": "2018-09-12T03:26:10.773664Z"
    },
    "slideshow": {
     "slide_type": "skip"
    }
   },
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 400x500 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "v = [2, 1]\n",
    "w = [math.sqrt(.25), math.sqrt(.75)]\n",
    "c = dot(v, w)\n",
    "vonw = scalar_multiply(c, w)\n",
    "o = [0,0]\n",
    "plt.figure(figsize=(4, 5), dpi = 100)\n",
    "\n",
    "plt.arrow(0, 0, v[0], v[1],\n",
    "          width=0.002, head_width=.1, length_includes_head=True)\n",
    "plt.annotate(\"v\", v, xytext=[v[0] + 0.01, v[1]])\n",
    "\n",
    "plt.arrow(0 ,0, w[0], w[1],\n",
    "          width=0.002, head_width=.1, length_includes_head=True)\n",
    "plt.annotate(\"w\", w, xytext=[w[0] - 0.1, w[1]])\n",
    "\n",
    "plt.arrow(0, 0, vonw[0], vonw[1], length_includes_head=True)\n",
    "plt.annotate(u\"(v•w)w\", vonw, xytext=[vonw[0] - 0.1, vonw[1] + 0.02])\n",
    "\n",
    "plt.arrow(v[0], v[1], vonw[0] - v[0], vonw[1] - v[1],\n",
    "          linestyle='dotted', length_includes_head=True)\n",
    "plt.scatter(*zip(v,w,o),marker='.')\n",
    "plt.axis('equal')\n",
    "\n",
    "plt.show()"
   ]
  }
 ],
 "metadata": {
  "celltoolbar": "Slideshow",
  "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"
  },
  "latex_envs": {
   "LaTeX_envs_menu_present": true,
   "autoclose": false,
   "autocomplete": true,
   "bibliofile": "biblio.bib",
   "cite_by": "apalike",
   "current_citInitial": 1,
   "eqLabelWithNumbers": true,
   "eqNumInitial": 1,
   "hotkeys": {
    "equation": "Ctrl-E",
    "itemize": "Ctrl-I"
   },
   "labels_anchors": false,
   "latex_user_defs": false,
   "report_style_numbering": false,
   "user_envs_cfg": false
  },
  "toc": {
   "base_numbering": 1,
   "nav_menu": {},
   "number_sections": false,
   "sideBar": true,
   "skip_h1_title": false,
   "title_cell": "Table of Contents",
   "title_sidebar": "Contents",
   "toc_cell": false,
   "toc_position": {},
   "toc_section_display": true,
   "toc_window_display": true
  }
 },
 "nbformat": 4,
 "nbformat_minor": 2
}