Chapter 4. Linear Algebra.ipynb

{
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  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "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": {},
   "source": [
    "Functions for working with vectors"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [],
   "source": [
    "v = [1, 2, 3]\n",
    "w = [2, 1, 5]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "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": 4,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[3, 3, 8]"
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "vector_add(v, w)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [],
   "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)]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[-1, 1, -2]"
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "vector_subtract(v, w)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [],
   "source": [
    "def vector_sum(vectors):\n",
    "    return reduce(vector_add, vectors)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[3, 3, 8]"
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "vectors = [v,w]\n",
    "vector_sum(vectors)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [],
   "source": [
    "def scalar_multiply(c, v):\n",
    "    return [c * v_i for v_i in v]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[5, 10, 15]"
      ]
     },
     "execution_count": 10,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "scalar_multiply(5, v)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "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))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[1.5, 1.5, 4.0]"
      ]
     },
     "execution_count": 12,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "vector_mean(vectors)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "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))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "19"
      ]
     },
     "execution_count": 14,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "dot(v, w)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {},
   "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": 16,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "14"
      ]
     },
     "execution_count": 16,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sum_of_squares(v)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {},
   "outputs": [],
   "source": [
    "def magnitude(v):\n",
    "    return math.sqrt(sum_of_squares(v))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "3.7416573867739413"
      ]
     },
     "execution_count": 18,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "magnitude(v)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {},
   "outputs": [],
   "source": [
    "def squared_distance(v, w):\n",
    "    return sum_of_squares(vector_subtract(v, w))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "6"
      ]
     },
     "execution_count": 20,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "squared_distance(v, w)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "metadata": {},
   "outputs": [],
   "source": [
    "def distance(v, w):\n",
    "   return math.sqrt(squared_distance(v, w))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "2.449489742783178"
      ]
     },
     "execution_count": 22,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "distance(v, w)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Functions for working with matrices"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "metadata": {},
   "outputs": [],
   "source": [
    "A =[\n",
    "    [1, 2, 3],\n",
    "    [4, 5, 6],\n",
    "    [7, 8, 9]\n",
    "]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "metadata": {},
   "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": 25,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(3, 3)"
      ]
     },
     "execution_count": 25,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "shape(A)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 26,
   "metadata": {},
   "outputs": [],
   "source": [
    "def get_row(A, i):\n",
    "    return A[i]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 27,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[1, 2, 3]"
      ]
     },
     "execution_count": 27,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "get_row(A, 0)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 28,
   "metadata": {},
   "outputs": [],
   "source": [
    "def get_column(A, j):\n",
    "    return [A_i[j] for A_i in A]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 29,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[3, 6, 9]"
      ]
     },
     "execution_count": 29,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "get_column(A, 2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 30,
   "metadata": {},
   "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",
    "    return [[entry_fn(i, j) for j in range(num_cols)]\n",
    "            for i in range(num_rows)]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 31,
   "metadata": {},
   "outputs": [],
   "source": [
    "def sqrt_index(i, j):\n",
    "    return i^j"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 32,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[[0, 1, 2, 3, 4],\n",
       " [1, 0, 3, 2, 5],\n",
       " [2, 3, 0, 1, 6],\n",
       " [3, 2, 1, 0, 7],\n",
       " [4, 5, 6, 7, 0]]"
      ]
     },
     "execution_count": 32,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "m1 = make_matrix(5, 5, sqrt_index)\n",
    "m1"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 33,
   "metadata": {},
   "outputs": [],
   "source": [
    "def is_diagonal(i, j):\n",
    "    \"\"\"1's on the 'diagonal', 0's everywhere else\"\"\"\n",
    "    return 1 if i == j else 0"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 34,
   "metadata": {},
   "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": 34,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "m2 = make_matrix(5, 5, is_diagonal)\n",
    "m2"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 35,
   "metadata": {},
   "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)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 36,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[[1, 1, 2, 3, 4],\n",
       " [1, 1, 3, 2, 5],\n",
       " [2, 3, 1, 1, 6],\n",
       " [3, 2, 1, 1, 7],\n",
       " [4, 5, 6, 7, 1]]"
      ]
     },
     "execution_count": 36,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "matrix_add(m1, m2)"
   ]
  }
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