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   "source": [
    "# Chapter 10 - Bilinear forms"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Page 363 Example 10.4"
   ]
  },
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     "text": [
      "a = [x1 x2]\n",
      "b = [y1 y2]\n",
      "f(a,b) = x1*y1 + x1*y2 + x2*y1 + x2*y2\n",
      "so, f(a,b) = \n",
      "[x1  x2]  *  |1   1|  *   |y1|\n",
      "              |1   1|     |y2|\n",
      "So the matrix of f in standard order basis B = {e1,e2} is:\n",
      "[f]B = \n",
      "[[1 1]\n",
      " [1 1]]\n",
      "P = \n",
      "[[ 1  1]\n",
      " [-1  1]]\n",
      "Thus, [f]B = P*[f]B*P\n",
      "[f]B = \n",
      "[[ 1 -1]\n",
      " [-1  1]]\n"
     ]
    }
   ],
   "source": [
    "from numpy import array,transpose\n",
    "print 'a = [x1 x2]'\n",
    "print 'b = [y1 y2]'\n",
    "print 'f(a,b) = x1*y1 + x1*y2 + x2*y1 + x2*y2'\n",
    "print 'so, f(a,b) = '\n",
    "print '[x1  x2]  *  |1   1|  *   |y1|'\n",
    "print '              |1   1|     |y2|'\n",
    "print 'So the matrix of f in standard order basis B = {e1,e2} is:'\n",
    "fb = array([[1, 1],[1, 1]])\n",
    "print '[f]B = \\n',fb\n",
    "P = array([[1 ,1],[-1, 1]])\n",
    "print 'P = \\n',P\n",
    "print 'Thus, [f]B'' = P''*[f]B*P'\n",
    "fb1 = transpose(P) * fb * P\n",
    "print '[f]B'' = \\n',fb1"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Page 365 Example 10.5"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {
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   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "n =  24.0\n",
      "a =  100.0\n",
      "b =  40.0\n",
      "f(a,b) =  4000.0\n",
      "f is non-degenerate billinear form on R**n.\n"
     ]
    }
   ],
   "source": [
    "from numpy import random,transpose\n",
    "n = round(random.randint(2,90))\n",
    "a = round(random.randint(1,n) * 10)#\n",
    "b = round(random.randint(1,n) * 10)#\n",
    "print 'n = ',n\n",
    "print 'a = ',a\n",
    "print 'b = ',b\n",
    "f = a * transpose(b)\n",
    "print 'f(a,b) = ',f\n",
    "print 'f is non-degenerate billinear form on R**n.'\n",
    "#end"
   ]
  }
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