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{
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{
"cell_type": "markdown",
"metadata": {},
"source": [
"# Chapter 10 - Bilinear forms"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Page 363 Example 10.4"
]
},
{
"cell_type": "code",
"execution_count": 1,
"metadata": {
"collapsed": false
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"outputs": [
{
"name": "stdout",
"output_type": "stream",
"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": [
"import numpy as np\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 = np.array([[1, 1],[1, 1]])\n",
"print '[f]B = \\n',fb\n",
"P = np.array([[1 ,1],[-1, 1]])\n",
"print 'P = \\n',P\n",
"print 'Thus, [f]B'' = P''*[f]B*P'\n",
"fb1 = np.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": {
"collapsed": false
},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"n = 56.0\n",
"a = 410.0\n",
"b = 70.0\n",
"f(a,b) = 28700.0\n",
"f is non-degenerate billinear form on R**n.\n"
]
}
],
"source": [
"import numpy as np\n",
"n = round(np.random.randint(2,90))\n",
"a = round(np.random.randint(1,n) * 10)#\n",
"b = round(np.random.randint(1,n) * 10)#\n",
"print 'n = ',n\n",
"print 'a = ',a\n",
"print 'b = ',b\n",
"f = a * np.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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