{
"cells": [
{
"cell_type": "markdown",
"metadata": {},
"source": [
"# Chapter 8 - Inner product spaces"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Page 271 Example 8.1"
]
},
{
"cell_type": "code",
"execution_count": 16,
"metadata": {
"collapsed": false
},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"n = 5\n",
"a = [[ 1. 4. 2. 8. 8.]]\n",
"b = [[ 10. 8. 3. 1. 8.]]\n",
"Then, (a|b) = \n",
"\n",
"[[ 10. 40. 20. 80. 80.]\n",
" [ 8. 32. 16. 64. 64.]\n",
" [ 3. 12. 6. 24. 24.]\n",
" [ 1. 4. 2. 8. 8.]\n",
" [ 8. 32. 16. 64. 64.]]\n"
]
}
],
"source": [
"import numpy as np\n",
"n=np.random.randint(2,9)\n",
"a=np.random.rand(1,n)\n",
"b=np.random.rand(1,n)\n",
"for i in range(0,n):\n",
" a[0,i]=round(a[0,i]*10)\n",
" b[0,i]=round(b[0,i]*10)\n",
"print 'n = ',n\n",
"print 'a = ',a\n",
"print 'b = ',b\n",
"print 'Then, (a|b) = \\n\\n',a*np.transpose(b)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Page 271 Example 8.2"
]
},
{
"cell_type": "code",
"execution_count": 17,
"metadata": {
"collapsed": false
},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"a = [[ 4. 3.]]\n",
"b = [[ 7. 5.]]\n",
"Then, a|b = 47.0\n"
]
}
],
"source": [
"import numpy as np\n",
"a=np.random.rand(1,2)\n",
"b=np.random.rand(1,2)\n",
"for i in range(0,2):\n",
" a[0,i]=round(a[0,i]*10)\n",
" b[0,i]=round(b[0,i]*10)\n",
"print 'a = ',a\n",
"print 'b = ',b\n",
"x1 = a[0,0]#\n",
"x2 = a[0,1]#\n",
"y1 = b[0,0]#\n",
"y2 = b[0,1]#\n",
"t = x1*y1 - x2*y1 - x1*y2 + 4*x2*y2#\n",
"print 'Then, a|b = ',t"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Page 307 Example 8.28"
]
},
{
"cell_type": "code",
"execution_count": 18,
"metadata": {
"collapsed": false
},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"x1 and x2 are two real nos. i.e., x1**2 + x2**2 = 1\n",
"x1 = 0.383547227589\n",
"x2 = 0.923521263539\n",
"B = \n",
"[[ 0.38354723 0.92352126 0. ]\n",
" [ 0. 1. 0. ]\n",
" [ 0. 0. 1. ]]\n",
"Applying Gram-Schmidt process to B:\n",
"a1 = [ 0.38354723 0.92352126 0. ]\n",
"a2 = [-0.35421402 0.14710848 0. ]\n",
"a3 = [0 0 1]\n",
"U = \n",
"[[[ 0.38354723 0.92352126 0. ]]\n",
"\n",
" [[-0.92352126 0.38354723 0. ]]\n",
"\n",
" [[ 0. 0. 1. ]]]\n",
"M = \n",
"[[ 1. 0. 0. ]\n",
" [-2.40784236 2.60724085 0. ]\n",
" [ 0. 0. 1. ]]\n",
"inverse(M) * U = [[[ 3.83547228e-01 -4.25822963e-17 0.00000000e+00]\n",
" [ 3.54214020e-01 3.54214020e-01 0.00000000e+00]\n",
" [ 0.00000000e+00 0.00000000e+00 0.00000000e+00]]\n",
"\n",
" [[ -9.23521264e-01 -1.76848356e-17 0.00000000e+00]\n",
" [ -8.52891524e-01 1.47108476e-01 0.00000000e+00]\n",
" [ -0.00000000e+00 0.00000000e+00 0.00000000e+00]]\n",
"\n",
" [[ 0.00000000e+00 -0.00000000e+00 0.00000000e+00]\n",
" [ 0.00000000e+00 0.00000000e+00 0.00000000e+00]\n",
" [ 0.00000000e+00 0.00000000e+00 1.00000000e+00]]]\n",
"So, B = inverse(M) * U\n"
]
}
],
"source": [
"import numpy as np\n",
"print 'x1 and x2 are two real nos. i.e., x1**2 + x2**2 = 1'\n",
"x1 = np.random.rand()\n",
"x2 = np.sqrt(1 - x1**2)\n",
"print 'x1 = ',x1\n",
"print 'x2 = ',x2\n",
"B = np.array([[x1, x2, 0],[0, 1, 0],[0, 0, 1]])\n",
"print 'B = \\n',B\n",
"print 'Applying Gram-Schmidt process to B:'\n",
"a1 = np.array([x1, x2, 0])\n",
"a2 = np.array([0 ,1 ,0]) - x2 * np.array([x1 ,x2 ,0])\n",
"a3 = np.array([0, 0, 1])\n",
"print 'a1 = ',a1\n",
"print 'a2 = ',a2\n",
"print 'a3 = ',a3\n",
"U = np.array([[a1],[a2/x1],[a3]])\n",
"print 'U = \\n',U\n",
"M = np.array([[1, 0, 0],[-x2/x1, 1/x1, 0],[0, 0, 1]])\n",
"print 'M = \\n',M\n",
"print 'inverse(M) * U = ',np.linalg.inv(M) * U\n",
"print 'So, B = inverse(M) * U'"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Page 278 Example 8.9"
]
},
{
"cell_type": "code",
"execution_count": 19,
"metadata": {
"collapsed": false
},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"(x,y) = [[ 5. 3.]]\n",
"(-y,x) = [-3.0, 5.0]\n",
"Inner product of these vectors is:\n",
"(x,y)|(-y,x) = 0.0\n",
"So, these are orthogonal.\n",
"------------------------------------------\n",
"If inner product is defined as:\n",
"(x1,x2)|(y1,y2) = x1y1- x2y1 - x1y2 + 4x2y2\n",
"Then, (x,y)|(-y,x) = -x*y+y**2-x**2+4*x*y = 0 if,\n",
"y = 1/2(-3 + sqrt(13))*x\n",
"or\n",
"y = 1/2(-3 - sqrt(13))*x\n",
"Hence,\n",
"[[ 5. 3.]]\n",
"is orthogonal to\n",
"[-3.0, 5.0]\n"
]
}
],
"source": [
"import numpy as np\n",
"#a = round(rand(1,2) * 10)#\n",
"a=np.random.rand(1,2)\n",
"for j in [0,1]:\n",
" a[0,j]=round(a[0,j]*10)\n",
"\n",
"x = a[0,0]\n",
"y = a[0,1]\n",
"b = [-y, x]#\n",
"print '(x,y) = ',a\n",
"print '(-y,x) = ',b\n",
"print 'Inner product of these vectors is:'\n",
"t = -x*y + y*x#\n",
"print '(x,y)|(-y,x) = ',t\n",
"\n",
"print 'So, these are orthogonal.'\n",
"print '------------------------------------------'\n",
"print 'If inner product is defined as:'\n",
"print '(x1,x2)|(y1,y2) = x1y1- x2y1 - x1y2 + 4x2y2'\n",
"print 'Then, (x,y)|(-y,x) = -x*y+y**2-x**2+4*x*y = 0 if,'\n",
"print 'y = 1/2(-3 + sqrt(13))*x'\n",
"print 'or'\n",
"print 'y = 1/2(-3 - sqrt(13))*x'\n",
"print 'Hence,'\n",
"if y == (1./2*(-3 + np.sqrt(13))*x) or (1./2*(-3 - np.sqrt(13))*x):\n",
" print a\n",
" print 'is orthogonal to'\n",
" print b\n",
"else:\n",
" print a\n",
" print 'is not orthogonal to'\n",
" print b\n"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Page 282 Example 8.12"
]
},
{
"cell_type": "code",
"execution_count": 20,
"metadata": {
"collapsed": false
},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"b1 = [3 0 4]\n",
"b2 = [-1 0 7]\n",
"b3 = [ 2 9 11]\n",
"Applying the Gram-Schmidt process to b1,b2,b3:\n",
"a1 = [3 0 4]\n",
"a2 = [2 0 3]\n",
"a3 = [2 9 4]\n",
"{a1,a2,a3} are mutually orthogonal and hence forms orthogonal basis for R**3\n",
"Any arbitrary vector {x1,x2,x3} in R**3 can be expressed as:\n",
"y = {x1,x2,x3} = (3*x1 + 4*x3)/25*a1 + (-4*x1 + 3*x3)/25*a2 + x2/9*a3\n",
"x1 = 1\n",
"x2 = 2\n",
"x3 = 3\n",
"y = [0 0 0]\n",
"i.e. y = [x1 x2 x3], according to above equation.\n",
"Hence, we get the orthonormal basis as:\n",
", [ 0.6 0. 0.8]\n",
", [ 0.4 0. 0.6]\n",
"[ 0.22222222 1. 0.44444444]\n"
]
}
],
"source": [
"import numpy as np\n",
"b1 = np.array([3, 0, 4])\n",
"b2 = np.array([-1 ,0 ,7])\n",
"b3 = np.array([2 ,9 ,11])\n",
"print 'b1 = ',b1\n",
"print 'b2 = ',b2\n",
"print 'b3 = ',b3\n",
"print 'Applying the Gram-Schmidt process to b1,b2,b3:'\n",
"a1 = b1\n",
"print 'a1 = ',a1\n",
"a2 = b2-(np.transpose((b2*np.transpose(b1)))/25*b1)\n",
"print 'a2 = ',a2\n",
"a3 = b3-(np.transpose(b3*np.transpose(b1))/25*b1) - (np.transpose(b3*np.transpose(a2))/25*a2)\n",
"print 'a3 = ',a3\n",
"print '{a1,a2,a3} are mutually orthogonal and hence forms orthogonal basis for R**3'\n",
"print 'Any arbitrary vector {x1,x2,x3} in R**3 can be expressed as:'\n",
"print 'y = {x1,x2,x3} = (3*x1 + 4*x3)/25*a1 + (-4*x1 + 3*x3)/25*a2 + x2/9*a3'\n",
"x1 = 1#\n",
"x2 = 2#\n",
"x3 = 3#\n",
"y = (3*x1 + 4*x3)/25*a1 + (-4*x1 + 3*x3)/25*a2 + x2/9*a3#\n",
"print 'x1 = ',x1\n",
"print 'x2 = ',x2\n",
"print 'x3 = ',x3\n",
"print 'y = ',y\n",
"print 'i.e. y = [x1 x2 x3], according to above equation.'\n",
"print 'Hence, we get the orthonormal basis as:'\n",
"\n",
"print ',',1./5*a1\n",
"print ',',1./5*a2\n",
"print 1/9.*a3"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Page 283 Example 8.13"
]
},
{
"cell_type": "code",
"execution_count": 21,
"metadata": {
"collapsed": false
},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"A = [[ 1.25598176 1.81258697]\n",
" [ 0.6193707 0.80341686]]\n",
"b1 = [ 1.25598176 1.81258697]\n",
"b2 = [ 0.6193707 0.80341686]\n",
"Applying the orthogonalization process to b1,b2:\n",
"[1.255981755902444, 1.8125869670307564] a1 = \n",
"[] a2 = \n",
"a2 is not equal to zero if and only if b1 and b2 are linearly independent.\n",
"That is, if determinant of A is non-zero.\n"
]
}
],
"source": [
"import numpy as np\n",
"A = np.random.rand(2,2)\n",
"A[0,:] = A[0,:] + 1# #so b1 is not equal to zero\n",
"a = A[0,0]\n",
"b = A[0,1]\n",
"c = A[1,0]\n",
"d = A[1,1]\n",
"b1 = A[0,:]\n",
"b2 = A[1,:]\n",
"print 'A = ',A\n",
"print 'b1 = ',b1\n",
"print 'b2 = ',b2\n",
"print 'Applying the orthogonalization process to b1,b2:'\n",
"\n",
"a1 = [a, b]\n",
"a2 = (np.linalg.det(A)/(a**2 + b**2))*[-np.transpose(b), np.transpose(a)]\n",
"print a1,'a1 = '\n",
"print a2,'a2 = '\n",
"print 'a2 is not equal to zero if and only if b1 and b2 are linearly independent.'\n",
"print 'That is, if determinant of A is non-zero.'"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Page 286 Example 8.14"
]
},
{
"cell_type": "code",
"execution_count": 22,
"metadata": {
"collapsed": false
},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"v = [-10 2 8]\n",
"u = [ 3 12 -1]\n",
"Orthogonal projection of v1 on subspace W spanned by v2 is given by:\n",
"[-3 0 1]\n",
"Orthogonal projection of R**3 on W is the linear transformation E given by:\n",
"(x1,x2,x3) -> (3*x1 + 12*x2 - x3)/%d * (3 12 -1) 154\n",
"Rank(E) = 1\n",
"Nullity(E) = 2\n"
]
}
],
"source": [
"import numpy as np\n",
"v = np.array([-10 ,2 ,8])\n",
"u = np.array([3, 12, -1])\n",
"print 'v = ',v\n",
"print 'u = ',u\n",
"print 'Orthogonal projection of v1 on subspace W spanned by v2 is given by:'\n",
"a = (np.transpose(u*np.transpose(v)))/(u[0]**2 + u[1]**2 + u[2]**2) * u\n",
"print a\n",
"print 'Orthogonal projection of R**3 on W is the linear transformation E given by:'\n",
"print '(x1,x2,x3) -> (3*x1 + 12*x2 - x3)/%d * (3 12 -1)',(u[0]**2 + u[1]**2 + u[2]**2)\n",
"print 'Rank(E) = 1'\n",
"print 'Nullity(E) = 2'"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Page 288 Example 8.15"
]
},
{
"cell_type": "code",
"execution_count": 23,
"metadata": {
"collapsed": false
},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"f = (sqrt(2)*cos(2*pi*t) + sqrt(2)*sin(4*pi*t))**2\n",
"Integration (f dt) in limits 0 to 1 = 2.0\n"
]
}
],
"source": [
"from sympy.mpmath import quad,cos,sin,pi,sqrt\n",
"\n",
"#part c\n",
"print 'f = (sqrt(2)*cos(2*pi*t) + sqrt(2)*sin(4*pi*t))**2'\n",
"print 'Integration (f dt) in limits 0 to 1 = ',\n",
"x0 = 0#\n",
"x1 = 1#\n",
"X = quad(lambda t:(sqrt(2)*cos(2*pi*t) + sqrt(2)*sin(4*pi*t))**2,[x0,x1])\n",
"print X"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Page 294 Example 8.17"
]
},
{
"cell_type": "code",
"execution_count": 24,
"metadata": {
"collapsed": false
},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"Matrix of projection E in orthonormal basis is:\n",
"A = 1/154 * [[ 9 36 -3]\n",
" [ 36 144 -12]\n",
" [ -3 -12 1]]\n",
"A* = [[ 9 36 -3]\n",
" [ 36 144 -12]\n",
" [ -3 -12 1]]\n",
"Since, E = E* and A = A*, then A is also the matrix of E*\n",
"a1 = [154, 0, 0]\n",
"a2 = [145, -36, 3]\n",
"a3 = [-36, 10, 12]\n",
"{a1,a2,a3} is the basis.\n",
"Ea1 = [9, 36, -3]\n",
"Ea2 = [0, 0, 0]\n",
"Ea3 = [0, 0, 0]\n",
"Matrix B of E in the basis is:\n",
"B = \n",
"[[-1 0 0]\n",
" [-1 0 0]\n",
" [ 0 0 0]]\n",
"B* = \n",
"[[-1 -1 0]\n",
" [ 0 0 0]\n",
" [ 0 0 0]]\n",
"Since, B is not equal to B*, B is not the matrix of E*\n"
]
}
],
"source": [
"import numpy as np\n",
"#Equation given in example 14 is used.\n",
"def transform(x,y,z):\n",
" x1 = 3*x#\n",
" x2 = 12*y#\n",
" x3 = -z#\n",
" m = [x1 ,x2, x3]\n",
" return m\n",
"\n",
"print 'Matrix of projection E in orthonormal basis is:'\n",
"t1 = transform(3,3,3)#\n",
"t2 = transform(12,12,12)#\n",
"t3 = transform(-1,-1,-1)#\n",
"A = np.vstack([t1,t2,t3])#[t1# t2# t3]#\n",
"print 'A = 1/154 * ',A\n",
"\n",
"A1 = np.transpose(np.conj(A))\n",
"print 'A* = ',A1\n",
"print 'Since, E = E* and A = A*, then A is also the matrix of E*'\n",
"a1 = [154, 0, 0]#\n",
"a2 = [145 ,-36, 3]#\n",
"a3 = [-36 ,10 ,12]#\n",
"print 'a1 = ',a1\n",
"print 'a2 = ',a2\n",
"print 'a3 = ',a3\n",
"print '{a1,a2,a3} is the basis.'\n",
"Ea1 = [9 ,36 ,-3]#\n",
"Ea2 = [0 ,0, 0]#\n",
"Ea3 = [0 ,0 ,0]#\n",
"print 'Ea1 = ',Ea1\n",
"print 'Ea2 = ',Ea2\n",
"print 'Ea3 = ',Ea3\n",
"B = np.array([[-1, 0, 0],[-1, 0 ,0],[0, 0, 0]])\n",
"print 'Matrix B of E in the basis is:'\n",
"print 'B = \\n',B\n",
"B1 = np.transpose(np.conj(B))\n",
"print 'B* = \\n',B1\n",
"print 'Since, B is not equal to B*, B is not the matrix of E*'"
]
}
],
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