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Diffstat (limited to 'Linear_Algebra_by_K._Hoffman_and_R._Kunze/Chapter8.ipynb')
| -rw-r--r--[-rwxr-xr-x] | Linear_Algebra_by_K._Hoffman_and_R._Kunze/Chapter8.ipynb | 171 |
1 files changed, 85 insertions, 86 deletions
diff --git a/Linear_Algebra_by_K._Hoffman_and_R._Kunze/Chapter8.ipynb b/Linear_Algebra_by_K._Hoffman_and_R._Kunze/Chapter8.ipynb index 6f1bfe72..6759a185 100755..100644 --- a/Linear_Algebra_by_K._Hoffman_and_R._Kunze/Chapter8.ipynb +++ b/Linear_Algebra_by_K._Hoffman_and_R._Kunze/Chapter8.ipynb @@ -16,7 +16,7 @@ }, { "cell_type": "code", - "execution_count": 16, + "execution_count": 1, "metadata": { "collapsed": false }, @@ -25,31 +25,30 @@ "name": "stdout", "output_type": "stream", "text": [ - "n = 5\n", - "a = [[ 1. 4. 2. 8. 8.]]\n", - "b = [[ 10. 8. 3. 1. 8.]]\n", + "n = 4\n", + "a = [[ 1. 5. 4. 1.]]\n", + "b = [[ 8. 6. 6. 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" + "[[ 8. 40. 32. 8.]\n", + " [ 6. 30. 24. 6.]\n", + " [ 6. 30. 24. 6.]\n", + " [ 8. 40. 32. 8.]]\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", + "from numpy import array,random,transpose\n", + "n=random.randint(2,9)\n", + "a=random.rand(1,n)\n", + "b=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)" + "print 'Then, (a|b) = \\n\\n',a*transpose(b)" ] }, { @@ -61,7 +60,7 @@ }, { "cell_type": "code", - "execution_count": 17, + "execution_count": 2, "metadata": { "collapsed": false }, @@ -70,16 +69,16 @@ "name": "stdout", "output_type": "stream", "text": [ - "a = [[ 4. 3.]]\n", - "b = [[ 7. 5.]]\n", - "Then, a|b = 47.0\n" + "a = [[ 9. 4.]]\n", + "b = [[ 1. 9.]]\n", + "Then, a|b = 68.0\n" ] } ], "source": [ - "import numpy as np\n", - "a=np.random.rand(1,2)\n", - "b=np.random.rand(1,2)\n", + "from numpy import array,random,transpose\n", + "a=random.rand(1,2)\n", + "b=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", @@ -102,7 +101,7 @@ }, { "cell_type": "code", - "execution_count": 18, + "execution_count": 3, "metadata": { "collapsed": false }, @@ -112,62 +111,62 @@ "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", + "x1 = 0.248003219206\n", + "x2 = 0.968759208092\n", "B = \n", - "[[ 0.38354723 0.92352126 0. ]\n", + "[[ 0.24800322 0.96875921 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", + "a1 = [ 0.24800322 0.96875921 0. ]\n", + "a2 = [-0.2402554 0.0615056 0. ]\n", "a3 = [0 0 1]\n", "U = \n", - "[[[ 0.38354723 0.92352126 0. ]]\n", + "[[[ 0.24800322 0.96875921 0. ]]\n", "\n", - " [[-0.92352126 0.38354723 0. ]]\n", + " [[-0.96875921 0.24800322 0. ]]\n", "\n", " [[ 0. 0. 1. ]]]\n", "M = \n", "[[ 1. 0. 0. ]\n", - " [-2.40784236 2.60724085 0. ]\n", + " [-3.90623642 4.03220572 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", + "inverse(M) * U = [[[ 0.24800322 -0. -0. ]\n", + " [ 0.2402554 0.2402554 0. ]\n", + " [ 0. 0. 0. ]]\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", + " [[-0.96875921 -0. -0. ]\n", + " [-0.9384944 0.0615056 0. ]\n", + " [-0. 0. 0. ]]\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", + " [[ 0. -0. -0. ]\n", + " [ 0. 0. 0. ]\n", + " [ 0. 0. 1. ]]]\n", "So, B = inverse(M) * U\n" ] } ], "source": [ - "import numpy as np\n", + "from numpy import array,random,transpose,linalg,sqrt\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", + "x1 = random.rand()\n", + "x2 = 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", + "B = 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", + "a1 = array([x1, x2, 0])\n", + "a2 = array([0 ,1 ,0]) - x2 * array([x1 ,x2 ,0])\n", + "a3 = 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", + "U = 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", + "M = 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 'inverse(M) * U = ',linalg.inv(M) * U\n", "print 'So, B = inverse(M) * U'" ] }, @@ -180,7 +179,7 @@ }, { "cell_type": "code", - "execution_count": 19, + "execution_count": 4, "metadata": { "collapsed": false }, @@ -189,8 +188,8 @@ "name": "stdout", "output_type": "stream", "text": [ - "(x,y) = [[ 5. 3.]]\n", - "(-y,x) = [-3.0, 5.0]\n", + "(x,y) = [[ 7. 4.]]\n", + "(-y,x) = [-4.0, 7.0]\n", "Inner product of these vectors is:\n", "(x,y)|(-y,x) = 0.0\n", "So, these are orthogonal.\n", @@ -202,16 +201,16 @@ "or\n", "y = 1/2(-3 - sqrt(13))*x\n", "Hence,\n", - "[[ 5. 3.]]\n", + "[[ 7. 4.]]\n", "is orthogonal to\n", - "[-3.0, 5.0]\n" + "[-4.0, 7.0]\n" ] } ], "source": [ - "import numpy as np\n", + "from numpy import array,random,transpose,linalg,sqrt\n", "#a = round(rand(1,2) * 10)#\n", - "a=np.random.rand(1,2)\n", + "a=random.rand(1,2)\n", "for j in [0,1]:\n", " a[0,j]=round(a[0,j]*10)\n", "\n", @@ -233,7 +232,7 @@ "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", + "if y == (1./2*(-3 + sqrt(13))*x) or (1./2*(-3 - sqrt(13))*x):\n", " print a\n", " print 'is orthogonal to'\n", " print b\n", @@ -252,7 +251,7 @@ }, { "cell_type": "code", - "execution_count": 20, + "execution_count": 5, "metadata": { "collapsed": false }, @@ -284,19 +283,19 @@ } ], "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", + "from numpy import array,random,transpose,linalg,sqrt\n", + "b1 = array([3, 0, 4])\n", + "b2 = array([-1 ,0 ,7])\n", + "b3 = 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", + "a2 = b2-(transpose((b2*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", + "a3 = b3-(transpose(b3*transpose(b1))/25*b1) - (transpose(b3*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", @@ -326,7 +325,7 @@ }, { "cell_type": "code", - "execution_count": 21, + "execution_count": 6, "metadata": { "collapsed": false }, @@ -335,12 +334,12 @@ "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", + "A = [[ 1.83351494 1.26265003]\n", + " [ 0.46651205 0.76790774]]\n", + "b1 = [ 1.83351494 1.26265003]\n", + "b2 = [ 0.46651205 0.76790774]\n", "Applying the orthogonalization process to b1,b2:\n", - "[1.255981755902444, 1.8125869670307564] a1 = \n", + "[1.8335149394280341, 1.2626500316837608] 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" @@ -348,8 +347,8 @@ } ], "source": [ - "import numpy as np\n", - "A = np.random.rand(2,2)\n", + "from numpy import array,random,transpose,linalg,sqrt\n", + "A = 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", @@ -363,7 +362,7 @@ "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", + "a2 = (linalg.det(A)/(a**2 + b**2))*[-transpose(b), 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", @@ -379,7 +378,7 @@ }, { "cell_type": "code", - "execution_count": 22, + "execution_count": 7, "metadata": { "collapsed": false }, @@ -400,13 +399,13 @@ } ], "source": [ - "import numpy as np\n", - "v = np.array([-10 ,2 ,8])\n", - "u = np.array([3, 12, -1])\n", + "from numpy import array,random,transpose,linalg,sqrt\n", + "v = array([-10 ,2 ,8])\n", + "u = 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", + "a = (transpose(u*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", @@ -423,7 +422,7 @@ }, { "cell_type": "code", - "execution_count": 23, + "execution_count": 8, "metadata": { "collapsed": false }, @@ -438,7 +437,7 @@ } ], "source": [ - "from sympy.mpmath import quad,cos,sin,pi,sqrt\n", + "from 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", @@ -458,7 +457,7 @@ }, { "cell_type": "code", - "execution_count": 24, + "execution_count": 9, "metadata": { "collapsed": false }, @@ -496,7 +495,7 @@ } ], "source": [ - "import numpy as np\n", + "from numpy import vstack,array,transpose,conj\n", "#Equation given in example 14 is used.\n", "def transform(x,y,z):\n", " x1 = 3*x#\n", @@ -509,10 +508,10 @@ "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", + "A = vstack([t1,t2,t3])#[t1# t2# t3]#\n", "print 'A = 1/154 * ',A\n", "\n", - "A1 = np.transpose(np.conj(A))\n", + "A1 = transpose(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", @@ -528,10 +527,10 @@ "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", + "B = 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", + "B1 = transpose(conj(B))\n", "print 'B* = \\n',B1\n", "print 'Since, B is not equal to B*, B is not the matrix of E*'" ] |