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Diffstat (limited to 'Linear_Algebra_by_K._Hoffman_and_R._Kunze/Chapter3.ipynb')
| -rw-r--r--[-rwxr-xr-x] | Linear_Algebra_by_K._Hoffman_and_R._Kunze/Chapter3.ipynb | 128 |
1 files changed, 61 insertions, 67 deletions
diff --git a/Linear_Algebra_by_K._Hoffman_and_R._Kunze/Chapter3.ipynb b/Linear_Algebra_by_K._Hoffman_and_R._Kunze/Chapter3.ipynb index 2e2e4be2..4007526a 100755..100644 --- a/Linear_Algebra_by_K._Hoffman_and_R._Kunze/Chapter3.ipynb +++ b/Linear_Algebra_by_K._Hoffman_and_R._Kunze/Chapter3.ipynb @@ -41,7 +41,7 @@ } ], "source": [ - "import numpy as np\n", + "from numpy import array\n", "a1 = [1, 2]#\n", "a2 = [3 ,4]#\n", "print 'a1 = ',a1\n", @@ -55,7 +55,7 @@ "print 'Now, we find scalars c1 and c2 for that we know T(c1a1 + c2a2) = c1(Ta1) + c2(Ta2))'\n", "print 'if(1,0) = c1(1,2) + c2(3,4), then '\n", "#c = inv([a1#a2]') * [1#0]#\n", - "c=np.array([a1,a2]).dot(np.array([[1],[0]]))\n", + "c=array([a1,a2]).dot(array([[1],[0]]))\n", "c1 = c[0,0]\n", "c2 = c[1,0]\n", "print 'c1 = ',c1\n", @@ -84,27 +84,27 @@ "name": "stdout", "output_type": "stream", "text": [ - "x1 = 5\n", - "x2 = 2\n", - "T(5,2) = [7, 5]\n", + "x1 = 3\n", + "x2 = 8\n", + "T(3,8) = [11, 3]\n", "If, T(x1,x2) = 0, then\n", "x1 = x2 = 0\n", "So, T is non-singular\n", "z1,z2 are two scalars in F\n", "z1 = 0\n", - "z2 = 8\n", - "So, x1 = 8\n", - "x2 = -8\n", + "z2 = 4\n", + "So, x1 = 4\n", + "x2 = -4\n", "Hence, T is onto.\n", - "inverse(T) = [8, -8]\n" + "inverse(T) = [4, -4]\n" ] } ], "source": [ - "import numpy as np\n", + "from numpy import array,random\n", "#x = round(rand(1,2) * 10)#\n", - "x1 = np.random.randint(1,9)\n", - "x2 = np.random.randint(1,9)\n", + "x1 = random.randint(1,9)\n", + "x2 = random.randint(1,9)\n", "T = [x1+x2 ,x1]\n", "print 'x1 = ',x1\n", "print 'x2 = ',x2\n", @@ -115,8 +115,8 @@ "print 'So, T is non-singular'\n", "print 'z1,z2 are two scalars in F'\n", "\n", - "z1 = np.random.randint(0,9)\n", - "z2 = np.random.randint(0,9)\n", + "z1 = random.randint(0,9)\n", + "z2 = random.randint(0,9)\n", "print 'z1 = ',z1\n", "print 'z2 = ',z2\n", "x1 = z2#\n", @@ -183,7 +183,7 @@ }, { "cell_type": "code", - "execution_count": 5, + "execution_count": 4, "metadata": { "collapsed": false }, @@ -207,14 +207,14 @@ } ], "source": [ - "import numpy as np\n", - "import sympy as sp\n", + "from numpy import array,zeros\n", + "from sympy import Symbol,diff\n", "print 'Differentiation operator D is defined as:'\n", - "D = np.zeros([4,4])\n", - "x=sp.Symbol('x')\n", + "D = zeros([4,4])\n", + "x=Symbol('x')\n", "for i in range(1,5):\n", " t= i-1#\n", - " f = sp.diff(x**t,'x')\n", + " f = diff(x**t,'x')\n", " print '(Df%d)(x) = '%(i),\n", " print f\n", " if not(i == 1):\n", @@ -234,7 +234,7 @@ }, { "cell_type": "code", - "execution_count": 6, + "execution_count": 5, "metadata": { "collapsed": false }, @@ -263,19 +263,19 @@ } ], "source": [ - "import numpy as np\n", + "from numpy import array,transpose,linalg\n", "print 'T is a linear operator on R**2 defined as T(x1,x2) = (x1,0)'\n", "print 'So, the matrix T in standard ordered basis B = {e1,e2} is '\n", - "T = np.array([[1, 0],[0, 0]])\n", + "T = array([[1, 0],[0, 0]])\n", "print '[T]B = ',T\n", "print 'Let B'' is the ordered basis for R**2 consisting of vectors:'\n", - "E1 = np.array([1, 1])\n", - "E2 = np.array([2 ,1])\n", + "E1 = array([1, 1])\n", + "E2 = array([2 ,1])\n", "print 'E1 = ',E1\n", "print 'E2 = ',E2\n", - "P = np.transpose(([E1,E2]))\n", + "P = transpose(([E1,E2]))\n", "print 'So, matrix P = \\n',P\n", - "Pinv=np.linalg.inv(P)\n", + "Pinv=linalg.inv(P)\n", "print 'P inverse = \\n',Pinv\n", "T1 = Pinv*T*P#\n", "print 'So, matrix T in ordered basis B'' is [T]B'' = \\n',T1" @@ -290,7 +290,7 @@ }, { "cell_type": "code", - "execution_count": 8, + "execution_count": 6, "metadata": { "collapsed": false }, @@ -316,24 +316,24 @@ } ], "source": [ - "import sympy as sp\n", - "t = sp.Symbol(\"t\")\n", + "from sympy import Symbol, Matrix\n", + "t = Symbol(\"t\")\n", "print 'g1 = f1'\n", "print 'g2 = t*f1 + f2'\n", "print 'g3 = t**2*f1 + 2*t*f2 + f3'\n", "print 'g4 = t**3*f1 + 3*t**2*f2 + 3*t*f3 + f4'\n", - "P = sp.Matrix(([1, t, t**2, t**3],[0 ,1 ,2*t, 3*t**2],[0, 0, 1, 3*t],[0, 0, 0, 1]))\n", + "P = Matrix(([1, t, t**2, t**3],[0 ,1 ,2*t, 3*t**2],[0, 0, 1, 3*t],[0, 0, 0, 1]))\n", "print 'P = \\n',P\n", "\n", - "print 'inverse P = \\n',sp.Matrix.inv(P)\n", + "print 'inverse P = \\n',Matrix.inv(P)\n", "\n", "\n", "\n", "print 'Matrix of differentiation operator D in ordered basis B is:'# #As found in example 15\n", - "D = sp.Matrix(([0, 1, 0, 0],[0, 0, 2, 0],[0, 0, 0, 3],[0, 0, 0, 0]))\n", + "D = Matrix(([0, 1, 0, 0],[0, 0, 2, 0],[0, 0, 0, 3],[0, 0, 0, 0]))\n", "print 'D = \\n',D\n", "print 'Matrix of D in ordered basis B'' is:'\n", - "print 'inverse(P) * D * P = ',sp.Matrix.inv(P)*D*P\n" + "print 'inverse(P) * D * P = ',Matrix.inv(P)*D*P\n" ] }, { @@ -345,7 +345,7 @@ }, { "cell_type": "code", - "execution_count": 9, + "execution_count": 7, "metadata": { "collapsed": false }, @@ -354,47 +354,41 @@ "name": "stdout", "output_type": "stream", "text": [ - "n = 8\n", + "n = 5\n", "A = \n", - "[[ 7. 3. 0. 4. 6. 8. 4. 4.]\n", - " [ 6. 4. 2. 8. 7. 8. 1. 7.]\n", - " [ 7. 0. 9. 3. 10. 9. 3. 8.]\n", - " [ 7. 5. 10. 1. 8. 6. 6. 5.]\n", - " [ 8. 8. 9. 9. 1. 9. 10. 4.]\n", - " [ 6. 3. 5. 2. 2. 4. 8. 4.]\n", - " [ 5. 1. 1. 2. 6. 9. 9. 5.]\n", - " [ 8. 6. 9. 9. 8. 9. 1. 2.]]\n", + "[[ 5. 9. 7. 8. 6.]\n", + " [ 3. 3. 2. 7. 6.]\n", + " [ 5. 1. 3. 2. 6.]\n", + " [ 1. 2. 10. 4. 1.]\n", + " [ 9. 0. 7. 5. 7.]]\n", "Trace of A:\n", - "tr(A) = 37.0\n", + "tr(A) = 22.0\n", "--------------------------------\n", - "c = 3\n", + "c = 2\n", "B = \n", - "[[ 4. 6. 10. 5. 8. 4. 1. 9.]\n", - " [ 9. 9. 3. 6. 3. 8. 2. 6.]\n", - " [ 1. 6. 0. 7. 7. 2. 8. 4.]\n", - " [ 5. 5. 9. 7. 9. 3. 9. 9.]\n", - " [ 7. 7. 10. 6. 1. 1. 7. 4.]\n", - " [ 0. 3. 10. 9. 5. 2. 8. 4.]\n", - " [ 1. 8. 2. 4. 5. 4. 4. 8.]\n", - " [ 7. 0. 1. 8. 2. 7. 4. 7.]]\n", + "[[ 6. 8. 8. 2. 4.]\n", + " [ 7. 6. 4. 3. 7.]\n", + " [ 6. 9. 8. 4. 8.]\n", + " [ 1. 4. 8. 4. 6.]\n", + " [ 10. 8. 2. 1. 6.]]\n", "Trace of B:\n", - "tr(B) = 34.0\n", - "tr(cA + B) = 145.0\n" + "tr(B) = 30.0\n", + "tr(cA + B) = 74.0\n" ] } ], "source": [ - "import numpy as np\n", + "from numpy import array,random\n", "def trace_matrix(M,n):\n", " tr=0\n", " for i in range(0,n):\n", " tr = tr + M[i,i]#\n", " return tr\n", "#n = round(rand() * 10 + 2)#\n", - "n=np.random.randint(1,9)\n", + "n=random.randint(1,9)\n", "print 'n = ',n\n", "#A = round(rand(n,n) * 10)#\n", - "A=np.random.rand(n,n)\n", + "A=random.rand(n,n)\n", "for x in range(0,n):\n", " for y in range(0,n):\n", " A[x,y]=round(A[x,y]*10)\n", @@ -407,10 +401,10 @@ "print 'tr(A) = ',tr1\n", "print '--------------------------------'\n", "#c = round(rand() * 10 + 2)#\n", - "c=np.random.randint(2,9)\n", + "c=random.randint(2,9)\n", "print 'c = ',c\n", "\n", - "B=np.random.rand(n,n)\n", + "B=random.rand(n,n)\n", "for x in range(0,n):\n", " for y in range(0,n):\n", " B[x,y]=round(B[x,y]*10)\n", @@ -431,7 +425,7 @@ }, { "cell_type": "code", - "execution_count": 10, + "execution_count": 8, "metadata": { "collapsed": false }, @@ -488,9 +482,9 @@ } ], "source": [ - "import numpy as np\n", + "from numpy import array\n", "print 'Matrix represented by given linear functionals on R**4:'\n", - "A = np.array([[1, 2 ,2 ,1],[0, 2, 0, 1],[-2 ,0 ,-4, 3]])\n", + "A = array([[1, 2 ,2 ,1],[0, 2, 0, 1],[-2 ,0 ,-4, 3]])\n", "print 'A = \\n',A\n", "T = A #Temporary matrix to store A\n", "print 'To find Row reduced echelon matrix of A given by R:'\n", @@ -532,7 +526,7 @@ }, { "cell_type": "code", - "execution_count": 11, + "execution_count": 9, "metadata": { "collapsed": false }, @@ -567,7 +561,7 @@ } ], "source": [ - "import numpy as np\n", + "from numpy import array\n", "print 'W be the subspace of R**5 spanned by vectors:'\n", "a1 = [2, -2, 3 ,4 ,-1]#\n", "a2 = [-1, 1, 2, 5, 2]#\n", @@ -578,7 +572,7 @@ "print 'a3 = ',a3\n", "print 'a4 = ',a4\n", "print 'Matrix A by the row vectors a1,a2,a3,a4 will be:'\n", - "A = np.array([a1,a2,a3,a4])\n", + "A = array([a1,a2,a3,a4])\n", "print 'A = \\n',A\n", "print 'After Applying row transformations, we get the row reduced echelon matrix R of A'\n", "\n", |