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author | kinitrupti | 2017-05-12 18:40:35 +0530 |
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committer | kinitrupti | 2017-05-12 18:40:35 +0530 |
commit | d36fc3b8f88cc3108ffff6151e376b619b9abb01 (patch) | |
tree | 9806b0d68a708d2cfc4efc8ae3751423c56b7721 /Linear_Algebra_by_K._Hoffman_and_R._Kunze/Chapter2.ipynb | |
parent | 1b1bb67e9ea912be5c8591523c8b328766e3680f (diff) | |
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Revised list of TBCs
Diffstat (limited to 'Linear_Algebra_by_K._Hoffman_and_R._Kunze/Chapter2.ipynb')
-rw-r--r--[-rwxr-xr-x] | Linear_Algebra_by_K._Hoffman_and_R._Kunze/Chapter2.ipynb | 104 |
1 files changed, 37 insertions, 67 deletions
diff --git a/Linear_Algebra_by_K._Hoffman_and_R._Kunze/Chapter2.ipynb b/Linear_Algebra_by_K._Hoffman_and_R._Kunze/Chapter2.ipynb index cac42c12..5659e3a2 100755..100644 --- a/Linear_Algebra_by_K._Hoffman_and_R._Kunze/Chapter2.ipynb +++ b/Linear_Algebra_by_K._Hoffman_and_R._Kunze/Chapter2.ipynb @@ -101,9 +101,9 @@ } ], "source": [ - "import numpy as np\n", + "from numpy import array\n", "\n", - "A = np.array([[1, 2, 0 ,3 ,0],[0, 0, 1, 4, 0],[0, 0, 0, 0, 1]])\n", + "A = array([[1, 2, 0 ,3 ,0],[0, 0, 1, 4, 0],[0, 0, 0, 0, 1]])\n", "print 'A = \\n',A\n", "print 'The subspace of F**5 spanned by a1 a2 a3(row vectors of A) is called row space of A.'\n", "a1 = A[0,:]\n", @@ -113,7 +113,7 @@ "print 'a2 = ',a2\n", "print 'a3 = ',a3\n", "print 'And, it is also the row space of B.'\n", - "B = np.array([[1, 2, 0, 3, 0],[0, 0, 1, 4, 0],[0, 0, 0, 0, 1],[-4, -8, 1 ,-8, 0]])\n", + "B = array([[1, 2, 0, 3, 0],[0, 0, 1, 4, 0],[0, 0, 0, 0, 1],[-4, -8, 1 ,-8, 0]])\n", "print 'B = \\n',B" ] }, @@ -137,36 +137,24 @@ "text": [ "V is the space of all polynomial functions over F.\n", "S contains the functions as:\n", - "n = 17\n", + "n = 5\n", "f0(x) = 1\n", "f1(x) = x\n", "f2(x) = x**2\n", "f3(x) = x**3\n", "f4(x) = x**4\n", - "f5(x) = x**5\n", - "f6(x) = x**6\n", - "f7(x) = x**7\n", - "f8(x) = x**8\n", - "f9(x) = x**9\n", - "f10(x) = x**10\n", - "f11(x) = x**11\n", - "f12(x) = x**12\n", - "f13(x) = x**13\n", - "f14(x) = x**14\n", - "f15(x) = x**15\n", - "f16(x) = x**16\n", "Then, V is the subspace spanned by set S.\n" ] } ], "source": [ - "import sympy as sp\n", - "import numpy as np\n", + "from sympy import Symbol\n", + "from numpy import random\n", "print 'V is the space of all polynomial functions over F.'\n", "print 'S contains the functions as:'\n", - "x = sp.Symbol(\"x\")\n", + "x = Symbol(\"x\")\n", "#n = round(rand()*10)#\n", - "n=np.random.randint(0,19)\n", + "n=random.randint(0,19)\n", "print 'n = ',n\n", "for i in range (0,n):\n", " f = x**i#\n", @@ -208,11 +196,11 @@ } ], "source": [ - "import numpy as np\n", - "a1 = np.array([3 ,0, -3])\n", - "a2 = np.array([-1 ,1 ,2])\n", - "a3 = np.array([4 ,2, -2])\n", - "a4 = np.array([2 ,1, 1])\n", + "from numpy import array\n", + "a1 = array([3 ,0, -3])\n", + "a2 = array([-1 ,1 ,2])\n", + "a3 = array([4 ,2, -2])\n", + "a4 = array([2 ,1, 1])\n", "print 'a1 = ',a1\n", "print 'a2 = ',a2\n", "print 'a3 = ',a3\n", @@ -248,27 +236,9 @@ "output_type": "stream", "text": [ "S is the subset of F**n consisting of n vectors.\n", - "n = 10\n", + "n = 1\n", "e1 = \n", - "[ 1. 0. 0. 0. 0. 0. 0. 0. 0. 0.]\n", - "e2 = \n", - "[ 0. 1. 0. 0. 0. 0. 0. 0. 0. 0.]\n", - "e3 = \n", - "[ 0. 0. 1. 0. 0. 0. 0. 0. 0. 0.]\n", - "e4 = \n", - "[ 0. 0. 0. 1. 0. 0. 0. 0. 0. 0.]\n", - "e5 = \n", - "[ 0. 0. 0. 0. 1. 0. 0. 0. 0. 0.]\n", - "e6 = \n", - "[ 0. 0. 0. 0. 0. 1. 0. 0. 0. 0.]\n", - "e7 = \n", - "[ 0. 0. 0. 0. 0. 0. 1. 0. 0. 0.]\n", - "e8 = \n", - "[ 0. 0. 0. 0. 0. 0. 0. 1. 0. 0.]\n", - "e9 = \n", - "[ 0. 0. 0. 0. 0. 0. 0. 0. 1. 0.]\n", - "e10 = \n", - "[ 0. 0. 0. 0. 0. 0. 0. 0. 0. 1.]\n", + "[ 1.]\n", "x1,x2,x3...xn are the scalars in F\n", "Putting a = x1*e1 + x2*e2 + x3*e3 + .... + xn*en\n", "So, a = (x1,x2,x3,...,xn)\n", @@ -280,12 +250,12 @@ } ], "source": [ - "import numpy as np\n", + "from numpy import array,random,identity\n", "print 'S is the subset of F**n consisting of n vectors.'\n", "#n = round(rand() *10 + 1)#\\\n", - "n=np.random.randint(0,19)\n", + "n=random.randint(0,19)\n", "print 'n = ',n\n", - "I = np.identity(n)\n", + "I = identity(n)\n", "for i in range(0,n):\n", " e = I[i,:]\n", " print 'e%d = '%(i+1)\n", @@ -342,17 +312,17 @@ } ], "source": [ - "import numpy as np\n", - "P = np.array([[-1, 4, 5],[ 0, 2, -3],[ 0, 0, 8]])\n", + "from numpy import array,transpose,linalg\n", + "P = array([[-1, 4, 5],[ 0, 2, -3],[ 0, 0, 8]])\n", "print 'P = \\n',P\n", - "print '\\ninverse(P) = \\n',np.linalg.inv(P)\n", + "print '\\ninverse(P) = \\n',linalg.inv(P)\n", "a1 = P[:,0]\n", "a2 = P[:,1]\n", "a3 = P[:,2]\n", "print 'The vectors forming basis of F**3 are a1'', a2'', a3'''\n", - "print \"a1' = \\n\",np.transpose(a1)\n", - "print \"\\na2' = \\n\",np.transpose(a2)\n", - "print \"\\na3' = \\n\",np.transpose(a3)\n", + "print \"a1' = \\n\",transpose(a1)\n", + "print \"\\na2' = \\n\",transpose(a2)\n", + "print \"\\na3' = \\n\",transpose(a3)\n", "print 'The coordinates x1'',x2'',x3'' of vector a = [x1,x2,x3] is given by inverse(P)*[x1# x2# x3]'\n", "t = -10*a1 - 1./2*a2 - a3#\n", "print 'And, -10*a1'' - 1/2*a2'' - a3'' = ',t" @@ -470,7 +440,7 @@ } ], "source": [ - "import numpy as np\n", + "from numpy import array,identity,rank,vstack\n", "a1 = [1 ,2 ,2, 1]#\n", "a2 = [0 ,2 ,0 ,1]#\n", "a3 = [-2, 0, -4, 3]#\n", @@ -480,12 +450,12 @@ "print 'a3 = ',a3\n", "print 'The matrix A from these vectors will be:'\n", "#A = [a1],[a2], [a3]]\n", - "A=np.array([a1,a2,a3])\n", + "A=array([a1,a2,a3])\n", "print 'A = \\n',A\n", "\n", "print 'Finding Row reduced echelon matrix of A that is given by R'\n", "print 'And applying same operations on identity matrix Q such that R = QA'\n", - "Q = np.identity(3)\n", + "Q = identity(3)\n", "print 'Q = \\n',Q\n", "T = A# #Temporary matrix to store A\n", "print 'Applying row transformations on A and Q,we get'\n", @@ -520,7 +490,7 @@ "print 'A = \\n',A\n", "print 'Q = \\n',Q\n", "#part a\n", - "print 'rank of R = ',np.rank(R)\n", + "print 'rank of R = ',rank(R)\n", "\n", "print 'Since, Rank of R is 3, so a1, a2, a3 are independent'\n", "#part b\n", @@ -542,19 +512,19 @@ "print 'Since a1'' a2'' a3'' are all of the form (y1 y2 y3 y4) with y3 = 2*y1, hence they are in W.'\n", "print 'So, they are independent.'\n", "#part d\n", - "c = np.array([c1,c2,c3])\n", - "P = np.identity(3)\n", + "c = array([c1,c2,c3])\n", + "P = identity(3)\n", "for i in range(0,3):\n", " b1 = c[i,0]\n", " b2 = c[i,1]\n", " b4 = c[i,3]\n", - " x1 = np.array([b1, b2, b4]) * Q[:,0]\n", - " x2 = np.array([b1, b2, b4])*Q[:,1]\n", - " x3 = np.array([b1, b2, b4])*Q[:,2]\n", + " x1 = array([b1, b2, b4]) * Q[:,0]\n", + " x2 = array([b1, b2, b4])*Q[:,1]\n", + " x3 = array([b1, b2, b4])*Q[:,2]\n", " \n", "\n", "print 'Required matrix P such that X = PX'' is:'\n", - "P=np.vstack([x1,x2,x3])\n", + "P=vstack([x1,x2,x3])\n", "print 'P = \\n',P\n", "#print x1" ] @@ -686,13 +656,13 @@ } ], "source": [ - "import numpy as np\n", - "A = np.array([[1, 2, 0, 3, 0],[1, 2, -1, -1, 0],[0 ,0 ,1 ,4 ,0],[2, 4 ,1 ,10, 1],[0 ,0 ,0 ,0 ,1]])\n", + "from numpy import array,identity\n", + "A = array([[1, 2, 0, 3, 0],[1, 2, -1, -1, 0],[0 ,0 ,1 ,4 ,0],[2, 4 ,1 ,10, 1],[0 ,0 ,0 ,0 ,1]])\n", "print 'A = \\n',A\n", "#part a\n", "T = A# #Temporary storing A in T\n", "print 'Taking an identity matrix P:'\n", - "P = np.identity(5)\n", + "P = identity(5)\n", "print 'P = \\n',P\n", "print 'Applying row transformations on P and A to get a row reduced echelon matrix R:'\n", "print 'R2 = R2 - R1 and R4 = R4 - 2* R1'\n", |