{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Chapter 2 - Vector spaces"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Page 37 Example 2.8"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "a1 =  [1, 2, 0, 3, 0]\n",
      "a2 =  [0, 0, 1, 4, 0]\n",
      "a3 =  [0, 0, 0, 0, 1]\n",
      "By theorem 3, vector a is in subspace W of F**5 spanned by a1, a2, a3\n",
      "if and only if there exist scalars c1, c2, c3 such that\n",
      "a= c1a1 + c2a2 + c3a3\n",
      "So, a = (c1,2*c1,c2,3c1+4c2,c3)\n",
      "c1 =  -3\n",
      "c2 =  1\n",
      "c3 =  2\n",
      "Therefore, a =  [0, 0, 1, 4, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1]\n",
      "This shows, a is in W\n",
      "And (2,4,6,7,8) is not in W as there is no value of c1 c2 c3 that satisfies the equation\n"
     ]
    }
   ],
   "source": [
    "\n",
    "a1 = [1, 2 ,0 ,3, 0]#\n",
    "a2 =[0, 0 ,1 ,4 ,0]#\n",
    "a3 = [0 ,0 ,0 ,0, 1]#\n",
    "print 'a1 = ',a1\n",
    "print 'a2 = ',a2\n",
    "print 'a3 = ',a3\n",
    "print 'By theorem 3, vector a is in subspace W of F**5 spanned by a1, a2, a3'\n",
    "print 'if and only if there exist scalars c1, c2, c3 such that'\n",
    "print 'a= c1a1 + c2a2 + c3a3'\n",
    "print 'So, a = (c1,2*c1,c2,3c1+4c2,c3)'\n",
    "c1 = -3#\n",
    "c2 = 1#\n",
    "c3 = 2#\n",
    "a = c1*a1 + c2*a2 + c3*a3#\n",
    "print 'c1 = ',c1\n",
    "print 'c2 = ',c2\n",
    "print 'c3 = ',c3\n",
    "print 'Therefore, a = ',a\n",
    "print 'This shows, a is in W'\n",
    "print 'And (2,4,6,7,8) is not in W as there is no value of c1 c2 c3 that satisfies the equation'"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Page 38 Example 2.10"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "A = \n",
      "[[1 2 0 3 0]\n",
      " [0 0 1 4 0]\n",
      " [0 0 0 0 1]]\n",
      "The subspace of F**5 spanned by a1 a2 a3(row vectors of A) is called row space of A.\n",
      "a1 =  [1 2 0 3 0]\n",
      "a2 =  [0 0 1 4 0]\n",
      "a3 =  [0 0 0 0 1]\n",
      "And, it is also the row space of B.\n",
      "B = \n",
      "[[ 1  2  0  3  0]\n",
      " [ 0  0  1  4  0]\n",
      " [ 0  0  0  0  1]\n",
      " [-4 -8  1 -8  0]]\n"
     ]
    }
   ],
   "source": [
    "from numpy import array\n",
    "\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",
    "a2 = A[1,:]\n",
    "a3 = A[2,:]\n",
    "print 'a1 = ',a1\n",
    "print 'a2 = ',a2\n",
    "print 'a3 = ',a3\n",
    "print 'And, it is also the row space of B.'\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"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Page 39 Example 2.11"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "V is the space of all polynomial functions over F.\n",
      "S contains the functions as:\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",
      "Then, V is the subspace spanned by set S.\n"
     ]
    }
   ],
   "source": [
    "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 = Symbol(\"x\")\n",
    "#n = round(rand()*10)#\n",
    "n=random.randint(0,19)\n",
    "print 'n = ',n\n",
    "for i in range (0,n):\n",
    "    f = x**i#\n",
    "    print 'f%d(x) = '%(i,),f\n",
    "    \n",
    "\n",
    "print 'Then, V is the subspace spanned by set S.'"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Page 41 Example 2.12"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "a1 =  [ 3  0 -3]\n",
      "a2 =  [-1  1  2]\n",
      "a3 =  [ 4  2 -2]\n",
      "a4 =  [2 1 1]\n",
      " Since, 2 * a1 + 2 * a2 - a3 + 0 * a4 =  [0 0 0] = 0\n",
      "a1,a2,a3,a4 are linearly independent\n",
      "Now, e1 =  [1, 0, 0]\n",
      "e2 =  [0, 1, 0]\n",
      "e3 =  [0, 0, 1]\n",
      "Also, e1,e2,e3 are linearly independent.\n"
     ]
    }
   ],
   "source": [
    "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",
    "print 'a4 = ',a4\n",
    "t = 2 * a1 + 2 * a2 - a3 + 0 * a4\n",
    "print ' Since, 2 * a1 + 2 * a2 - a3 + 0 * a4 = ',t,'= 0'\n",
    "print 'a1,a2,a3,a4 are linearly independent'\n",
    "e1 = [1, 0, 0]#\n",
    "e2 = [0 ,1 ,0]#\n",
    "e3 = [0 ,0, 1]#\n",
    "print  'Now, e1 = ',e1\n",
    "print  'e2 = ',e2\n",
    "print  'e3 = ',e3\n",
    "print 'Also, e1,e2,e3 are linearly independent.'"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Page 41 Example 2.13"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "S is the subset of F**n consisting of n vectors.\n",
      "n =  1\n",
      "e1 = \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",
      "Therefore, e1,e2..,en span F**n\n",
      "a = 0 if x1 = x2 = x3 = .. = xn = 0\n",
      "So,e1,e2,e3,..,en are linearly independent.\n",
      "The set S = {e1,e2,..,en} is called standard basis of F**n\n"
     ]
    }
   ],
   "source": [
    "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=random.randint(0,19)\n",
    "print 'n = ',n\n",
    "I = identity(n)\n",
    "for i in range(0,n):\n",
    "    e = I[i,:]\n",
    "    print 'e%d = '%(i+1)\n",
    "    print e\n",
    "\n",
    "print 'x1,x2,x3...xn are the scalars in F'\n",
    "print 'Putting a = x1*e1 + x2*e2 + x3*e3 + .... + xn*en'\n",
    "print 'So, a = (x1,x2,x3,...,xn)'\n",
    "print 'Therefore, e1,e2..,en span F**n'\n",
    "print 'a = 0 if x1 = x2 = x3 = .. = xn = 0'\n",
    "print 'So,e1,e2,e3,..,en are linearly independent.'\n",
    "print 'The set S = {e1,e2,..,en} is called standard basis of F**n'"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Page 54 Example 2.20"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "P = \n",
      "[[-1  4  5]\n",
      " [ 0  2 -3]\n",
      " [ 0  0  8]]\n",
      "\n",
      "inverse(P) = \n",
      "[[-1.      2.      1.375 ]\n",
      " [ 0.      0.5     0.1875]\n",
      " [ 0.      0.      0.125 ]]\n",
      "The vectors forming basis of F**3 are a1, a2, a3\n",
      "a1' = \n",
      "[-1  0  0]\n",
      "\n",
      "a2' = \n",
      "[4 2 0]\n",
      "\n",
      "a3' = \n",
      "[ 5 -3  8]\n",
      "The coordinates x1,x2,x3 of vector a = [x1,x2,x3] is given by inverse(P)*[x1# x2# x3]\n",
      "And, -10*a1 - 1/2*a2 - a3 =  [ 3.  2. -8.]\n"
     ]
    }
   ],
   "source": [
    "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',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\",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"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Page 60 Example 2.21"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Given row vectors are:\n",
      "a1 =  [1, 2, 2, 1]\n",
      "a2 =  [0, 2, 0, 1]\n",
      "a3 =  [-2, 0, -4, 3]\n",
      "The matrix A from these vectors will be:\n",
      "A = \n",
      "[[ 1  2  2  1]\n",
      " [ 0  2  0  1]\n",
      " [-2  0 -4  3]]\n",
      "Finding Row reduced echelon matrix of A that is given by R\n",
      "And applying same operations on identity matrix Q such that R = QA\n",
      "Q = \n",
      "[[ 1.  0.  0.]\n",
      " [ 0.  1.  0.]\n",
      " [ 0.  0.  1.]]\n",
      "Applying row transformations on A and Q,we get\n",
      "R1 = R1-R2\n",
      "A = \n",
      "[[ 1  0  2  0]\n",
      " [ 0  2  0  1]\n",
      " [-2  0 -4  3]]\n",
      "Q = \n",
      "[[ 1. -1.  0.]\n",
      " [ 0.  1.  0.]\n",
      " [ 0.  0.  1.]]\n",
      "R3 = R3 + 2*R1\n",
      "A = \n",
      "[[1 0 2 0]\n",
      " [0 2 0 1]\n",
      " [0 0 0 3]]\n",
      "Q = \n",
      "[[ 1. -1.  0.]\n",
      " [ 0.  1.  0.]\n",
      " [ 2. -2.  1.]]\n",
      "R3 = R3/3\n",
      "A = \n",
      "[[1 0 2 0]\n",
      " [0 2 0 1]\n",
      " [0 0 0 1]]\n",
      "Q = \n",
      "[[ 1.         -1.          0.        ]\n",
      " [ 0.          1.          0.        ]\n",
      " [ 0.66666667 -0.66666667  0.33333333]]\n",
      "R2 = R2/2\n",
      "A = \n",
      "[[1 0 2 0]\n",
      " [0 1 0 0]\n",
      " [0 0 0 1]]\n",
      "Q = \n",
      "[[ 1.         -1.          0.        ]\n",
      " [ 0.          5.          0.        ]\n",
      " [ 0.66666667 -0.66666667  0.33333333]]\n",
      "R2 = R2 - 1/2*R3\n",
      "A = \n",
      "[[1 0 2 0]\n",
      " [0 1 0 0]\n",
      " [0 0 0 1]]\n",
      "Q = \n",
      "[[ 1.         -1.          0.        ]\n",
      " [-0.33333333  5.33333333 -0.16666667]\n",
      " [ 0.66666667 -0.66666667  0.33333333]]\n",
      "Row reduced echelon matrix:\n",
      "A = \n",
      "[[1 0 2 0]\n",
      " [0 1 0 0]\n",
      " [0 0 0 1]]\n",
      "Q = \n",
      "[[ 1.         -1.          0.        ]\n",
      " [-0.33333333  5.33333333 -0.16666667]\n",
      " [ 0.66666667 -0.66666667  0.33333333]]\n",
      "rank of R =  2\n",
      "Since, Rank of R is 3, so a1, a2, a3 are independent\n",
      "Now, basis for W can be given by row vectors of R i.e. p1,p2,p3\n",
      "b is any vector in W. b = [b1 b2 b3 b4]\n",
      "Span of vectors p1,p2,p3 consist of vector b with b3 = 2*b1\n",
      "So,b = b1p1 + b2p2 + b4p3\n",
      "And,[p1 p2 p3] = R = Q*A\n",
      "So, b = [b1 b2 b3]* Q * A\n",
      "hence, b = x1a1 + x2a2 + x3a3 where x1 = [b1 b2 b4] * Q(1) and so on\n",
      "Now, given 3 vectors a1 a2 a3:\n",
      "a1 =  [1, 0, 2, 0]\n",
      "a2 =  [0, 2, 0, 1]\n",
      "a3 =  [0, 0, 0, 3]\n",
      "Since a1 a2 a3 are all of the form (y1 y2 y3 y4) with y3 = 2*y1, hence they are in W.\n",
      "So, they are independent.\n",
      "Required matrix P such that X = PX is:\n",
      "P = \n",
      "[[ 0. -0.  2.]\n",
      " [-0.  0. -2.]\n",
      " [ 0. -0.  1.]]\n"
     ]
    }
   ],
   "source": [
    "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",
    "print 'Given row vectors are:'\n",
    "print 'a1 = ',a1\n",
    "print 'a2 = ',a2\n",
    "print 'a3 = ',a3\n",
    "print 'The matrix A from these vectors will be:'\n",
    "#A = [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 = 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",
    "print 'R1 = R1-R2'\n",
    "A[0,:] = A[0,:] - A[1,:]\n",
    "Q[0,:] = Q[0,:] - Q[1,:]\n",
    "print 'A = \\n',A\n",
    "print 'Q = \\n',Q\n",
    "print 'R3 = R3 + 2*R1'\n",
    "A[2,:] = A[2,:] + 2*A[0,:]\n",
    "Q[2,:] = Q[2,:] + 2*Q[0,:]\n",
    "print 'A = \\n',A\n",
    "print 'Q = \\n',Q\n",
    "print 'R3 = R3/3'\n",
    "A[2,:] = 1./3*A[2,:]\n",
    "Q[2,:] = 1./3*Q[2,:]\n",
    "print 'A = \\n',A\n",
    "print 'Q = \\n',Q\n",
    "print 'R2 = R2/2'\n",
    "A[1,:] = 1./2*A[1,:]\n",
    "Q[1,:] = 10/2*Q[1,:]\n",
    "print 'A = \\n',A\n",
    "print 'Q = \\n',Q\n",
    "print 'R2 = R2 - 1/2*R3'\n",
    "A[1,:] = A[1,:] - 1./2*A[2,:]\n",
    "Q[1,:] = Q[1,:] - 1./2*Q[2,:]\n",
    "print 'A = \\n',A\n",
    "print 'Q = \\n',Q\n",
    "R = A#\n",
    "A = T#\n",
    "print 'Row reduced echelon matrix:'\n",
    "print 'A = \\n',A\n",
    "print 'Q = \\n',Q\n",
    "#part a\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",
    "print 'Now, basis for W can be given by row vectors of R i.e. p1,p2,p3'\n",
    "print 'b is any vector in W. b = [b1 b2 b3 b4]'\n",
    "print 'Span of vectors p1,p2,p3 consist of vector b with b3 = 2*b1'\n",
    "print 'So,b = b1p1 + b2p2 + b4p3'\n",
    "print 'And,[p1 p2 p3] = R = Q*A'\n",
    "print 'So, b = [b1 b2 b3]* Q * A'\n",
    "print 'hence, b = x1a1 + x2a2 + x3a3 where x1 = [b1 b2 b4] * Q(1) and so on' # #Equation 1\n",
    "#part c\n",
    "print 'Now, given 3 vectors a1'' a2'' a3'':'\n",
    "c1 = [1, 0, 2, 0]#\n",
    "c2 = [0 ,2 ,0, 1]#\n",
    "c3 = [0 ,0 ,0 ,3]#\n",
    "print 'a1'' = ',c1\n",
    "print 'a2'' = ',c2\n",
    "print 'a3'' = ',c3\n",
    "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 = 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 = 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=vstack([x1,x2,x3])\n",
    "print 'P = \\n',P\n",
    "#print x1"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Page 63 Example 2.22"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "A = \n",
      "[[ 1  2  0  3  0]\n",
      " [ 1  2 -1 -1  0]\n",
      " [ 0  0  1  4  0]\n",
      " [ 2  4  1 10  1]\n",
      " [ 0  0  0  0  1]]\n",
      "Taking an identity matrix P:\n",
      "P = \n",
      "[[ 1.  0.  0.  0.  0.]\n",
      " [ 0.  1.  0.  0.  0.]\n",
      " [ 0.  0.  1.  0.  0.]\n",
      " [ 0.  0.  0.  1.  0.]\n",
      " [ 0.  0.  0.  0.  1.]]\n",
      "Applying row transformations on P and A to get a row reduced echelon matrix R:\n",
      "R2 = R2 - R1 and R4 = R4 - 2* R1\n",
      "A = \n",
      "[[ 1  2  0  3  0]\n",
      " [ 0  0 -1 -4  0]\n",
      " [ 0  0  1  4  0]\n",
      " [ 0  0  1  4  1]\n",
      " [ 0  0  0  0  1]]\n",
      "P = \n",
      "[[ 1.  0.  0.  0.  0.]\n",
      " [-1.  1.  0.  0.  0.]\n",
      " [ 0.  0.  1.  0.  0.]\n",
      " [-2.  0.  0.  1.  0.]\n",
      " [ 0.  0.  0.  0.  1.]]\n",
      "R2 = -R2 , R3 = R3 - R1 + R2 and R4  = R4 - R1 + R2\n",
      "A = \n",
      "[[1 2 0 3 0]\n",
      " [0 0 1 4 0]\n",
      " [0 0 0 0 0]\n",
      " [0 0 0 0 1]\n",
      " [0 0 0 0 1]]\n",
      "P = \n",
      "[[ 1.  0.  0.  0.  0.]\n",
      " [ 1. -1. -0. -0. -0.]\n",
      " [-1.  1.  1.  0.  0.]\n",
      " [-3.  1.  0.  1.  0.]\n",
      " [-3.  1.  0.  1.  0.]]\n",
      "Mutually interchanging R3, R4 and R5\n",
      "Row reduced echelon matrix R = \n",
      "[[1 2 0 3 0]\n",
      " [0 0 1 4 0]\n",
      " [0 0 0 0 1]\n",
      " [0 0 0 0 1]\n",
      " [0 0 0 0 0]]\n",
      "Invertible Matrix P = \n",
      "[[ 1.  0.  0.  0.  0.]\n",
      " [ 1. -1. -0. -0. -0.]\n",
      " [-3.  1.  0.  1.  0.]\n",
      " [-3.  1.  0.  1.  0.]\n",
      " [ 0.  0.  0.  0.  0.]]\n",
      "Invertible matrix P is not unique. There can be many that depends on operations used to reduce A\n",
      "-----------------------------------------\n",
      "For the basis of row space W of A, we can take the non-zero rows of R\n",
      "It can be given by p1, p2, p3\n",
      "p1 =  [1 2 0 3 0]\n",
      "p2 =  [0 0 1 4 0]\n",
      "p3 =  [0 0 0 0 1]\n",
      "-----------------------------------------\n",
      "The row space W consists of vectors of the form:\n",
      "b = c1p1 + c2p2 + c3p3\n",
      "i.e. b = (c1,2*c1,c2,3*c1+4*c2,c3) where, c1 c2 c3 are scalars.\n",
      "So, if b2 = 2*b1 and b4 = 3*b1 + 4*b3  =>  (b1,b2,b3,b4,b5) = b1p1 + b3p2 + b5p3\n",
      "then,(b1,b2,b3,b4,b5) is in W\n",
      "-----------------------------------------\n",
      "The coordinate matrix of the vector (b1,2*b1,b2,3*b1+4*b2,b3) in the basis (p1,p2,p3) is column matrix of b1,b2,b3 such that:\n",
      "  b1\n",
      "  b2\n",
      "  b3\n",
      "-----------------------------------------\n",
      "Now, to write each vector in W as a linear combination of rows of A:\n",
      "Let b = (b1,b2,b3,b4,b5) and if b is in W, then\n",
      "we know,b = (b1,2*b1,b3,3*b1 + 4*b3,b5)  =>  [b1,b3,b5,0,0]*R\n",
      "=> b = [b1,b3,b5,0,0] * P*A  =>  b = [b1+b3,-b3,0,0,b5] * A\n",
      "if b = (-5,-10,1,-11,20)\n",
      "b = ( [-4, -1, 0, 0, 20] ) *[ [[1 2 0 3 0]\n",
      " [0 0 1 4 0]\n",
      " [0 0 0 0 1]\n",
      " [0 0 0 0 1]\n",
      " [0 0 0 0 0]] ]\n",
      "-----------------------------------------\n",
      "The equations in system RX = 0 are given by R * [x1 x2 x3 x4 x5]\n",
      "i.e., x1 + 2*x2 + 3*x4\n",
      "x3 + 4*x4\n",
      "x5\n",
      "so, V consists of all columns of the form\n",
      "[ X=\n",
      "  -2*x2 - 3*x4\n",
      "  x2\n",
      "  -4*x4\n",
      "  x4\n",
      "  0\n",
      "where x2 and x4  are arbitrary ]\n",
      "-----------------------------------------\n",
      "Let x2 = 1,x4 = 0 then the given column forms a basis of V\n",
      "[[-2], [1], [0], [0], [0]]\n",
      "Similarly,if x2 = 0,x4 = 1 then the given column forms a basis of V\n",
      "[[-3], [0], [-4], [1], [0]]\n",
      "-----------------------------------------\n",
      "The equation AX = Y has solutions X if and only if\n",
      "-y1 + y2 + y3 = 0\n",
      "-3*y1 + y2 + y4 -y5 = 0\n",
      "where, Y = (y1 y2 y3 y4 y5)\n"
     ]
    }
   ],
   "source": [
    "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 = 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",
    "A[1,:] = A[1,:] - A[0,:]\n",
    "P[1,:] = P[1,:] - P[0,:]\n",
    "A[3,:] = A[3,:] - 2 * A[0,:]\n",
    "P[3,:] = P[3,:] - 2 * P[0,:]\n",
    "print 'A = \\n',A\n",
    "print 'P = \\n',P\n",
    "print 'R2 = -R2 , R3 = R3 - R1 + R2 and R4  = R4 - R1 + R2'\n",
    "A[1,:] = -A[1,:]\n",
    "P[1,:] = -P[1,:]\n",
    "A[2,:] = A[2,:] - A[1,:]\n",
    "P[2,:] = P[2,:] - P[1,:]\n",
    "A[3,:] = A[3,:] - A[1,:]\n",
    "P[3:] = P[3,:] - P[1,:]\n",
    "print 'A = \\n',A\n",
    "print 'P = \\n',P\n",
    "print 'Mutually interchanging R3, R4 and R5'\n",
    "x = A[2,:]\n",
    "A[2,:] = A[4,:]\n",
    "y = A[3,:]\n",
    "A[3,:] = x#\n",
    "A[4,:] = y - A[2,:]\n",
    "x = P[2,:]\n",
    "P[2,:] = P[4,:]\n",
    "y = P[3,:]\n",
    "P[3,:] = x#\n",
    "P[4,:] = y - P[2,:]\n",
    "R = A#\n",
    "A = T#\n",
    "print 'Row reduced echelon matrix R = \\n',R\n",
    "print 'Invertible Matrix P = \\n',P\n",
    "print 'Invertible matrix P is not unique. There can be many that depends on operations used to reduce A'\n",
    "print '-----------------------------------------'\n",
    "#part b\n",
    "print 'For the basis of row space W of A, we can take the non-zero rows of R'\n",
    "print 'It can be given by p1, p2, p3'\n",
    "p1 = R[0,:]\n",
    "p2 = R[1,:]\n",
    "p3 = R[2,:]\n",
    "print 'p1 = ',p1\n",
    "print 'p2 = ',p2\n",
    "print 'p3 = ',p3\n",
    "print '-----------------------------------------'\n",
    "#part c\n",
    "print 'The row space W consists of vectors of the form:'\n",
    "print 'b = c1p1 + c2p2 + c3p3'\n",
    "print 'i.e. b = (c1,2*c1,c2,3*c1+4*c2,c3) where, c1 c2 c3 are scalars.'\n",
    "print 'So, if b2 = 2*b1 and b4 = 3*b1 + 4*b3  =>  (b1,b2,b3,b4,b5) = b1p1 + b3p2 + b5p3'\n",
    "print 'then,(b1,b2,b3,b4,b5) is in W'\n",
    "print '-----------------------------------------'\n",
    "#part d\n",
    "print 'The coordinate matrix of the vector (b1,2*b1,b2,3*b1+4*b2,b3) in the basis (p1,p2,p3) is column matrix of b1,b2,b3 such that:'\n",
    "print '  b1'\n",
    "print '  b2'\n",
    "print '  b3'\n",
    "print '-----------------------------------------'\n",
    "#part e\n",
    "print 'Now, to write each vector in W as a linear combination of rows of A:'\n",
    "print 'Let b = (b1,b2,b3,b4,b5) and if b is in W, then'\n",
    "print 'we know,b = (b1,2*b1,b3,3*b1 + 4*b3,b5)  =>  [b1,b3,b5,0,0]*R'\n",
    "print '=> b = [b1,b3,b5,0,0] * P*A  =>  b = [b1+b3,-b3,0,0,b5] * A'\n",
    "print 'if b = (-5,-10,1,-11,20)'\n",
    "b1 = -5#\n",
    "b2 = -10#\n",
    "b3 = 1#\n",
    "b4 = -11#\n",
    "b5 = 20#\n",
    "x = [b1 + b3,-b3,0,0,b5]#\n",
    "print 'b = (',x,')','*[',A,']'\n",
    "print '-----------------------------------------'\n",
    "#part f\n",
    "print 'The equations in system RX = 0 are given by R * [x1 x2 x3 x4 x5]'\n",
    "print 'i.e., x1 + 2*x2 + 3*x4'\n",
    "print 'x3 + 4*x4'\n",
    "print 'x5'\n",
    "print 'so, V consists of all columns of the form'\n",
    "print '[','X='\n",
    "print '  -2*x2 - 3*x4'\n",
    "print '  x2'\n",
    "print '  -4*x4'\n",
    "print '  x4'\n",
    "print '  0'\n",
    "print 'where x2 and x4  are arbitrary',']'\n",
    "print '-----------------------------------------'\n",
    "#part g\n",
    "print 'Let x2 = 1,x4 = 0 then the given column forms a basis of V'\n",
    "\n",
    "x2 = 1#\n",
    "x4 = 0#\n",
    "print [[-2*x2-3*x4],[ x2],[ -4*x4],[ x4],[ 0]]\n",
    "print 'Similarly,if x2 = 0,x4 = 1 then the given column forms a basis of V'\n",
    "x2 = 0#\n",
    "x4 = 1#\n",
    "print [[-2*x2-3*x4],[ x2],[ -4*x4],[ x4],[ 0]]\n",
    "print '-----------------------------------------'\n",
    "#part h\n",
    "print 'The equation AX = Y has solutions X if and only if'\n",
    "print '-y1 + y2 + y3 = 0'\n",
    "print '-3*y1 + y2 + y4 -y5 = 0'\n",
    "print 'where, Y = (y1 y2 y3 y4 y5)'"
   ]
  }
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