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 A  "sample_notebooks/Raj Kumar/Ch5.ipynb"

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+{
+ "cells": [
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "# Chapter 5 - Vectors & Matrices"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "## Ex5.2 Pg 103"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 52,
+   "metadata": {
+    "collapsed": false
+   },
+   "outputs": [
+    {
+     "name": "stdout",
+     "output_type": "stream",
+     "text": [
+      "u+v =  [[ 3 -2  4]]\n",
+      "5*u =  [[ 10  15 -20]]\n",
+      "-v =  [[-1  5 -8]]\n",
+      "2*u-3*v =  [[  1  21 -32]]\n",
+      "dot product of the two vectors, k = u.v =  [[-45]]\n",
+      "norm or length of the vector u =  5.3852\n"
+     ]
+    }
+   ],
+   "source": [
+    "from __future__ import division\n",
+    "from numpy import mat\n",
+    "from numpy.linalg import det\n",
+    "from numpy import mat, add, dot, multiply, inner\n",
+    "u=mat([[2,3,-4]])\n",
+    "v=mat([[1,-5,8]])\n",
+    "print \"u+v = \",add(u,v)\n",
+    "print \"5*u = \",multiply(5,u)\n",
+    "print \"-v = \",multiply(-1,v)\n",
+    "print \"2*u-3*v = \",add(multiply(2,u),multiply(-3,v))\n",
+    "print 'dot product of the two vectors, k = u.v = ',inner(u,v)\n",
+    "from numpy.linalg import norm\n",
+    "l=norm(u)# \n",
+    "print 'norm or length of the vector u = ',round(l,4)\n"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "## Ex5.3 Pg 104 "
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 26,
+   "metadata": {
+    "collapsed": false
+   },
+   "outputs": [
+    {
+     "name": "stdout",
+     "output_type": "stream",
+     "text": [
+      "2*u-3*v =\n",
+      "[[ 1]\n",
+      " [ 9]\n",
+      " [-2]]\n",
+      "The dot product of the two vectors u and v is: [[20]]\n",
+      "norm or length of the vector u =  7.0711\n"
+     ]
+    }
+   ],
+   "source": [
+    "u=mat([[5],[3],[-4]])\n",
+    "v=mat([[3],[-1],[-2]])\n",
+    "print \"2*u-3*v =\\n\",add(multiply(2,u),multiply(-3,v))\n",
+    "print 'The dot product of the two vectors u and v is:', sum(multiply(u,v))\n",
+    "l=norm(u)#\n",
+    "print 'norm or length of the vector u = ',round(l,4)"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "## Ex5.5 Pg 105"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 33,
+   "metadata": {
+    "collapsed": false
+   },
+   "outputs": [
+    {
+     "name": "stdout",
+     "output_type": "stream",
+     "text": [
+      "The addition of the two matrices A and B is:\n",
+      "[[ 5  4 11]\n",
+      " [ 1  1 -2]]\n",
+      "\n",
+      "The multiplication of a vector with a scalar is:\n",
+      "[[ 3 -6  9]\n",
+      " [ 0 12 15]]\n",
+      "\n",
+      "2*A-3*B = \n",
+      "[[-10 -22 -18]\n",
+      " [ -3  17  31]]\n"
+     ]
+    }
+   ],
+   "source": [
+    "A=mat([[1,-2,3],[0,4,5]])\n",
+    "B=mat([[4,6,8],[1,-3,-7]])\n",
+    "k=add(A,B)\n",
+    "print 'The addition of the two matrices A and B is:\\n',k\n",
+    "m=multiply(3,A)\n",
+    "print '\\nThe multiplication of a vector with a scalar is:\\n',m\n",
+    "p=add(multiply(2,A),multiply(-3,B))\n",
+    "print \"\\n2*A-3*B = \\n\",p"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "## Ex5.6 Pg 106"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 36,
+   "metadata": {
+    "collapsed": false
+   },
+   "outputs": [
+    {
+     "name": "stdout",
+     "output_type": "stream",
+     "text": [
+      "product of a and b is :  [[8]]\n",
+      "product of p and q is: [[32]]\n"
+     ]
+    }
+   ],
+   "source": [
+    "a=mat([[7,-4,5]])\n",
+    "b=mat([[3,2,-1]])\n",
+    "k=inner(a,b)\n",
+    "print 'product of a and b is : ',k\n",
+    "p=mat([[6,-1,8,3]])\n",
+    "q=mat([[4,-9,-2,5]])\n",
+    "l=inner(p,q)\n",
+    "print 'product of p and q is:',l"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "## Ex5.7 Pg 107"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 40,
+   "metadata": {
+    "collapsed": false
+   },
+   "outputs": [
+    {
+     "name": "stdout",
+     "output_type": "stream",
+     "text": [
+      "A*B = \n",
+      "[[ 17  -6  14]\n",
+      " [ -1   2 -14]]\n",
+      "A*B = \n",
+      "[[ 5  2]\n",
+      " [15 10]]\n",
+      "B*A = \n",
+      "[[23 34]\n",
+      " [-6 -8]]\n",
+      "matrix mulitplication is not commutative since AB may not be equal to BA\n"
+     ]
+    }
+   ],
+   "source": [
+    "A=mat([[1 ,3],[2, -1]])\n",
+    "B=mat([[2, 0, -4],[5, -2, 6]])\n",
+    "print \"A*B = \\n\", dot(A,B)\n",
+    "A=mat([[1, 2],[3, 4]])\n",
+    "B=mat([[5, 6],[0, -2]])\n",
+    "print \"A*B = \\n\",dot(A,B)\n",
+    "print \"B*A = \\n\", dot(B,A)\n",
+    "print 'matrix mulitplication is not commutative since AB may not be equal to BA'\n"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "## Ex5.8 Pg 109"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 47,
+   "metadata": {
+    "collapsed": false
+   },
+   "outputs": [
+    {
+     "name": "stdout",
+     "output_type": "stream",
+     "text": [
+      "for the function f(x)=2x**2-3x+5,f(A) is :\n",
+      "[[ 16. -18.]\n",
+      " [-27.  61.]]\n",
+      "for the function g(x)=x**2+3x-10,g(A) is\n",
+      "[[ 0.  0.]\n",
+      " [ 0.  0.]]\n"
+     ]
+    }
+   ],
+   "source": [
+    "from numpy import identity as idt\n",
+    "A=mat([[1, 2],[3, -4]])\n",
+    "A2=dot(A,A)  #multiplying A by itself\n",
+    "A3=dot(A2,A)\n",
+    "f=add(add(multiply(2,A2),multiply(-3,A)),multiply(5,idt(2)))\n",
+    "print 'for the function f(x)=2x**2-3x+5,f(A) is :\\n',f\n",
+    "g=add(add(A2,multiply(3,A)),multiply(-10,idt(2)))\n",
+    "print 'for the function g(x)=x**2+3x-10,g(A) is\\n',g"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "## Ex5.9 Pg 110"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 50,
+   "metadata": {
+    "collapsed": false
+   },
+   "outputs": [
+    {
+     "name": "stdout",
+     "output_type": "stream",
+     "text": [
+      "A*b = \n",
+      "[[1 0 0]\n",
+      " [0 1 0]\n",
+      " [0 0 1]]\n",
+      "since A*B is identity matrix,A and B are invertible and inverse of each other\n"
+     ]
+    }
+   ],
+   "source": [
+    "A=mat([[1 ,0 ,2],[2 ,-1, 3],[4, 1, 8]])\n",
+    "B=mat([[-11, 2 ,2],[-4, 0 ,1],[6, -1, -1]])\n",
+    "print \"A*b = \\n\",dot(A,B)\n",
+    "print 'since A*B is identity matrix,A and B are invertible and inverse of each other'"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "## Ex5.10 Pg 111"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 53,
+   "metadata": {
+    "collapsed": false
+   },
+   "outputs": [
+    {
+     "name": "stdout",
+     "output_type": "stream",
+     "text": [
+      "determinant of A 7.0\n",
+      "determinant of B 16.0\n",
+      "determinant of C -81.0\n"
+     ]
+    }
+   ],
+   "source": [
+    "A=mat([[5 ,4],[2, 3]])\n",
+    "print 'determinant of A',det(A)\n",
+    "B=mat([[2, 1],[-4, 6]])\n",
+    "print 'determinant of B',det(B)\n",
+    "C=mat([[2, 1, 3],[4, 6, -1],[5 ,1 ,0]])\n",
+    "print 'determinant of C',det(C)"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "## Ex5.13 Pg 115"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 60,
+   "metadata": {
+    "collapsed": false
+   },
+   "outputs": [
+    {
+     "name": "stdout",
+     "output_type": "stream",
+     "text": [
+      "x =  [[ 2.]]\n",
+      "y =  [[-1.]]\n",
+      "z =  [[ 3.]]\n"
+     ]
+    }
+   ],
+   "source": [
+    "A=mat([[1, 2, 1],[2, 5, -1],[3, -2, -1]])     #left hand side of the system of equations\n",
+    "B=mat([[3] ,[-4] ,[5]])                   #right hand side or the constants in the equations\n",
+    "from numpy import divide\n",
+    "from numpy.linalg import solve\n",
+    "X=divide(A,B) #   #unique solution for the system of equations\n",
+    "X = solve(A, B)\n",
+    "print \"x = \",X[0]\n",
+    "print \"y = \",X[1]\n",
+    "print \"z = \",X[2]"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "## Ex5.14 Pg 116"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 64,
+   "metadata": {
+    "collapsed": false
+   },
+   "outputs": [
+    {
+     "name": "stdout",
+     "output_type": "stream",
+     "text": [
+      "Inverse of A = \n",
+      "[[-11.   2.   2.]\n",
+      " [ -4.   0.   1.]\n",
+      " [  6.  -1.  -1.]]\n"
+     ]
+    }
+   ],
+   "source": [
+    "A=mat([[1 ,0 ,2],[2, -1, 3],[4, 1, 8]])\n",
+    "A_inv = A**-1\n",
+    "print \"Inverse of A = \\n\", A_inv"
+   ]
+  }
+ ],
+ "metadata": {
+  "kernelspec": {
+   "display_name": "Python 2",
+   "language": "python",
+   "name": "python2"
+  },
+  "language_info": {
+   "codemirror_mode": {
+    "name": "ipython",
+    "version": 2
+   },
+   "file_extension": ".py",
+   "mimetype": "text/x-python",
+   "name": "python",
+   "nbconvert_exporter": "python",
+   "pygments_lexer": "ipython2",
+   "version": "2.7.9"
+  }
+ },
+ "nbformat": 4,
+ "nbformat_minor": 0
+}