{
 "metadata": {
  "language": "Julia",
  "name": "",
  "signature": "sha256:94fa37b5577564aa4793a1cc812ed8efd58a7115e79eebf371ff84293e5b43af"
 },
 "nbformat": 3,
 "nbformat_minor": 0,
 "worksheets": [
  {
   "cells": [
    {
     "cell_type": "heading",
     "level": 4,
     "metadata": {},
     "source": [
      "Instructions: You may use any computer language you like, but if you are not using Julia, you will need to set things up yourself."
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "a. Using the matrix squaring operator create a \"triangular\" matrix with 1 on the main diagonal, 2 above, etc.\n",
      "$M(n) = \\left( \\begin{array}{ccccc} 1  & 2 & \\ldots & n-1 & n\\\\ & 1 & 2  & \\ldots & n-1   \\\\ & & \\ddots & \\ddots  \\\\  &&& 1 & 2 \\\\ &&&& 1\\end{array} \\right)$"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "M(n) = triu(ones(n,n))^2 \n",
      "M(5)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 1
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "{M(n) for n=1:6}"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 2
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "b. You very likely have heard of the  <a href=\"http://en.wikipedia.org/wiki/Triangular_number\"> triangular numbers:   </a>\n",
      "$$T_n = 1+2+\\ldots+n.$$"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "#Here they are\n",
      "cumsum(1:10)'"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 3
    },
    {
     "cell_type": "heading",
     "level": 4,
     "metadata": {},
     "source": [
      "Don't use cumsum, or sum or \"+\", just matrix operations to create the matrix that has the triangular numbers on the diagonals:\n",
      "Explain roughly (not too formal a proof), why your idea works."
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "$M(n) = \\left( \\begin{array}{ccccc} 1  & 3 & \\ldots & (n-1)n/2 & n(n+1)/2\\\\ & 1 & 3  & \\ldots & (n-1)n/2   \\\\ & & \\ddots & \\ddots  \\\\  &&& 1 & 3 \\\\ &&&& 1\\end{array} \\right)$"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 4
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "c. Don't stop.  Keep going, and get the  <a href=\"http://en.wikipedia.org/wiki/Tetrahedral_number\">tetrahedral numbers. </a> Explain briefly why this worked."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 5
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "e) Let $A=\\left[\\begin{array}{c} 1\\\\ 2\\\\ 3\\\\ 1\\end{array}\\right]$ and $B=\\left[\\begin{array}{ccc} -1& 2& 1& 4\\end{array}\\right]$. Compute $(AB)^{10}$ by using Julia. Explain why it's possible to get this without  a computer!"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "A=[1;2;3;1]"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 2,
       "text": [
        "4-element Array{Int64,1}:\n",
        " 1\n",
        " 2\n",
        " 3\n",
        " 1"
       ]
      }
     ],
     "prompt_number": 2
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "B=[-1 2 1 4]"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 3,
       "text": [
        "1x4 Array{Int64,2}:\n",
        " -1  2  1  4"
       ]
      }
     ],
     "prompt_number": 3
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "(A*B)^10"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 8
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "B*A"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 4,
       "text": [
        "1-element Array{Int64,1}:\n",
        " 10"
       ]
      }
     ],
     "prompt_number": 4
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [],
     "language": "python",
     "metadata": {},
     "outputs": []
    }
   ],
   "metadata": {}
  }
 ]
}