(************** Content-type: application/mathematica ************** CreatedBy='Mathematica 5.1' Mathematica-Compatible Notebook This notebook can be used with any Mathematica-compatible application, such as Mathematica, MathReader or Publicon. The data for the notebook starts with the line containing stars above. To get the notebook into a Mathematica-compatible application, do one of the following: * Save the data starting with the line of stars above into a file with a name ending in .nb, then open the file inside the application; * Copy the data starting with the line of stars above to the clipboard, then use the Paste menu command inside the application. Data for notebooks contains only printable 7-bit ASCII and can be sent directly in email or through ftp in text mode. Newlines can be CR, LF or CRLF (Unix, Macintosh or MS-DOS style). NOTE: If you modify the data for this notebook not in a Mathematica- compatible application, you must delete the line below containing the word CacheID, otherwise Mathematica-compatible applications may try to use invalid cache data. For more information on notebooks and Mathematica-compatible applications, contact Wolfram Research: web: http://www.wolfram.com email: info@wolfram.com phone: +1-217-398-0700 (U.S.) Notebook reader applications are available free of charge from Wolfram Research. *******************************************************************) (*CacheID: 232*) (*NotebookFileLineBreakTest NotebookFileLineBreakTest*) (*NotebookOptionsPosition[ 45237, 1401]*) (*NotebookOutlinePosition[ 45941, 1426]*) (* CellTagsIndexPosition[ 45897, 1422]*) (*WindowFrame->Normal*) Notebook[{ Cell["Quiz", "Title"], Cell["Wednesday, April 26, 2006", "Subtitle"], Cell["\<\ Topic: Using the reduced row echelon form to give bases of the null space, \ row space, and column space.\ \>", "Subsubsection"], Cell[TextData[{ "Instructions: open this notebook in ", StyleBox["Mathematica", FontSlant->"Italic"], ". Select (Kernel, Evaluation, Evaluate Notebook) from the menus. This will \ generate a new quiz with solutions. Select (File, Print...) to print out your \ quiz. Each quiz is different (some are easier than others) so try a dozen to \ make sure you understand everything that can happen. None of the problems \ require any significant amount of computation." }], "Text"], Cell[CellGroupData[{ Cell["\<\ The code that generates the quiz can be viewed, but it isn't necessary to do \ so.\ \>", "Text"], Cell[BoxData[{ \(\(Off[General::"\", Solve::"\"];\)\), "\[IndentingNewLine]", \(\(Clear[S, v];\)\), "\[IndentingNewLine]", \(\(dimV = Random[Integer, {4, 9}];\)\), "\[IndentingNewLine]", \(\(v[i_] := \(v[i] = Table[Random[], {dimV}]\);\)\), "\[IndentingNewLine]", \(\(dimW = Random[Integer, {1, dimV}];\)\), "\[IndentingNewLine]", \(\(NumberOfVectors = Random[Integer, {dimW, dimV}];\)\), "\[IndentingNewLine]", \(\(basis = Table[v[i], {i, dimW}];\)\), "\[IndentingNewLine]", \(\(S[i_] := \(S[i] = First[Subsets[ Range[dimW], {1, dimW}, {Random[ Integer, {1, 2^dimW - 1}]}]]\);\)\), "\[IndentingNewLine]", \(\(multiplier = Join[IdentityMatrix[dimW], Table[Table[If[MemberQ[S[j], i], Random[], 0], {i, 1, dimW}], {j, dimW + 1, NumberOfVectors}]];\)\), "\[IndentingNewLine]", \(\(perm = 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Here's how it comes out of the \ reduce row echelon form." }], "Text"], Cell["\<\ First, identify the free variables (columns of the rref that do not contain \ pivots).\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(Complement[Range[dimV], PivotColumnNumbers]\)], "Input"], Cell[BoxData[ \({9}\)], "Output"] }, Open ]], Cell["\<\ Each nonzero row of the rref corresponds to an equation with exactly one \ bound variable (bound=not free). Solve each equation for its bound \ variable.\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(Table[ Solve[R[\([i]\)] . 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Note that if ", Cell[BoxData[ \(TraditionalForm\`x\_7\)]], " is a free variable, then the seventh component of each of the vectors \ will be 0, except for the one multiplied by ", Cell[BoxData[ \(TraditionalForm\`x\_7\)]], ", which will have seventh component equal to 1." }], "Text"], Cell["Here's the final answer for the matrix in this quiz:", "Text"], Cell[CellGroupData[{ Cell[BoxData[{ \(\(freevars = \((\(x\_# &\)\ /@ Complement[Range[NumberOfColumns], PivotColumnNumbers])\);\)\), "\[IndentingNewLine]", \(\(ns = MatrixForm[Table[x\_i, {i, NumberOfColumns}]] /. Flatten[Table[ Solve[R[\([i]\)] . 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Prove or disprove the following two statements:\n\t\t(1)\t\ ", Cell[BoxData[ \(TraditionalForm\`U \[Intersection] V\)]], ", the set of all vectors in both ", Cell[BoxData[ \(TraditionalForm\`U\)]], " and ", Cell[BoxData[ \(TraditionalForm\`V\)]], ", is a subspace of ", Cell[BoxData[ \(TraditionalForm\`W\)]], ".\n\t\t(2)\t", Cell[BoxData[ \(TraditionalForm\`U \[Union] V\)]], ", the set of all vectors in either ", Cell[BoxData[ \(TraditionalForm\`U\)]], " or ", Cell[BoxData[ \(TraditionalForm\`V\)]], ", is a subspace of ", Cell[BoxData[ \(TraditionalForm\`W\)]], "." }], "Subsubsection"], Cell[TextData[{ "The set ", Cell[BoxData[ \(TraditionalForm\`U \[Union] V\)]], " is not a subspace. For example, let ", Cell[BoxData[ \(TraditionalForm\`U = Span {\((1, 0)\)}\)]], ", and ", Cell[BoxData[ \(TraditionalForm\`V = Span {\((0, 1)\)}\)]], ". Since ", Cell[BoxData[ \(TraditionalForm\`U \[Union] V\)]], " is a subset of the real vector space ", Cell[BoxData[ \(TraditionalForm\`W\)]], ", we need only to check if it is closed under addition and multiplication \ by a scalar. It is not closed under addition: (1,0)+(0,1)=(1,1) is a vector \ that is not in ", Cell[BoxData[ \(TraditionalForm\`U \[Union] V\)]], ", although it is the sum of two vectors in ", Cell[BoxData[ \(TraditionalForm\`U \[Union] V\)]], "." }], "Text"], Cell[TextData[{ "The ", Cell[BoxData[ \(TraditionalForm\`U \[Intersection] V\)]], " is a subspace. Since it is a subset of the real vector space ", Cell[BoxData[ \(TraditionalForm\`W\)]], ", we need only to check if it is closed under addition and multiplication \ by a scalar. Suppose that ", Cell[BoxData[ \(TraditionalForm\`\(x\& \[RightVector] \)\)]], " and ", Cell[BoxData[ \(TraditionalForm\`\(y\& \[RightVector] \)\)]], " are both in ", Cell[BoxData[ \(TraditionalForm\`U\)]], " and in ", Cell[BoxData[ \(TraditionalForm\`V\)]], ". Then ", Cell[BoxData[ \(TraditionalForm\`\(x\& \[RightVector] \) + \(y\& \[RightVector] \ \)\)]], " is in ", Cell[BoxData[ \(TraditionalForm\`U\)]], " because ", Cell[BoxData[ \(TraditionalForm\`U\)]], " is a subspace, and ", Cell[BoxData[ \(TraditionalForm\`\(x\& \[RightVector] \) + \(y\& \[RightVector] \ \)\)]], " is in ", Cell[BoxData[ \(TraditionalForm\`V\)]], " because ", Cell[BoxData[ \(TraditionalForm\`V\)]], " is a subspace. Thus, ", Cell[BoxData[ \(TraditionalForm\`\(x\& \[RightVector] \) + \(y\& \[RightVector] \ \)\)]], " is in ", Cell[BoxData[ FormBox[ RowBox[{" ", FormBox[\(U \[Intersection] V\), "TraditionalForm"]}], TraditionalForm]]], " because it is in both ", Cell[BoxData[ \(TraditionalForm\`U\)]], " and ", Cell[BoxData[ \(TraditionalForm\`V\)]], ". Likewise, let ", Cell[BoxData[ \(TraditionalForm\`c\)]], " be any real number. The vector ", Cell[BoxData[ \(TraditionalForm\`c\ \(x\& \[RightVector] \)\)]], " is in ", Cell[BoxData[ \(TraditionalForm\`U\)]], " because ", Cell[BoxData[ \(TraditionalForm\`\(x\& \[RightVector] \)\)]], " is in ", Cell[BoxData[ \(TraditionalForm\`U\)]], " and ", Cell[BoxData[ \(TraditionalForm\`U\)]], " is a subspace, and likewise ", Cell[BoxData[ \(TraditionalForm\`c\ \(x\& \[RightVector] \)\)]], " is in ", Cell[BoxData[ \(TraditionalForm\`V\)]], " because ", Cell[BoxData[ \(TraditionalForm\`\(x\& \[RightVector] \)\)]], " is in ", Cell[BoxData[ \(TraditionalForm\`V\)]], " and ", Cell[BoxData[ \(TraditionalForm\`V\)]], " is a subspace. Therefore, ", Cell[BoxData[ \(TraditionalForm\`c\ \(x\& \[RightVector] \)\)]], " is in both ", Cell[BoxData[ \(TraditionalForm\`U\)]], " and ", Cell[BoxData[ \(TraditionalForm\`V\)]], ", and so is in ", Cell[BoxData[ \(TraditionalForm\`U \[Intersection] V\)]], "." }], "Text"] }, Open ]] }, Open ]] }, FrontEndVersion->"5.1 for Microsoft Windows", ScreenRectangle->{{0, 1024}, {0, 685}}, CellGrouping->Manual, WindowSize->{835, 592}, WindowMargins->{{2, Automatic}, {Automatic, 1}}, ShowSelection->True, Magnification->1 ] (******************************************************************* Cached data follows. If you edit this Notebook file directly, not using Mathematica, you must remove the line containing CacheID at the top of the file. 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