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Cell[TextData[{
  "                                          ",
  StyleBox["Vector Spaces, II -- Two, Three, ... N, ... Infinity",
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  "\n\n          Using ",
  StyleBox["Mathematica",
    FontSlant->"Italic"],
  "    we can experiment with some crude images.  They are crude in two \
ways--first, they are essentially monochromatic, black-and-white, rather than \
in color; and, second, they are very low resolution.  We are using crude \
images because we can work with them simply but the ideas we develop with \
crude images work equally well with more refined images."
}], "Special1",
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  "          The procedure  ",
  StyleBox["ShowImage",
    FontWeight->"Bold"],
  "  defined in the closed initialization cell above displays an image \
defined by a matrix.  It has three arguments\n          \n               1.  \
An (n by k)- matrix,   ",
  StyleBox["A",
    FontWeight->"Bold"],
  ",  whose entries give  the intensity of light for each pixel in an image \
with  n  lines, each having  k  pixels.  0 represents black and 1 represents \
the maximum intensity on your particular screen.  If an entry is outside the \
range ",
  StyleBox[" [0, 1]",
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  "  then  ",
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  " if it is ",
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  ".  These colors are meant to indicate errors, not as part of an image in \
color.\n               \n               2.  ",
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  "  --  the number of lines in the image.\n               \n               \
3.  ",
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  "  --  the number of pixels in each line. \n               \n          The \
following cell illustrates how  ",
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  "  is used.  ",
  StyleBox["Evaluate it now. ",
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        \t\t\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ {0.33, 0.44, 0.55, 0.66}, \n
        \t\t\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ {0.77, 0.88, 0.99, 2.00}}\n\), 
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  "\" and the other showing the letter \"",
  StyleBox["H",
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  "\".  The cell below defines matrices representing these two images and \
displays them using the procedure  ",
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  "  gradually changing to ",
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  ".  Notice that the key algebraic expression\n     \n                       \
                 ",
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  " is exactly the same algebraic expression that we used to move from one \
point to another point  in the plane.  ",
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  "  Then, if your computer supports animation, animate the resulting frames \
to produce a movie."
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  "                                                             \n            \
                                                 Exercises     \n     \n     \
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    FontSize->9,
    FontColor->RGBColor[0, 0, 1]],
  
  Cell[StyleData["CellLabel", "Presentation"],
    FontSize->12],
  
  Cell[StyleData["CellLabel", "Condensed"],
    FontSize->9],
  
  Cell[StyleData["CellLabel", "Printout"],
    FontFamily->"Courier",
    FontSize->8,
    FontSlant->"Italic",
    FontColor->GrayLevel[0]]
  }, Closed]]
  }, Closed]],
  
  Cell[CellGroupData[{
  
  Cell["Formulas and Programming", "Section"],
  
  Cell[CellGroupData[{
  
  Cell[StyleData["InlineFormula"],
    CellMargins->{{10, 4}, {0, 8}},
    CellHorizontalScrolling->True,
    ScriptLevel->1,
    SingleLetterItalics->True],
  
  Cell[StyleData["InlineFormula", "Presentation"],
    CellMargins->{{24, 10}, {10, 10}},
    LineSpacing->{1, 5}],
  
  Cell[StyleData["InlineFormula", "Condensed"],
    CellMargins->{{8, 10}, {6, 6}},
    LineSpacing->{1, 1}],
  
  Cell[StyleData["InlineFormula", "Printout"],
    CellMargins->{{2, 0}, {6, 6}}]
  }, Closed]],
  
  Cell[CellGroupData[{
  
  Cell[StyleData["DisplayFormula"],
    CellMargins->{{42, Inherited}, {Inherited, Inherited}},
    CellHorizontalScrolling->True,
    ScriptLevel->0,
    SingleLetterItalics->True,
    StyleMenuListing->None,
    UnderoverscriptBoxOptions->{LimitsPositioning->True}],
  
  Cell[StyleData["DisplayFormula", "Presentation"],
    LineSpacing->{1, 5}],
  
  Cell[StyleData["DisplayFormula", "Condensed"],
    LineSpacing->{1, 1}],
  
  Cell[StyleData["DisplayFormula", "Printout"]]
  }, Closed]]
  }, Closed]],
  
  Cell[CellGroupData[{
  
  Cell["Styles for Headers and Footers", "Section"],
  
  Cell[StyleData["Header"],
    CellMargins->{{7, 0}, {4, 1}},
    Evaluatable->False,
    PageBreakWithin->Automatic,
    AspectRatioFixed->True,
    StyleMenuListing->None,
    FontFamily->"Times",
    FontSize->12,
    FontSlant->"Italic"],
  
  Cell[StyleData["Footer"],
    CellMargins->{{7, 0}, {0, 4}},
    Evaluatable->False,
    PageBreakWithin->Automatic,
    TextAlignment->Center,
    AspectRatioFixed->True,
    StyleMenuListing->None,
    FontFamily->"Times",
    FontSize->12,
    FontSlant->"Italic"],
  
  Cell[StyleData["PageNumber"],
    CellMargins->{{0, 0}, {4, 1}},
    StyleMenuListing->None,
    FontFamily->"Times",
    FontSize->10]
  }, Closed]],
  
  Cell[CellGroupData[{
  
  Cell["Palette Styles", "Section"],
  
  Cell["\<\
The cells below define styles that define standard \
ButtonFunctions, for use in palette buttons.\
\>", "Text"],
  
  Cell[StyleData["Paste"],
    StyleMenuListing->None,
    ButtonStyleMenuListing->Automatic,
    ButtonBoxOptions->{ButtonFunction:>(FrontEndExecute[ {
        FrontEnd`NotebookApply[ 
          FrontEnd`InputNotebook[ ], #, After]}]&)}],
  
  Cell[StyleData["Evaluate"],
    StyleMenuListing->None,
    ButtonStyleMenuListing->Automatic,
    ButtonBoxOptions->{ButtonFunction:>(FrontEndExecute[ {
        FrontEnd`NotebookApply[ 
          FrontEnd`InputNotebook[ ], #, All], 
        SelectionEvaluate[ 
          FrontEnd`InputNotebook[ ], All]}]&)}],
  
  Cell[StyleData["EvaluateCell"],
    StyleMenuListing->None,
    ButtonStyleMenuListing->Automatic,
    ButtonBoxOptions->{ButtonFunction:>(FrontEndExecute[ {
        FrontEnd`NotebookApply[ 
          FrontEnd`InputNotebook[ ], #, All], 
        FrontEnd`SelectionMove[ 
          FrontEnd`InputNotebook[ ], All, Cell, 1], 
        FrontEnd`SelectionEvaluateCreateCell[ 
          FrontEnd`InputNotebook[ ], All]}]&)}],
  
  Cell[StyleData["CopyEvaluate"],
    StyleMenuListing->None,
    ButtonStyleMenuListing->Automatic,
    ButtonBoxOptions->{ButtonFunction:>(FrontEndExecute[ {
        FrontEnd`SelectionCreateCell[ 
          FrontEnd`InputNotebook[ ], All], 
        FrontEnd`NotebookApply[ 
          FrontEnd`InputNotebook[ ], #, All], 
        FrontEnd`SelectionEvaluate[ 
          FrontEnd`InputNotebook[ ], All]}]&)}],
  
  Cell[StyleData["CopyEvaluateCell"],
    StyleMenuListing->None,
    ButtonStyleMenuListing->Automatic,
    ButtonBoxOptions->{ButtonFunction:>(FrontEndExecute[ {
        FrontEnd`SelectionCreateCell[ 
          FrontEnd`InputNotebook[ ], All], 
        FrontEnd`NotebookApply[ 
          FrontEnd`InputNotebook[ ], #, All], 
        FrontEnd`SelectionEvaluateCreateCell[ 
          FrontEnd`InputNotebook[ ], All]}]&)}]
  }, Closed]],
  
  Cell[CellGroupData[{
  
  Cell["Hyperlink Styles", "Section"],
  
  Cell["\<\
The cells below define styles useful for making hypertext \
ButtonBoxes.  The \"Hyperlink\" style is for links within the same Notebook, \
or between Notebooks.\
\>", "Text"],
  
  Cell[CellGroupData[{
  
  Cell[StyleData["Hyperlink"],
    StyleMenuListing->None,
    ButtonStyleMenuListing->Automatic,
    FontColor->RGBColor[0, 0, 1],
    FontVariations->{"Underline"->True},
    ButtonBoxOptions->{ButtonFunction:>(FrontEndExecute[ {
        FrontEnd`NotebookLocate[ #2]}]&),
    Active->True,
    ButtonNote->ButtonData}],
  
  Cell[StyleData["Hyperlink", "Presentation"]],
  
  Cell[StyleData["Hyperlink", "Condensed"]],
  
  Cell[StyleData["Hyperlink", "Printout"],
    FontColor->GrayLevel[0],
    FontVariations->{"Underline"->False}]
  }, Closed]],
  
  Cell["\<\
The following styles are for linking automatically to the on-line \
help system.\
\>", "Text"],
  
  Cell[CellGroupData[{
  
  Cell[StyleData["MainBookLink"],
    StyleMenuListing->None,
    ButtonStyleMenuListing->Automatic,
    FontColor->RGBColor[0, 0, 1],
    FontVariations->{"Underline"->True},
    ButtonBoxOptions->{ButtonFunction:>(FrontEndExecute[ {
        FrontEnd`HelpBrowserLookup[ "MainBook", #]}]&),
    Active->True,
    ButtonFrame->"None"}],
  
  Cell[StyleData["MainBookLink", "Presentation"]],
  
  Cell[StyleData["MainBookLink", "Condensed"]],
  
  Cell[StyleData["MainBookLink", "Printout"],
    FontColor->GrayLevel[0],
    FontVariations->{"Underline"->False}]
  }, Closed]],
  
  Cell[CellGroupData[{
  
  Cell[StyleData["AddOnsLink"],
    StyleMenuListing->None,
    ButtonStyleMenuListing->Automatic,
    FontFamily->"Courier",
    FontColor->RGBColor[0, 0, 1],
    FontVariations->{"Underline"->True},
    ButtonBoxOptions->{ButtonFunction:>(FrontEndExecute[ {
        FrontEnd`HelpBrowserLookup[ "AddOns", #]}]&),
    Active->True,
    ButtonFrame->"None"}],
  
  Cell[StyleData["AddOnsLink", "Presentation"]],
  
  Cell[StyleData["AddOnsLink", "Condensed"]],
  
  Cell[StyleData["AddOnLink", "Printout"],
    FontColor->GrayLevel[0],
    FontVariations->{"Underline"->False}]
  }, Closed]],
  
  Cell[CellGroupData[{
  
  Cell[StyleData["RefGuideLink"],
    StyleMenuListing->None,
    ButtonStyleMenuListing->Automatic,
    FontFamily->"Courier",
    FontColor->RGBColor[0, 0, 1],
    FontVariations->{"Underline"->True},
    ButtonBoxOptions->{ButtonFunction:>(FrontEndExecute[ {
        FrontEnd`HelpBrowserLookup[ "RefGuideLink", #]}]&),
    Active->True,
    ButtonFrame->"None"}],
  
  Cell[StyleData["RefGuideLink", "Presentation"]],
  
  Cell[StyleData["RefGuideLink", "Condensed"]],
  
  Cell[StyleData["RefGuideLink", "Printout"],
    FontColor->GrayLevel[0],
    FontVariations->{"Underline"->False}]
  }, Closed]],
  
  Cell[CellGroupData[{
  
  Cell[StyleData["GettingStartedLink"],
    StyleMenuListing->None,
    ButtonStyleMenuListing->Automatic,
    FontColor->RGBColor[0, 0, 1],
    FontVariations->{"Underline"->True},
    ButtonBoxOptions->{ButtonFunction:>(FrontEndExecute[ {
        FrontEnd`HelpBrowserLookup[ "GettingStarted", #]}]&),
    Active->True,
    ButtonFrame->"None"}],
  
  Cell[StyleData["GettingStartedLink", "Presentation"]],
  
  Cell[StyleData["GettingStartedLink", "Condensed"]],
  
  Cell[StyleData["GettingStartedLink", "Printout"],
    FontColor->GrayLevel[0],
    FontVariations->{"Underline"->False}]
  }, Closed]],
  
  Cell[CellGroupData[{
  
  Cell[StyleData["OtherInformationLink"],
    StyleMenuListing->None,
    ButtonStyleMenuListing->Automatic,
    FontColor->RGBColor[0, 0, 1],
    FontVariations->{"Underline"->True},
    ButtonBoxOptions->{ButtonFunction:>(FrontEndExecute[ {
        FrontEnd`HelpBrowserLookup[ "OtherInformation", #]}]&),
    Active->True,
    ButtonFrame->"None"}],
  
  Cell[StyleData["OtherInformationLink", "Presentation"]],
  
  Cell[StyleData["OtherInformationLink", "Condensed"]],
  
  Cell[StyleData["OtherInformationLink", "Printout"],
    FontColor->GrayLevel[0],
    FontVariations->{"Underline"->False}]
  }, Closed]]
  }, Closed]],
  
  Cell[CellGroupData[{
  
  Cell["Placeholder Styles", "Section"],
  
  Cell["\<\
The cells below define styles useful for making placeholder \
objects in palette templates.\
\>", "Text"],
  
  Cell[CellGroupData[{
  
  Cell[StyleData["Placeholder"],
    Editable->False,
    Selectable->False,
    StyleBoxAutoDelete->True,
    Placeholder->True,
    StyleMenuListing->None],
  
  Cell[StyleData["Placeholder", "Presentation"]],
  
  Cell[StyleData["Placeholder", "Condensed"]],
  
  Cell[StyleData["Placeholder", "Printout"]]
  }, Closed]],
  
  Cell[CellGroupData[{
  
  Cell[StyleData["SelectionPlaceholder"],
    Editable->False,
    Selectable->False,
    StyleBoxAutoDelete->True,
    Placeholder->Primary,
    StyleMenuListing->None,
    DrawHighlighted->True],
  
  Cell[StyleData["SelectionPlaceholder", "Presentation"]],
  
  Cell[StyleData["SelectionPlaceholder", "Condensed"]],
  
  Cell[StyleData["SelectionPlaceholder", "Printout"]]
  }, Closed]]
  }, Closed]],
  
  Cell[CellGroupData[{
  
  Cell["FormatType Styles", "Section"],
  
  Cell["\<\
The cells below define styles that are mixed in with the styles \
of most cells.  If a cell's FormatType matches the name of one of the styles \
defined below, then that style is applied between the cell's style and its \
own options.\
\>", "Text"],
  
  Cell[StyleData["CellExpression"],
    PageWidth->Infinity,
    CellMargins->{{6, Inherited}, {Inherited, Inherited}},
    ShowCellLabel->False,
    ShowSpecialCharacters->False,
    AllowInlineCells->False,
    AutoItalicWords->{},
    StyleMenuListing->None,
    FontFamily->"Courier",
    Background->GrayLevel[1]],
  
  Cell[StyleData["InputForm"],
    AllowInlineCells->False,
    StyleMenuListing->None,
    FontFamily->"Courier"],
  
  Cell[StyleData["OutputForm"],
    PageWidth->Infinity,
    TextAlignment->Left,
    LineSpacing->{1, -5},
    StyleMenuListing->None,
    FontFamily->"Courier"],
  
  Cell[StyleData["StandardForm"],
    LineSpacing->{1.25, 0},
    StyleMenuListing->None,
    FontFamily->"Courier"],
  
  Cell[StyleData["TraditionalForm"],
    LineSpacing->{1.25, 0},
    SingleLetterItalics->True,
    TraditionalFunctionNotation->True,
    DelimiterMatching->None,
    StyleMenuListing->None],
  
  Cell["\<\
The style defined below is mixed in to any cell that is in an \
inline cell within another.\
\>", "Text"],
  
  Cell[StyleData["InlineCell"],
    TextAlignment->Left,
    ScriptLevel->1,
    StyleMenuListing->None],
  
  Cell[StyleData["InlineCellEditing"],
    StyleMenuListing->None,
    Background->RGBColor[1, 0.749996, 0.8]]
  }, Closed]]
  }, Open  ]],
  
  Cell[StyleData["Special1"],
    CellMargins->{{7, Inherited}, {Inherited, Inherited}},
    Evaluatable->False,
    CellGroupingRules->{"NormalGrouping", 79},
    CellHorizontalScrolling->False,
    PageBreakWithin->Automatic,
    AspectRatioFixed->True,
    FontFamily->"Times",
    FontSize->12]
  }],
MacintoshSystemPageSetup->"\<\
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]


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End of Mathematica Notebook file.
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