(*^

::[	frontEndVersion = "Macintosh Mathematica Notebook Front End Version 2.1";
	macintoshStandardFontEncoding; 
	paletteColors = 128; automaticGrouping; currentKernel; 
	fontset = title, inactive, noPageBreakBelow, nohscroll, preserveAspect, groupLikeTitle, center, M7, bold, e8,  24, "Times"; ;
	fontset = subtitle, inactive, noPageBreakBelow, nohscroll, preserveAspect, groupLikeTitle, center, M7, bold, e6,  18, "Times"; ;
	fontset = subsubtitle, inactive, noPageBreakBelow, nohscroll, preserveAspect, groupLikeTitle, center, M7, italic, e6,  14, "Times"; ;
	fontset = section, inactive, noPageBreakBelow, nohscroll, preserveAspect, groupLikeSection, grayBox, M22, bold, a20,  18, "Times"; ;
	fontset = subsection, inactive, noPageBreakBelow, nohscroll, preserveAspect, groupLikeSection, blackBox, M19, bold, a15,  14, "Times"; ;
	fontset = subsubsection, inactive, noPageBreakBelow, nohscroll, preserveAspect, groupLikeSection, whiteBox, M18, bold, a12,  12, "Times"; ;
	fontset = text, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7,  12, "Times"; ;
	fontset = smalltext, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7,  10, "Times"; ;
	fontset = input, noPageBreakInGroup, nowordwrap, preserveAspect, groupLikeInput, M42, N23, bold, L-5,  12, "Courier"; ;
	fontset = output, output, inactive, noPageBreakInGroup, nowordwrap, preserveAspect, groupLikeOutput, M42, N23, L-5,  12, "Courier"; ;
	fontset = message, inactive, noPageBreakInGroup, nowordwrap, preserveAspect, groupLikeOutput, M42, N23, R65535, L-5,  12, "Courier"; ;
	fontset = print, inactive, noPageBreakInGroup, nowordwrap, preserveAspect, groupLikeOutput, M42, N23, L-5,  12, "Courier"; ;
	fontset = info, inactive, noPageBreakInGroup, nowordwrap, preserveAspect, groupLikeOutput, M42, N23, B65535, L-5,  12, "Courier"; ;
	fontset = postscript, PostScript, formatAsPostScript, output, inactive, noPageBreakInGroup, nowordwrap, preserveAspect, groupLikeGraphics, M7, l34, w282, h287,  12, "Courier"; ;
	fontset = name, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, italic,  10, "Geneva"; ;
	fontset = header, inactive, noKeepOnOnePage, preserveAspect, M7,  12, "Times"; ;
	fontset = leftheader, inactive, L2,  12, "Times"; ;
	fontset = footer, inactive, noKeepOnOnePage, preserveAspect, center, M7,  12, "Times"; ;
	fontset = leftfooter, inactive, L2,  12, "Times"; ;
	fontset = help, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7,  10, "Times"; ;
	fontset = clipboard, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7,  12, "Times"; ;
	fontset = completions, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7,  12, "Times"; ;
	fontset = special1, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7,  12, "Times"; ;
	fontset = special2, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7,  12, "Times"; ;
	fontset = special3, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7,  12, "Times"; ;
	fontset = special4, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7,  12, "Times"; ;
	fontset = special5, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7,  12, "Times"; ;
]
:[font = special1; inactive; preserveAspect; ]
                        Transformations from One Vector Space to Another

The cell below defines two images that we will use to study the affects of transformations.  Evaluate it now. 
;[s]
4:0,1;72,0;167,2;182,0;185,-1;
3:2,13,9,Times,0,12,0,0,0;1,13,9,Times,1,12,0,0,0;1,13,9,Times,1,12,65535,0,0;
:[font = input; preserveAspect; ]
BoxPoints := {{0, 0}, {1, 0}, {1, 1}, {0, 1}}

Box := {{RGBColor[0.2, 0.2, 0.2],
         Thickness[0.01],
         Line[{AllPoints[[1]], AllPoints[[2]],
               AllPoints[[3]], AllPoints[[4]],
               AllPoints[[1]]}]},
        {RGBColor[1, 0, 0], 
         PointSize[0.04],
         Point[AllPoints[[1]]]},
        {RGBColor[0, 1, 0], 
         PointSize[0.04],
         Point[AllPoints[[2]]]},
        {RGBColor[0, 0, 1], 
         PointSize[0.04],
         Point[AllPoints[[3]]]},
        {RGBColor[1, 0, 1], 
         PointSize[0.04],
         Point[AllPoints[[4]]]}}
         
 DrawBox[] := Show[Graphics[Box],
                   PlotRange -> {{-3, 3}, {-3, 3}},
                   Axes -> None,
                   Frame -> True,
                   AspectRatio -> Automatic]
                   
 AllPoints := BoxPoints
 DrawBox[]
 
 AllPoints := Table[BoxPoints[[k]] + {-2, 1}, 
                    {k, 1, Length[BoxPoints]}]
                    
 DrawBox[]
 
 ZPoints := {{0, 0, 0}, {0, 0, 2}, {0, 0, 4},
             {0, 0, 6}, {0, 0, 8}, {0, 0, 10}}
             
 Zaxis := {{RGBColor[0.2, 0.2, 0.2],
            Thickness[0.01],
            Line[{AllPoints[[1]], AllPoints[[6]]}]},
           {RGBColor[1, 0, 0],
            PointSize[0.02],
            Point[AllPoints[[1]]]},
           {RGBColor[0, 0, 1],
            PointSize[0.02],
            Point[AllPoints[[2]]]},
           {RGBColor[1, 0, 0],
            PointSize[0.02],
            Point[AllPoints[[3]]]},
           {RGBColor[0, 0, 1],
            PointSize[0.02],
            Point[AllPoints[[4]]]},
           {RGBColor[1, 0, 0],
            PointSize[0.02],
            Point[AllPoints[[5]]]},
           {RGBColor[0, 0, 1],
            PointSize[0.02],
            Point[AllPoints[[6]]]}}
            
 DrawZaxis[] := Show[Graphics[Zaxis],
                   PlotRange -> {{-10, 10}, {-10, 10}},
                   Axes -> None,
                   Frame -> True,
                   AspectRatio -> Automatic]
                   

            
  u := {1/Sqrt[3], 1/Sqrt[3], 1/Sqrt[3]}
  v := {-1/Sqrt[2], 1/Sqrt[2], 0}
  
  AllPoints := Table[{ZPoints[[k]].u, ZPoints[[k]].v},
                     {k, 1, Length[ZPoints]}]
                     
  DrawZaxis[]
           
        
^*)