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Cell[CellGroupData[{
Cell["Inner Products", "Title"],

Cell[TextData[{
  "Example 2.1 (Page 5)\nDr. Van Fleet\n\nCommands Used (Linked to Help \
Browser): \n\t",
  ButtonBox["MatrixForm",
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  "\n\t",
  ButtonBox["Transpose",
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  "\n\t",
  ButtonBox["Dot(.)",
    ButtonStyle->"MainBookLink"],
  "\n\t",
  ButtonBox["Clear",
    ButtonStyle->"MainBookLink"],
  "\n\t",
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Cell[TextData[{
  "Defining Vectors in ",
  StyleBox["Mathematica",
    FontSlant->"Italic"],
  " "
}], "Subsubtitle"],

Cell[TextData[{
  "We will compute the following inner products using ",
  StyleBox["Mathematica",
    FontSlant->"Italic"],
  ".  First, we will define two vectors.  We will call these vectors ",
  StyleBox["v",
    FontWeight->"Bold"],
  " and ",
  StyleBox["w",
    FontWeight->"Bold"],
  ".  ",
  "Creating vectors in ",
  StyleBox["Mathematica",
    FontSlant->"Italic"],
  " ",
  "is quite simple.  A vector in ",
  StyleBox["Mathematica",
    FontSlant->"Italic"],
  " is a ",
  "list",
  ".  Elements of a list are separated by commas and enclosed with curly \
brackets, i.e., {}. ",
  "These curly brackets  define a list, which is essentially a column of a \
matrix.  To represent a row, you must create a list of lists by using two \
sets of curly brackets.  This will be portrayed more clearly when we use \
matrices in later examples. "
}], "Text"],

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same length!",
  FontWeight->"Bold",
  FontSlant->"Italic"]], "SmallText"],

Cell[BoxData[{
    \(\(v = {1, 4, \(-1\)};\)\), "\[IndentingNewLine]", 
    \(\(w = {3, 2, \(-2\)};\)\)}], "Input"],

Cell[TextData[{
  "To check that you have correctly defined the two vectors, you can simply \
type ",
  StyleBox["v",
    FontWeight->"Bold"],
  " in the input cell.  If you want to display the vector in our regular \
vector format, you can type the following after the vector: ",
  StyleBox["//MatrixForm.",
    FontWeight->"Bold"],
  "  You can see this by executing the cell below.  "
}], "Text"],

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    \(v\), "\[IndentingNewLine]", 
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Cell[TextData[{
  "To take the inner product of vectors in ",
  StyleBox["Mathematica",
    FontSlant->"Italic"],
  ", you must use a period.  If you use a regular multiplication sign, i.e., \
*, the two vectors will be multiplied. \n\nv.w = Tramspose[v] * w\n\nThis \
will be demonstrated with the two vectors ",
  StyleBox["v",
    FontWeight->"Bold"],
  " and ",
  StyleBox["w",
    FontWeight->"Bold"],
  " in the cell below.  To make sure that ",
  StyleBox["Mathematica",
    FontSlant->"Italic"],
  " is producing the correct answer, check by hand."
}], "Text"],

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Cell[TextData[{
  "Let's try another example with v={-1,3} and w={6,2}.  You can simply \
redefine ",
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    FontWeight->"Bold"],
  " and ",
  StyleBox["w",
    FontWeight->"Bold"],
  " without having to worry what the two vectors were defined previously in \
the notebook.  Although, you can use the ",
  StyleBox["Clear",
    FontWeight->"Bold"],
  " command, which will erase any previous definitions of any variables as \
long as you specify their names separated by commas.  Compute the inner \
product with the new vectors in the cell below."
}], "Text"],

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    \(\(Clear[v, w];\)\), "\[IndentingNewLine]", 
    \(v = \[IndentingNewLine]\(\(w\)\(=\)\)\)}], "Input"],

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Cell["Further Exploration", "Section"],

Cell[TextData[{
  "A really useful tool for creating lists is the ",
  StyleBox["Table",
    FontWeight->"Bold"],
  " command.  A Table command is a function, so its contents are enclosed \
with [ ].  For most of our purposes, the command takes two arguments.  These \
arguments are separated by commas.  The first is a function of the index \
variable, and the second is a list that gives the range of the index \
variable.  This list can take many different forms.  Execute the cell below \
to see some examples:"
}], "Text"],

Cell[BoxData[{
    \(v = Table[2*k, {k, 1, 6}]\), "\[IndentingNewLine]", 
    \(\(\(w = Table[k, {k, 2, 12, 2}]\)\(\[IndentingNewLine]\)
    \)\), "\[IndentingNewLine]", 
    \(\(n = 20;\)\), "\[IndentingNewLine]", 
    \(\(f[m_] := Sin[2*Pi*m/n];\)\), "\[IndentingNewLine]", 
    \(\(\(x = Table[f[m], {m, 1, n}]\)\(\[IndentingNewLine]\)
    \)\), "\[IndentingNewLine]", 
    \(z = Table[{k, k^2}, {k, 1, 5}]\)}], "Input"],

Cell["\<\
To review what occured in the previous cell, the first vector v is a simply \
table of the first six even positive integers.  We can generate exactly the \
same list, if we change the range on the index variable.  For w, we started \
at 2 and stopped at 12, but stepped by intervals of 2.

The next example showed how you can define a function (in this case the sine \
curve) and build a list of samples of the function.  Finally, the last \
example shows that you can even generate a list of order pairs.  We can also \
think of z as a 2 x 5 matrix.\
\>", "Text"]
}, Open  ]]
}, Open  ]]
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