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Notebook[{
Cell["Quiz", "Title"],

Cell["May 17, 2006", "Subtitle"],

Cell["Topic: Review of essentials", "Section"],

Cell["\<\
1)\tIf a system of equations has 2 different solutions then it has infinitely \
many solutions.\
\>", "Text"],

Cell[TextData[{
  "2)\tIf ",
  Cell[BoxData[
      \(TraditionalForm\`U\)]],
  " and ",
  Cell[BoxData[
      \(TraditionalForm\`V\)]],
  " are subspaces of ",
  Cell[BoxData[
      \(TraditionalForm\`\[DoubleStruckCapitalR]\^15\)]],
  ", then the intersection of ",
  Cell[BoxData[
      \(TraditionalForm\`U\)]],
  " and ",
  Cell[BoxData[
      \(TraditionalForm\`V\)]],
  " is infinite."
}], "Text"],

Cell[TextData[{
  "3)\tIf matrices ",
  Cell[BoxData[
      \(TraditionalForm\`A\)]],
  " and ",
  Cell[BoxData[
      \(TraditionalForm\`B\)]],
  " have the same reduced row echelon form, then ",
  Cell[BoxData[
      \(TraditionalForm\`A = B\)]],
  "."
}], "Text"],

Cell["\<\
4)\tElementary row operations do not change the span of the rows.\
\>", "Text"],

Cell["\<\
5)\tElementary row operations do not change the span of the columns.\
\>", "Text"],

Cell[TextData[{
  "6)\tIf ",
  Cell[BoxData[
      \(TraditionalForm\`\(y\& \[RightVector] \)\)]],
  " and ",
  Cell[BoxData[
      \(TraditionalForm\`\(z\& \[RightVector] \)\)]],
  " are solutions to the system ",
  Cell[BoxData[
      \(TraditionalForm\`A\ \(x\& \[RightVector] \) = \(b\& \[RightVector] \)\
\)]],
  " (the matrix ",
  Cell[BoxData[
      \(TraditionalForm\`A\)]],
  " and the vector ",
  Cell[BoxData[
      \(TraditionalForm\`\(b\& \[RightVector] \)\)]],
  " are fixed), then ",
  Cell[BoxData[
      \(TraditionalForm\`\(y\& \[RightVector] \) - \(z\& \[RightVector] \
\)\)]],
  " is a solution to the homogeneous system ",
  Cell[BoxData[
      \(TraditionalForm\`A\ \(x\& \[RightVector] \) = \(0\& \[RightVector] \)\
\)]],
  "."
}], "Text"],

Cell[TextData[{
  "7)\tIf the sum of the elements in each row of a square matrix ",
  Cell[BoxData[
      \(TraditionalForm\`M\)]],
  " is 13 (the first row adds up to 13, and the second row adds up to 13, et \
cetera), then 13 is an eigenvalue of ",
  Cell[BoxData[
      \(TraditionalForm\`M\)]],
  "."
}], "Text"],

Cell[TextData[{
  "8)\tThe vectors ",
  Cell[BoxData[
      \(TraditionalForm\`\((1, 1, 1)\)\^T\)]],
  ", ",
  Cell[BoxData[
      \(TraditionalForm\`\((2, 1, \(-3\))\)\^T\)]],
  ", ",
  Cell[BoxData[
      \(TraditionalForm\`\((4, \(-5\), 1)\)\^T\)]],
  " are orthonormal."
}], "Text"],

Cell[TextData[{
  "9)\tIf the null space of the square matrix ",
  Cell[BoxData[
      \(TraditionalForm\`M\)]],
  " is just the set ",
  Cell[BoxData[
      \(TraditionalForm\`{\(0\& \[RightVector] \)}\)]],
  ", then ",
  Cell[BoxData[
      \(TraditionalForm\`M\)]],
  " has an inverse."
}], "Text"],

Cell[TextData[{
  "10)\tLet ",
  Cell[BoxData[
      \(TraditionalForm\`L : \[DoubleStruckCapitalR]\^n \[Rule] \ \
\[DoubleStruckCapitalR]\^n\)]],
  " be a linear transformation. If ",
  Cell[BoxData[
      \(TraditionalForm\`L(\(\(x\_1\)\& \[RightVector] \)) = 
        L(\(\(x\_2\)\& \[RightVector] \))\)]],
  ", then the vectors ",
  Cell[BoxData[
      \(TraditionalForm\`\(\(x\_1\)\& \[RightVector] \)\)]],
  " and ",
  Cell[BoxData[
      \(TraditionalForm\`\(\(x\_2\)\& \[RightVector] \)\)]],
  " must be equal."
}], "Text"],

Cell[TextData[{
  "11)\tIf ",
  Cell[BoxData[
      \(TraditionalForm\`S\)]],
  " is a subspace of a vector space ",
  Cell[BoxData[
      \(TraditionalForm\`V\)]],
  ", then ",
  Cell[BoxData[
      \(TraditionalForm\`S\)]],
  " is a vector space."
}], "Text"],

Cell[TextData[{
  "12)\t",
  Cell[BoxData[
      \(TraditionalForm\`\[DoubleStruckCapitalR]\^2\)]],
  " is a subspace of ",
  Cell[BoxData[
      \(TraditionalForm\`\[DoubleStruckCapitalR]\^3\)]],
  "."
}], "Text"],

Cell[TextData[{
  "13)\tThe dimensions of the null space of ",
  Cell[BoxData[
      \(TraditionalForm\`A\)]],
  " and of ",
  Cell[BoxData[
      \(TraditionalForm\`A\^T\)]],
  " are the same."
}], "Text"],

Cell[TextData[{
  "14)\tIf ",
  Cell[BoxData[
      \(TraditionalForm\`x\_1, \ x\_2, \  ... , \ x\_n\)]],
  " span ",
  Cell[BoxData[
      \(TraditionalForm\`\[DoubleStruckCapitalR]\^n\)]],
  ", then they are linearly independent."
}], "Text"],

Cell[TextData[{
  "15)\tIf ",
  Cell[BoxData[
      \(TraditionalForm\`x\_1, \ x\_2, \  ... , \ x\_n\)]],
  " span a vector space ",
  Cell[BoxData[
      \(TraditionalForm\`V\)]],
  ", then they are linearly independent."
}], "Text"],

Cell["16)\tA homogeneous linear system is always consistent.", "Text"],

Cell[TextData[{
  "17)\tIf ",
  Cell[BoxData[
      \(TraditionalForm\`A\)]],
  " and ",
  Cell[BoxData[
      \(TraditionalForm\`B\)]],
  " are nonsingular ",
  Cell[BoxData[
      \(TraditionalForm\`n\[Cross]n\)]],
  " matrices, then ",
  Cell[BoxData[
      \(TraditionalForm\`\((A - B)\)\^2 = A\^2 - 2  A\ B + B\^2\)]],
  "."
}], "Text"],

Cell[TextData[{
  "18)\tIf ",
  Cell[BoxData[
      \(TraditionalForm\`A\)]],
  " and ",
  Cell[BoxData[
      \(TraditionalForm\`B\)]],
  " are nonsingular  ",
  Cell[BoxData[
      \(TraditionalForm\`n\[Cross]n\)]],
  " matrices, then ",
  Cell[BoxData[
      \(TraditionalForm\`A\ B\)]],
  " is also nonsingular and ",
  Cell[BoxData[
      \(TraditionalForm\`\((A\ B)\)\^\(-1\) = B\^\(-1\)\ A\^\(-1\)\)]],
  "."
}], "Text"],

Cell[TextData[{
  "19)\tLet ",
  Cell[BoxData[
      \(TraditionalForm\`x\_1, \ x\_2, \ x\_3, \ x\_4\)]],
  " be linearly independent vectors in ",
  Cell[BoxData[
      \(TraditionalForm\`\[DoubleStruckCapitalR]\^4\)]],
  ", and ",
  Cell[BoxData[
      \(TraditionalForm\`\[Alpha], \ \[Beta], \ \[Chi], \ \[Delta]\)]],
  " are any real numbers. There is a 4\[Cross]4 matrix with eigenvectors ",
  Cell[BoxData[
      \(TraditionalForm\`x\_1, \ x\_2, \ x\_3, \ x\_4\)]],
  " and eigenvalues ",
  "\[Alpha], \[Beta], \[Chi], \[Delta]."
}], "Text"],

Cell[TextData[{
  "20)\tIf \[Lambda] is an eigenvalue of ",
  Cell[BoxData[
      \(TraditionalForm\`A\)]],
  ", then ",
  Cell[BoxData[
      \(TraditionalForm\`\[Lambda]\^2\)]],
  " is an eigenvalue of ",
  Cell[BoxData[
      \(TraditionalForm\`A\^2\)]],
  "."
}], "Text"]
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