(* Content-type: application/mathematica *) (*** Wolfram Notebook File ***) (* http://www.wolfram.com/nb *) (* CreatedBy='Mathematica 7.0' *) (*CacheID: 234*) (* Internal cache information: NotebookFileLineBreakTest NotebookFileLineBreakTest NotebookDataPosition[ 145, 7] NotebookDataLength[ 22348, 710] NotebookOptionsPosition[ 20314, 634] NotebookOutlinePosition[ 20678, 650] CellTagsIndexPosition[ 20635, 647] WindowFrame->Normal*) (* Beginning of Notebook Content *) Notebook[{ Cell[CellGroupData[{ Cell[BoxData[{ RowBox[{ RowBox[{"F", "=", RowBox[{"{", RowBox[{ RowBox[{"{", RowBox[{"0", ",", "0", ",", "0", ",", "0"}], "}"}], ",", RowBox[{"{", RowBox[{"0", ",", "0", ",", RowBox[{"1", "/", "R"}], ",", "0"}], "}"}], ",", RowBox[{"{", RowBox[{"0", ",", RowBox[{"-", "g"}], ",", "0", ",", RowBox[{"-", "g"}]}], "}"}], ",", RowBox[{"{", RowBox[{"0", ",", "0", ",", "0", ",", "0"}], "}"}]}], "}"}]}], ";"}], "\[IndentingNewLine]", RowBox[{"MatrixForm", "[", "F", "]"}]}], "Input", CellChangeTimes->{{3.4813613173748302`*^9, 3.4813613487189817`*^9}}], Cell[BoxData[ TagBox[ RowBox[{"(", "\[NoBreak]", GridBox[{ {"0", "0", "0", "0"}, {"0", "0", FractionBox["1", "R"], "0"}, {"0", RowBox[{"-", "g"}], "0", RowBox[{"-", "g"}]}, {"0", "0", "0", "0"} }, GridBoxAlignment->{ "Columns" -> {{Center}}, "ColumnsIndexed" -> {}, "Rows" -> {{Baseline}}, "RowsIndexed" -> {}}, GridBoxSpacings->{"Columns" -> { Offset[0.27999999999999997`], { Offset[0.7]}, Offset[0.27999999999999997`]}, "ColumnsIndexed" -> {}, "Rows" -> { Offset[0.2], { Offset[0.4]}, Offset[0.2]}, "RowsIndexed" -> {}}], "\[NoBreak]", ")"}], Function[BoxForm`e$, MatrixForm[BoxForm`e$]]]], "Output", CellChangeTimes->{3.4813613490158606`*^9}] }, Open ]], Cell["Part (a)", "Text", CellChangeTimes->{{3.481362889656213*^9, 3.481362912046981*^9}}], Cell[CellGroupData[{ Cell[BoxData[{ RowBox[{ RowBox[{"H", "=", RowBox[{"{", RowBox[{"{", RowBox[{"1", ",", "1", ",", "0", ",", "0"}], "}"}], "}"}]}], ";"}], "\[IndentingNewLine]", RowBox[{"MatrixForm", "[", "H", "]"}], "\[IndentingNewLine]", RowBox[{ RowBox[{"col1", "=", RowBox[{"Transpose", "[", "H", "]"}]}], ";"}]}], "Input", CellChangeTimes->{{3.4813607108800306`*^9, 3.481360716473888*^9}, { 3.481361012870204*^9, 3.481361021589121*^9}, {3.4813612024514847`*^9, 3.481361213436*^9}, {3.4813618223031683`*^9, 3.4813618268032265`*^9}}], Cell[BoxData[ TagBox[ RowBox[{"(", "\[NoBreak]", GridBox[{ {"1", "1", "0", "0"} }, GridBoxAlignment->{ "Columns" -> {{Center}}, "ColumnsIndexed" -> {}, "Rows" -> {{Baseline}}, "RowsIndexed" -> {}}, GridBoxSpacings->{"Columns" -> { Offset[0.27999999999999997`], { Offset[0.7]}, Offset[0.27999999999999997`]}, "ColumnsIndexed" -> {}, "Rows" -> { Offset[0.2], { Offset[0.4]}, Offset[0.2]}, "RowsIndexed" -> {}}], "\[NoBreak]", ")"}], Function[BoxForm`e$, MatrixForm[BoxForm`e$]]]], "Output", CellChangeTimes->{3.4813607920065885`*^9, 3.481361022604766*^9, 3.4813612137172537`*^9, 3.481361374203683*^9, 3.481361827318858*^9}] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"col2", "=", RowBox[{ RowBox[{"Transpose", "[", "F", "]"}], ".", RowBox[{"Transpose", "[", "H", "]"}]}]}]], "Input", CellChangeTimes->{{3.4813613060934362`*^9, 3.481361307827833*^9}, { 3.4813614611891713`*^9, 3.481361464142334*^9}, {3.481361622972492*^9, 3.4813616292069473`*^9}, {3.481361666488674*^9, 3.4813616693324604`*^9}}], Cell[BoxData[ RowBox[{"{", RowBox[{ RowBox[{"{", "0", "}"}], ",", RowBox[{"{", "0", "}"}], ",", RowBox[{"{", FractionBox["1", "R"], "}"}], ",", RowBox[{"{", "0", "}"}]}], "}"}]], "Output", CellChangeTimes->{ 3.4813613082965894`*^9, 3.4813614647985926`*^9, {3.481361625691277*^9, 3.481361629613202*^9}, 3.481361670066845*^9}] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"col3", "=", RowBox[{ RowBox[{"Transpose", "[", "F", "]"}], ".", RowBox[{"Transpose", "[", "F", "]"}], ".", RowBox[{"Transpose", "[", "H", "]"}]}]}]], "Input", CellChangeTimes->{{3.4813613060934362`*^9, 3.481361307827833*^9}, { 3.4813614611891713`*^9, 3.481361486408244*^9}, {3.4813615197836714`*^9, 3.4813615258306236`*^9}, {3.481361672410625*^9, 3.4813616739575195`*^9}}], Cell[BoxData[ RowBox[{"{", RowBox[{ RowBox[{"{", "0", "}"}], ",", RowBox[{"{", RowBox[{"-", FractionBox["g", "R"]}], "}"}], ",", RowBox[{"{", "0", "}"}], ",", RowBox[{"{", RowBox[{"-", FractionBox["g", "R"]}], "}"}]}], "}"}]], "Output", CellChangeTimes->{ 3.4813613082965894`*^9, {3.4813614647985926`*^9, 3.4813614875332584`*^9}, 3.4813615262368793`*^9, 3.4813616775669413`*^9}] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"col4", "=", RowBox[{ RowBox[{"Transpose", "[", "F", "]"}], ".", RowBox[{"Transpose", "[", "F", "]"}], ".", RowBox[{"Transpose", "[", "F", "]"}], ".", RowBox[{"Transpose", "[", "H", "]"}]}]}]], "Input", CellChangeTimes->{{3.4813613060934362`*^9, 3.481361307827833*^9}, { 3.4813614611891713`*^9, 3.481361486408244*^9}, {3.4813615197836714`*^9, 3.4813615258306236`*^9}, 3.481361646332166*^9, {3.4813616789107084`*^9, 3.4813616802857256`*^9}}], Cell[BoxData[ RowBox[{"{", RowBox[{ RowBox[{"{", "0", "}"}], ",", RowBox[{"{", "0", "}"}], ",", RowBox[{"{", RowBox[{"-", FractionBox["g", SuperscriptBox["R", "2"]]}], "}"}], ",", RowBox[{"{", "0", "}"}]}], "}"}]], "Output", CellChangeTimes->{ 3.4813613082965894`*^9, {3.4813614647985926`*^9, 3.4813614875332584`*^9}, 3.4813615262368793`*^9, 3.4813616473009286`*^9, 3.4813616805513544`*^9}] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[{ RowBox[{ RowBox[{"Z", "=", RowBox[{"{", RowBox[{ RowBox[{"{", RowBox[{"1", ",", "0", ",", "0", ",", "0"}], "}"}], ",", RowBox[{"{", RowBox[{"1", ",", "0", ",", FractionBox[ RowBox[{"-", "g"}], "R"], ",", "0"}], "}"}], ",", RowBox[{"{", RowBox[{"0", ",", FractionBox["1", "R"], ",", "0", ",", FractionBox[ RowBox[{"-", "g"}], SuperscriptBox["R", "2"]]}], "}"}], ",", RowBox[{"{", RowBox[{"0", ",", "0", ",", FractionBox[ RowBox[{"-", "g"}], "R"], ",", "0"}], "}"}]}], "}"}]}], ";"}], "\[IndentingNewLine]", RowBox[{"MatrixForm", "[", "Z", "]"}]}], "Input", CellChangeTimes->{{3.481361979383304*^9, 3.481362052634242*^9}}], Cell[BoxData[ TagBox[ RowBox[{"(", "\[NoBreak]", GridBox[{ {"1", "0", "0", "0"}, {"1", "0", RowBox[{"-", FractionBox["g", "R"]}], "0"}, {"0", FractionBox["1", "R"], "0", RowBox[{"-", FractionBox["g", SuperscriptBox["R", "2"]]}]}, {"0", "0", RowBox[{"-", FractionBox["g", "R"]}], "0"} }, GridBoxAlignment->{ "Columns" -> {{Center}}, "ColumnsIndexed" -> {}, "Rows" -> {{Baseline}}, "RowsIndexed" -> {}}, GridBoxSpacings->{"Columns" -> { Offset[0.27999999999999997`], { Offset[0.7]}, Offset[0.27999999999999997`]}, "ColumnsIndexed" -> {}, "Rows" -> { Offset[0.2], { Offset[0.4]}, Offset[0.2]}, "RowsIndexed" -> {}}], "\[NoBreak]", ")"}], Function[BoxForm`e$, MatrixForm[BoxForm`e$]]]], "Output", CellChangeTimes->{3.481362052931121*^9}] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"MatrixRank", "[", "Z", "]"}]], "Input", CellChangeTimes->{{3.4813626779048576`*^9, 3.4813626805298743`*^9}}], Cell[BoxData["3"], "Output", CellChangeTimes->{3.4813626810455027`*^9}] }, Open ]], Cell["Part (b)", "Text", CellChangeTimes->{{3.481362889656213*^9, 3.4813628937812395`*^9}}], Cell[CellGroupData[{ Cell[BoxData[{ RowBox[{ RowBox[{"H", "=", RowBox[{"{", RowBox[{ RowBox[{"{", RowBox[{"1", ",", "1", ",", "0", ",", "0"}], "}"}], ",", RowBox[{"{", RowBox[{"0", ",", "0", ",", "1", ",", "0"}], "}"}]}], "}"}]}], ";"}], "\[IndentingNewLine]", RowBox[{ RowBox[{"col1", "=", RowBox[{"Transpose", "[", "H", "]"}]}], ";"}], "\[IndentingNewLine]", RowBox[{"MatrixForm", "[", "col1", "]"}]}], "Input", CellChangeTimes->{{3.4813607108800306`*^9, 3.481360716473888*^9}, { 3.481361012870204*^9, 3.481361021589121*^9}, {3.4813612024514847`*^9, 3.481361213436*^9}, {3.4813618223031683`*^9, 3.4813618268032265`*^9}, { 3.481362923812681*^9, 3.4813629272814536`*^9}, {3.481362973234873*^9, 3.4813629794536624`*^9}}], Cell[BoxData[ TagBox[ RowBox[{"(", "\[NoBreak]", GridBox[{ {"1", "0"}, {"1", "0"}, {"0", "1"}, {"0", "0"} }, GridBoxAlignment->{ "Columns" -> {{Center}}, "ColumnsIndexed" -> {}, "Rows" -> {{Baseline}}, "RowsIndexed" -> {}}, GridBoxSpacings->{"Columns" -> { Offset[0.27999999999999997`], { Offset[0.7]}, Offset[0.27999999999999997`]}, "ColumnsIndexed" -> {}, "Rows" -> { Offset[0.2], { Offset[0.4]}, Offset[0.2]}, "RowsIndexed" -> {}}], "\[NoBreak]", ")"}], Function[BoxForm`e$, MatrixForm[BoxForm`e$]]]], "Output", CellChangeTimes->{3.4813607920065885`*^9, 3.481361022604766*^9, 3.4813612137172537`*^9, 3.481361374203683*^9, 3.481361827318858*^9, 3.481362928000208*^9, 3.4813629797505393`*^9}] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[{ RowBox[{ RowBox[{"col2", "=", RowBox[{ RowBox[{"Transpose", "[", "F", "]"}], ".", RowBox[{"Transpose", "[", "H", "]"}]}]}], ";"}], "\[IndentingNewLine]", RowBox[{"MatrixForm", "[", "col2", "]"}]}], "Input", CellChangeTimes->{{3.4813613060934362`*^9, 3.481361307827833*^9}, { 3.4813614611891713`*^9, 3.481361464142334*^9}, {3.481361622972492*^9, 3.4813616292069473`*^9}, {3.481361666488674*^9, 3.4813616693324604`*^9}, { 3.481362963562936*^9, 3.48136296893797*^9}}], Cell[BoxData[ TagBox[ RowBox[{"(", "\[NoBreak]", GridBox[{ {"0", "0"}, {"0", RowBox[{"-", "g"}]}, { FractionBox["1", "R"], "0"}, {"0", RowBox[{"-", "g"}]} }, GridBoxAlignment->{ "Columns" -> {{Center}}, "ColumnsIndexed" -> {}, "Rows" -> {{Baseline}}, "RowsIndexed" -> {}}, GridBoxSpacings->{"Columns" -> { Offset[0.27999999999999997`], { Offset[0.7]}, Offset[0.27999999999999997`]}, "ColumnsIndexed" -> {}, "Rows" -> { Offset[0.2], { Offset[0.4]}, Offset[0.2]}, "RowsIndexed" -> {}}], "\[NoBreak]", ")"}], Function[BoxForm`e$, MatrixForm[BoxForm`e$]]]], "Output", CellChangeTimes->{ 3.4813613082965894`*^9, 3.4813614647985926`*^9, {3.481361625691277*^9, 3.481361629613202*^9}, 3.481361670066845*^9, {3.4813629479378357`*^9, 3.481362969312972*^9}}] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[{ RowBox[{ RowBox[{"col3", "=", RowBox[{ RowBox[{"Transpose", "[", "F", "]"}], ".", RowBox[{"Transpose", "[", "F", "]"}], ".", RowBox[{"Transpose", "[", "H", "]"}]}]}], ";"}], "\[IndentingNewLine]", RowBox[{"MatrixForm", "[", "col3", "]"}]}], "Input", CellChangeTimes->{{3.4813613060934362`*^9, 3.481361307827833*^9}, { 3.4813614611891713`*^9, 3.481361486408244*^9}, {3.4813615197836714`*^9, 3.4813615258306236`*^9}, {3.481361672410625*^9, 3.4813616739575195`*^9}, { 3.4813629863443317`*^9, 3.4813629902662315`*^9}}], Cell[BoxData[ TagBox[ RowBox[{"(", "\[NoBreak]", GridBox[{ {"0", "0"}, { RowBox[{"-", FractionBox["g", "R"]}], "0"}, {"0", RowBox[{"-", FractionBox["g", "R"]}]}, { RowBox[{"-", FractionBox["g", "R"]}], "0"} }, GridBoxAlignment->{ "Columns" -> {{Center}}, "ColumnsIndexed" -> {}, "Rows" -> {{Baseline}}, "RowsIndexed" -> {}}, GridBoxSpacings->{"Columns" -> { Offset[0.27999999999999997`], { Offset[0.7]}, Offset[0.27999999999999997`]}, "ColumnsIndexed" -> {}, "Rows" -> { Offset[0.2], { Offset[0.4]}, Offset[0.2]}, "RowsIndexed" -> {}}], "\[NoBreak]", ")"}], Function[BoxForm`e$, MatrixForm[BoxForm`e$]]]], "Output", CellChangeTimes->{ 3.4813613082965894`*^9, {3.4813614647985926`*^9, 3.4813614875332584`*^9}, 3.4813615262368793`*^9, 3.4813616775669413`*^9, 3.4813629577972736`*^9, 3.4813629905787334`*^9}] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[{ RowBox[{ RowBox[{"col4", "=", RowBox[{ RowBox[{"Transpose", "[", "F", "]"}], ".", RowBox[{"Transpose", "[", "F", "]"}], ".", RowBox[{"Transpose", "[", "F", "]"}], ".", RowBox[{"Transpose", "[", "H", "]"}]}]}], ";"}], "\[IndentingNewLine]", RowBox[{"MatrixForm", "[", "col4", "]"}]}], "Input", CellChangeTimes->{{3.4813613060934362`*^9, 3.481361307827833*^9}, { 3.4813614611891713`*^9, 3.481361486408244*^9}, {3.4813615197836714`*^9, 3.4813615258306236`*^9}, 3.481361646332166*^9, {3.4813616789107084`*^9, 3.4813616802857256`*^9}, {3.4813629932350006`*^9, 3.481362996281895*^9}}], Cell[BoxData[ TagBox[ RowBox[{"(", "\[NoBreak]", GridBox[{ {"0", "0"}, {"0", FractionBox[ SuperscriptBox["g", "2"], "R"]}, { RowBox[{"-", FractionBox["g", SuperscriptBox["R", "2"]]}], "0"}, {"0", FractionBox[ SuperscriptBox["g", "2"], "R"]} }, GridBoxAlignment->{ "Columns" -> {{Center}}, "ColumnsIndexed" -> {}, "Rows" -> {{Baseline}}, "RowsIndexed" -> {}}, GridBoxSpacings->{"Columns" -> { Offset[0.27999999999999997`], { Offset[0.7]}, Offset[0.27999999999999997`]}, "ColumnsIndexed" -> {}, "Rows" -> { Offset[0.2], { Offset[0.4]}, Offset[0.2]}, "RowsIndexed" -> {}}], "\[NoBreak]", ")"}], Function[BoxForm`e$, MatrixForm[BoxForm`e$]]]], "Output", CellChangeTimes->{ 3.4813613082965894`*^9, {3.4813614647985926`*^9, 3.4813614875332584`*^9}, 3.4813615262368793`*^9, 3.4813616473009286`*^9, 3.4813616805513544`*^9, 3.48136296035979*^9, 3.4813629965943975`*^9}] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[{ RowBox[{ RowBox[{"ZT", "=", RowBox[{"{", RowBox[{ RowBox[{"{", RowBox[{"1", ",", "1", ",", "0", ",", "0"}], "}"}], ",", RowBox[{"{", RowBox[{"0", ",", "0", ",", "1", ",", "0"}], "}"}], ",", RowBox[{"{", RowBox[{"0", ",", "0", ",", FractionBox["1", "R"], ",", "0"}], "}"}], ",", RowBox[{"{", RowBox[{"0", ",", RowBox[{"-", "g"}], ",", "0", ",", RowBox[{"-", "g"}]}], "}"}], ",", RowBox[{"{", RowBox[{"0", ",", FractionBox[ RowBox[{"-", "g"}], "R"], ",", "0", ",", FractionBox[ RowBox[{"-", "g"}], "R"]}], "}"}], ",", RowBox[{"{", RowBox[{"0", ",", "0", ",", FractionBox[ RowBox[{"-", "g"}], "R"], ",", "0"}], "}"}], ",", RowBox[{"{", RowBox[{"0", ",", "0", ",", FractionBox[ RowBox[{"-", "g"}], SuperscriptBox["R", "2"]], ",", "0"}], "}"}], ",", RowBox[{"{", RowBox[{"0", ",", FractionBox[ SuperscriptBox["g", "2"], "R"], ",", "0", ",", FractionBox[ SuperscriptBox["g", "2"], "R"]}], "}"}]}], "}"}]}], ";"}], "\[IndentingNewLine]", RowBox[{ RowBox[{"Z", "=", RowBox[{"Transpose", "[", "ZT", "]"}]}], ";"}], "\[IndentingNewLine]", RowBox[{"MatrixForm", "[", "Z", "]"}]}], "Input", CellChangeTimes->{{3.481361979383304*^9, 3.481362052634242*^9}, { 3.4813636749644885`*^9, 3.4813637707463517`*^9}, {3.481365114889289*^9, 3.48136511732682*^9}}], Cell[BoxData[ TagBox[ RowBox[{"(", "\[NoBreak]", GridBox[{ {"1", "0", "0", "0", "0", "0", "0", "0"}, {"1", "0", "0", RowBox[{"-", "g"}], RowBox[{"-", FractionBox["g", "R"]}], "0", "0", FractionBox[ SuperscriptBox["g", "2"], "R"]}, {"0", "1", FractionBox["1", "R"], "0", "0", RowBox[{"-", FractionBox["g", "R"]}], RowBox[{"-", FractionBox["g", SuperscriptBox["R", "2"]]}], "0"}, {"0", "0", "0", RowBox[{"-", "g"}], RowBox[{"-", FractionBox["g", "R"]}], "0", "0", FractionBox[ SuperscriptBox["g", "2"], "R"]} }, GridBoxAlignment->{ "Columns" -> {{Center}}, "ColumnsIndexed" -> {}, "Rows" -> {{Baseline}}, "RowsIndexed" -> {}}, GridBoxSpacings->{"Columns" -> { Offset[0.27999999999999997`], { Offset[0.7]}, Offset[0.27999999999999997`]}, "ColumnsIndexed" -> {}, "Rows" -> { Offset[0.2], { Offset[0.4]}, Offset[0.2]}, "RowsIndexed" -> {}}], "\[NoBreak]", ")"}], Function[BoxForm`e$, MatrixForm[BoxForm`e$]]]], "Output", CellChangeTimes->{ 3.481362052931121*^9, {3.4813636985583897`*^9, 3.4813637095115843`*^9}, 3.481363771433856*^9, 3.481365117670574*^9}] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"MatrixRank", "[", "Z", "]"}]], "Input", CellChangeTimes->{{3.4813651335457773`*^9, 3.481365135827057*^9}}], Cell[BoxData["3"], "Output", CellChangeTimes->{3.4813651361395607`*^9}] }, Open ]], Cell["\<\ Part (c): We will have the same col1, col2, and col3 as before. The only \ thing that changes is the total matrix will not have col4\ \>", "Text", CellChangeTimes->{{3.481362889656213*^9, 3.4813628937812395`*^9}, 3.4813653923615904`*^9, {3.481367097854418*^9, 3.4813671308863025`*^9}}], Cell[CellGroupData[{ Cell[BoxData[{ RowBox[{ RowBox[{"ZT", "=", RowBox[{"{", RowBox[{ RowBox[{"{", RowBox[{"1", ",", "1", ",", "0", ",", "0"}], "}"}], ",", RowBox[{"{", RowBox[{"0", ",", "0", ",", "1", ",", "0"}], "}"}], ",", RowBox[{"{", RowBox[{"0", ",", "0", ",", FractionBox["1", "R"], ",", "0"}], "}"}], ",", RowBox[{"{", RowBox[{"0", ",", RowBox[{"-", "g"}], ",", "0", ",", RowBox[{"-", "g"}]}], "}"}], ",", RowBox[{"{", RowBox[{"0", ",", FractionBox[ RowBox[{"-", "g"}], "R"], ",", "0", ",", FractionBox[ RowBox[{"-", "g"}], "R"]}], "}"}], ",", RowBox[{"{", RowBox[{"0", ",", "0", ",", FractionBox[ RowBox[{"-", "g"}], "R"], ",", "0"}], "}"}]}], "}"}]}], ";"}], "\[IndentingNewLine]", RowBox[{ RowBox[{"Z", "=", RowBox[{"Transpose", "[", "ZT", "]"}]}], ";"}], "\[IndentingNewLine]", RowBox[{"MatrixForm", "[", "Z", "]"}]}], "Input", CellChangeTimes->{{3.481361979383304*^9, 3.481362052634242*^9}, { 3.4813636749644885`*^9, 3.4813637707463517`*^9}, {3.481365114889289*^9, 3.48136511732682*^9}, 3.4813671703401847`*^9}], Cell[BoxData[ TagBox[ RowBox[{"(", "\[NoBreak]", GridBox[{ {"1", "0", "0", "0", "0", "0"}, {"1", "0", "0", RowBox[{"-", "g"}], RowBox[{"-", FractionBox["g", "R"]}], "0"}, {"0", "1", FractionBox["1", "R"], "0", "0", RowBox[{"-", FractionBox["g", "R"]}]}, {"0", "0", "0", RowBox[{"-", "g"}], RowBox[{"-", FractionBox["g", "R"]}], "0"} }, GridBoxAlignment->{ "Columns" -> {{Center}}, "ColumnsIndexed" -> {}, "Rows" -> {{Baseline}}, "RowsIndexed" -> {}}, GridBoxSpacings->{"Columns" -> { Offset[0.27999999999999997`], { Offset[0.7]}, Offset[0.27999999999999997`]}, "ColumnsIndexed" -> {}, "Rows" -> { Offset[0.2], { Offset[0.4]}, Offset[0.2]}, "RowsIndexed" -> {}}], "\[NoBreak]", ")"}], Function[BoxForm`e$, MatrixForm[BoxForm`e$]]]], "Output", CellChangeTimes->{ 3.481362052931121*^9, {3.4813636985583897`*^9, 3.4813637095115843`*^9}, 3.481363771433856*^9, 3.481365117670574*^9, 3.4813671711995764`*^9}] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"MatrixRank", "[", "Z", "]"}]], "Input", CellChangeTimes->{{3.4813651335457773`*^9, 3.481365135827057*^9}}], Cell[BoxData["3"], "Output", CellChangeTimes->{3.4813651361395607`*^9, 3.4813671742308846`*^9}] }, Open ]] }, WindowSize->{616, 750}, WindowMargins->{{0, Automatic}, {Automatic, 0}}, ShowSelection->True, FrontEndVersion->"7.0 for Microsoft Windows (32-bit) (February 18, 2009)", StyleDefinitions->"Default.nb" ] (* End of Notebook Content *) (* Internal cache information *) (*CellTagsOutline CellTagsIndex->{} *) (*CellTagsIndex CellTagsIndex->{} *) (*NotebookFileOutline Notebook[{ Cell[CellGroupData[{ Cell[567, 22, 641, 18, 52, "Input"], Cell[1211, 42, 772, 23, 96, "Output"] }, Open ]], Cell[1998, 68, 90, 1, 29, "Text"], Cell[CellGroupData[{ Cell[2113, 73, 554, 13, 72, "Input"], Cell[2670, 88, 715, 18, 44, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[3422, 111, 367, 7, 31, "Input"], Cell[3792, 120, 355, 10, 47, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[4184, 135, 416, 8, 31, "Input"], Cell[4603, 145, 426, 13, 45, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[5066, 163, 493, 10, 31, "Input"], Cell[5562, 175, 434, 12, 45, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[6033, 192, 769, 23, 69, "Input"], Cell[6805, 217, 900, 28, 113, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[7742, 250, 133, 2, 31, "Input"], Cell[7878, 254, 72, 1, 30, "Output"] }, Open ]], Cell[7965, 258, 92, 1, 29, "Text"], Cell[CellGroupData[{ Cell[8082, 263, 759, 18, 72, "Input"], Cell[8844, 283, 805, 22, 86, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[9686, 310, 505, 10, 52, "Input"], Cell[10194, 322, 883, 26, 96, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[11114, 353, 560, 11, 52, "Input"], Cell[11677, 366, 964, 29, 110, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[12678, 400, 633, 12, 52, "Input"], Cell[13314, 414, 1027, 30, 123, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[14378, 449, 1517, 44, 129, "Input"], Cell[15898, 495, 1281, 38, 125, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[17216, 538, 131, 2, 31, "Input"], Cell[17350, 542, 72, 1, 30, "Output"] }, Open ]], Cell[17437, 546, 302, 5, 47, "Text"], Cell[CellGroupData[{ Cell[17764, 555, 1185, 33, 121, "Input"], Cell[18952, 590, 1079, 31, 112, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[20068, 626, 131, 2, 31, "Input"], Cell[20202, 630, 96, 1, 30, "Output"] }, Open ]] } ] *) (* End of internal cache information *)