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For more information on notebooks and Mathematica-compatible applications, contact Wolfram Research: web: http://www.wolfram.com email: info@wolfram.com phone: +1-217-398-0700 (U.S.) Notebook reader applications are available free of charge from Wolfram Research. *******************************************************************) (*CacheID: 232*) (*NotebookFileLineBreakTest NotebookFileLineBreakTest*) (*NotebookOptionsPosition[ 20524, 721]*) (*NotebookOutlinePosition[ 25983, 853]*) (* CellTagsIndexPosition[ 24812, 825]*) (*WindowFrame->Normal*) Notebook[{ Cell["Solving Linear Systems, Existence and Uniqueness", "Title", CellTags->"mtag:07:systems_of_linear_equations__existence_and_uniqueness"], Cell["Solving and Uniqueness", "Subtitle"], Cell["\<\ Consider the set of equations \t x + 2y + z + t = a \t-x + 4y - 2z = b \t x + 3y + 4z + 5t = c \t x + z + t = d We illustrate how to use a matrix representation to write these out and solve \ them\[Ellipsis]\t \ \>", "Subsection"], Cell[CellGroupData[{ Cell[TextData[{ "Start with the matrix of coefficients of the variables, ", StyleBox["mymatrix", FontWeight->"Bold"], ":" }], "Subsection"], Cell[BoxData[{ \(\(mymatrix\ = \ {\[IndentingNewLine]{1, 2, 1, 1}, \[IndentingNewLine]{\(-1\), 4, \(-2\), 0}, \[IndentingNewLine]{1, 3, 4, 5}, \[IndentingNewLine]{1, 0, 1, 1}};\)\), "\[IndentingNewLine]", \(mymatrix // MatrixForm\)}], "Input"] }, Open ]], Cell[CellGroupData[{ Cell[TextData[{ "The system of equations will only have a unique solution if the \ determinant of ", StyleBox["mymatrix", FontWeight->"Bold"], " is nonzero." }], "Subsection"], Cell[BoxData[ \(Det[mymatrix]\)], "Input", CellTags->{ "mtag:07:systems_of_linear_equations__existence_and_determinants", "mtag:07:Det[]__and_uniqueness"}] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{ "Now", " ", "define", " ", "vectors", " ", "for", " ", \(x\&\[RightVector]\), " ", "and", " ", \(b\&\[RightVector]\), " ", "in", " ", FormBox[\(A\+_\ x\&\[RightVector]\ = \ b\&\[RightVector]\), "TraditionalForm"]}]], "Subsection"], Cell[BoxData[ \(\(myx\ = {x, y, z, t};\)\)], "Input"], Cell[BoxData[ \(\(myb\ = \ {a, b, c, d};\)\)], "Input"] }, Open ]], Cell[CellGroupData[{ Cell["The left-hand side of the first equation will be", "Subsection"], Cell[BoxData[ \(\((mymatrix . myx)\)[\([1]\)]\)], "Input"] }, Open ]], Cell[CellGroupData[{ Cell["and the left-hand side of all four equations will be", "Subsection"], Cell[BoxData[{ \(\(lhs\ = \ mymatrix . myx;\)\), "\[IndentingNewLine]", \(lhs // MatrixForm\)}], "Input"] }, Open ]], Cell[CellGroupData[{ Cell[TextData[{ "Now define an indexed variable ", StyleBox["linsys", FontWeight->"Bold"], " with four entries, each being one of the equations in the system of \ interest:" }], "Subsection"], Cell[BoxData[ \(linsys[i_Integer]\ := \ lhs[\([i]\)]\ == \ myb[\([i]\)]\)], "Input"], Cell[BoxData[ \(linsys[2]\)], "Input"] }, Open ]], Cell[CellGroupData[{ Cell[TextData[{ "Solving the set of equations for the unknowns ", Cell[BoxData[ \(\(x\& \[Rule] \)\)], FontFamily->"Helvetica"] }], "Subsection", CellTags-> "mtag:07:systems_of_linear_equations__creating_from_matrices_and_vectors"], Cell[BoxData[ \(linsol\ = Solve[{linsys[1], linsys[2], \ linsys[3], \ linsys[4]}, myx]\)], "Input"] }, Open ]], Cell[CellGroupData[{ Cell[TextData[{ "Doing the same thing a different way, using ", StyleBox["Mathematica", FontSlant->"Italic"], "'s ", StyleBox["LinearSolve", FontWeight->"Bold"], " function:" }], "Subsection"], Cell[BoxData[ \(\(?LinearSolve\)\)], "Input"], Cell[BoxData[ \(LinearSolve[mymatrix, myb]\)], "Input", CellTags-> "mtag:07:LinearSolve[]_solutions_without_direct_computation_of_matrix_\ inverse"] }, Open ]], Cell[CellGroupData[{ Cell[TextData[{ "And yet another way, based on ", Cell[BoxData[ \(TraditionalForm\`x\&\[RightVector] = \(\(A\+_\^\(-1\)\) A\+_\ x\&\[RightVector]\ = \ \(A\+_\^\(-1\)\) b\&\[RightVector]\)\)]], " " }], "Subsection"], Cell[BoxData[ \(Inverse[mymatrix] . myb // MatrixForm\)], "Input", CellTags->"mtag:07:Inverse[]"] }, Open ]], Cell[CellGroupData[{ Cell["When determinants are zero", "Section"], Cell["\<\ Create a matrix with one row as a linear combination of the \ others\ \>", "Text"], Cell[BoxData[{ \(\(myzeromatrix = \[IndentingNewLine]{mymatrix[\([1]\)], \ \[IndentingNewLine]mymatrix[\([2]\)], \[IndentingNewLine]p* mymatrix[\([1]\)]\ + \ q*mymatrix[\([2]\)]\ + \ r*mymatrix[\([4]\)], \[IndentingNewLine]mymatrix[\([4]\)]};\)\), \ "\[IndentingNewLine]", \(myzeromatrix // MatrixForm\)}], "Input"], Cell[BoxData[ \(Det[myzeromatrix]\)], "Input"], Cell[BoxData[ \(LinearSolve[myzeromatrix, myb]\)], "Input"], Cell[TextData[{ "This was not expected to have a solution because one of four equations in \ the system was a linear combination of others in the system. Effectively, we \ were asking ", StyleBox["Mathematica", FontSlant->"Italic"], " to solve a system of ", StyleBox["three", FontSlant->"Italic"], " equations in four unknowns. The ", StyleBox["rank", FontSlant->"Italic"], " of a square matrix of coefficients is equal to the number of linearly \ independent equations in the system. The ", StyleBox["null space", FontSlant->"Italic"], " of the matrix will be empty when the equations are all linearly \ independent." }], "Text"], Cell[BoxData[{ \(MatrixRank[mymatrix]\), "\[IndentingNewLine]", \(MatrixRank[myzeromatrix]\)}], "Input", CellTags->"mtag:07:MatrixRank[]"], Cell[BoxData[{ \(NullSpace[mymatrix]\), "\[IndentingNewLine]", \(NullSpace[myzeromatrix]\)}], "Input", CellTags->"mtag:07:NullSpace[]"], Cell[CellGroupData[{ Cell[TextData[{ "Try solving this inhomogeneous system of equations using ", StyleBox["Solve", FontWeight->"Bold"], ":" }], "Subsection"], Cell[BoxData[ \(zerolhs = \ myzeromatrix . myx\)], "Input"], Cell[BoxData[ \(zerolinsys[i_Integer]\ := \ zerolhs[\([i]\)]\ == \ myb[\([i]\)]\)], "Input"], Cell[BoxData[ \(Table[zerolinsys[i], {i, 4}] // MatrixForm\)], "Input"], Cell[BoxData[ \(zerolinsol\ = Solve[Table[zerolinsys[i], {i, 4}], myx]\)], "Input"] }, Open ]], Cell[CellGroupData[{ Cell[TextData[{ "No solution, as expected, Let's see what happens if we ask ", StyleBox["Mathematica", FontSlant->"Italic"], " to solve the homogeneous problem:" }], "Subsection"], Cell[BoxData[ \(zerolinsol\ = Solve[Table[ zerolinsys[i] /. {a \[Rule] 0, b \[Rule] 0, c \[Rule] 0, d \[Rule] 0}, {i, 4}], myx]\)], "Input"] }, Open ]], Cell[CellGroupData[{ Cell[TextData[{ "In this case, ", StyleBox["Mathematica", FontSlant->"Italic"], " gives a relationship between the variables, but because there are fewer \ equations than variables, there is still no unique solution." }], "Subsection"], Cell["Determinants", "Subtitle"], Cell[BoxData[ \(Clear[rv]\)], "Input"] }, Open ]], Cell[CellGroupData[{ Cell["\<\ Start by building a routine to make vectors containing six random \ numbers on the interval {-1,1}:\ \>", "Subsection"], Cell[BoxData[ \(rv[i_]\ := \ \(rv[i]\ = Table[Random[Real, {\(-1\), 1}], {j, 6}]\)\)], "Input"], Cell[BoxData[ \(rv[1]\)], "Input"], Cell[BoxData[ \(rv[2]\)], "Input"] }, Open ]], Cell[CellGroupData[{ Cell[TextData[{ "Now use ", StyleBox["rv", 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The Octahedron is made up of eight \ polygons and the initial coordinates of the vertices were set to make a \ regular octahedron with its main diagonals parallel to axes ", StyleBox["x", FontSlant->"Italic"], ",", StyleBox["y", FontSlant->"Italic"], ",", StyleBox["z", FontSlant->"Italic"], ". 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In this first example, each ", StyleBox["z", FontSlant->"Italic"], " coordinate will be changed to ", StyleBox["-z", FontSlant->"Italic"], ":" }], "Subsection"], Cell[BoxData[{ \(\(Clear[tmat];\)\), "\[IndentingNewLine]", \(\(tmat = \ {{1, 0, 0}, {0, 1, 0}, \ {0, 0, \(-1\)}};\)\), "\[IndentingNewLine]", \(tmat // MatrixForm\)}], "Input"] }, Open ]], Cell[CellGroupData[{ Cell[TextData[{ "The ", StyleBox["Replace", FontWeight->"Bold"], " is used to modify the coordinates fed to the ", StyleBox["Polygon", FontWeight->"Bold"], " function in ", StyleBox["coloct", FontWeight->"Bold"], " by matrix multiplication of the vertex vectors:" }], "Subsection"], Cell[BoxData[ \(Show[ coloct /. {Polygon[{a_List\ , \ b_List\ , \ c_List}] \[Rule] Polygon[{tmat . a, tmat . b, tmat . c}]}, Lighting \[Rule] False]\)], "Input", CellTags->{ "mtag:07:Polyhedra__example_of_matrix_transformation", "mtag:07:Permutations[]"}] }, Open ]], Cell[CellGroupData[{ Cell[TextData[{ "The program ", StyleBox["seetrans", FontWeight->"Bold"], " does the same thing by feeding a matrix ", StyleBox["tranmat", FontWeight->"Bold"], " to operate on the vertex vectors before ", StyleBox["Polygon", FontWeight->"Bold"], " is executed:" }], "Subsection"], Cell[BoxData[ \(seetrans[tranmat_] := \ Show[coloct /. {Polygon[{a_List\ , \ b_List\ , \ c_List}] \[Rule] Polygon[{tranmat . a, tranmat . b, tranmat . c}]}, Lighting \[Rule] False]\)], "Input"] }, Open ]], Cell[CellGroupData[{ Cell[TextData[{ "When", " ", StyleBox["tranmat", FontWeight->"Bold"], " is the identity matrix, the octagon is rendered in its initial \ orientation." }], "Subsection"], Cell[BoxData[ \(seetrans[{{1, 0, 0}, {0, 1, 0}, {0, 0, 1}}]\)], "Input"] }, Open ]], Cell[CellGroupData[{ Cell[TextData[{ "The next command rotates the octagon by 45\[Degree] about the vertical (", StyleBox["z", FontSlant->"Italic"], ") axis\[Ellipsis]" }], "Subsection"], Cell[BoxData[ \(seetrans[{{Cos[Pi/4], Sin[Pi/4], 0}, {Sin[\(-Pi\)/4], Cos[Pi/4], 0}, {0, 0, 1}}]\)], "Input"] }, Open ]], Cell[CellGroupData[{ Cell[TextData[{ "And the next command both rotates it 180\[Degree] about ", StyleBox["x", FontSlant->"Italic"], " and ", StyleBox["y", FontSlant->"Italic"], " ", "and stretches it by a factor of 5 along ", StyleBox["z", FontSlant->"Italic"], "." }], "Subsection"], Cell[BoxData[ \(seetrans[{{\(-1\), 0, 0}, {0, \(-1\), 0}, {0, 0, 5}}]\)], "Input"] }, Open ]] }, Open ]] }, FrontEndVersion->"5.2 for Macintosh", ScreenRectangle->{{45, 1920}, {0, 1178}}, ScreenStyleEnvironment->"Presentation", WindowSize->{1400, 919}, WindowMargins->{{136, Automatic}, {Automatic, 53}}, WindowTitle->"Lecture 07 MIT 3.016 (Fall 2005) \[Copyright] W. 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