(************** Content-type: application/mathematica ************** CreatedBy='Mathematica 5.0' Mathematica-Compatible Notebook This notebook can be used with any Mathematica-compatible application, such as Mathematica, MathReader or Publicon. The data for the notebook starts with the line containing stars above. To get the notebook into a Mathematica-compatible application, do one of the following: * Save the data starting with the line of stars above into a file with a name ending in .nb, then open the file inside the application; * Copy the data starting with the line of stars above to the clipboard, then use the Paste menu command inside the application. Data for notebooks contains only printable 7-bit ASCII and can be sent directly in email or through ftp in text mode. Newlines can be CR, LF or CRLF (Unix, Macintosh or MS-DOS style). 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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[ 22531, 801]*) (*NotebookOutlinePosition[ 23201, 824]*) (* CellTagsIndexPosition[ 23157, 820]*) (*WindowFrame->Normal*) Notebook[{ Cell[CellGroupData[{ Cell["Linear Algebra", "Title", Evaluatable->False, TextAlignment->Center, TextJustification->0, AspectRatioFixed->True, Background->RGBColor[0, 1, 1]], Cell["\<\ This notebook is by Steven Amgott. Please send any questions or\ \[NonBreakingSpace]comments to samgott1@swarthmore.edu. Feel free to use and \ distribute this notebook, but keep this author information in any copy you \ use or distribute.\ \>", "SmallText"], Cell[TextData[{ "Buttons to create many symbols in ", StyleBox["Mathematica", FontSlant->"Italic"], " are available in the ", StyleBox["BasicInput Palette", FontColor->RGBColor[0, 0, 1]], ". This palette is often opened at the time ", StyleBox["Mathematica", FontSlant->"Italic"], " is started. If it is not up on the screen, it can be displayed using the \ ", StyleBox["Palette", FontColor->RGBColor[0, 0, 1]], " command in the ", StyleBox["File", FontColor->RGBColor[0, 0, 1]], " menu. You might wish to ", StyleBox["take a brief look at this palette", FontColor->RGBColor[1, 0, 1]], " to see what symbols it contains. If the computer you are using has the \ additional ", StyleBox["Linear Algebra ", FontColor->RGBColor[0, 0, 1]], "palette installed, you may also wish to look at it.\n\nTo open the closed \ group of cells for any of the sections or commands, you can ", StyleBox["double-click the bracket with a triangle at its bottom at the \ right edge of the cell", FontColor->RGBColor[1, 0, 1]], ".\n\nIf this is your first experience with ", StyleBox["Mathematica", FontSlant->"Italic"], ", or if you need a brief refresher before continuing, you might first go \ through the ", Cell[BoxData[ FormBox[ ButtonBox[\(Introduction\ to\ Mathematica\), ButtonData:>{"Introduction_to_Mathematica.nb", None}, ButtonStyle->"Hyperlink"], TraditionalForm]]], " notebook contained in the same folder as this notebook. In particular, \ remember that to evaluate a cell you need to press the ", StyleBox["enter", FontColor->RGBColor[0, 0, 1]], " key on the number pad at the lower right side of the keyboard and NOT the \ ", StyleBox["return", FontColor->RGBColor[0, 0, 1]], " key on a Macintosh or the ", StyleBox["Enter", FontColor->RGBColor[0, 0, 1]], " key next to the single and double quote marks on a Windows machine! ", "(The latter only produces a line break. You can, however, use the ", StyleBox["Shift", FontColor->RGBColor[0, 0, 1]], " and ", StyleBox["Enter", FontColor->RGBColor[0, 0, 1]], " keys on a Windows machine, or the ", StyleBox["shift", FontColor->RGBColor[0, 0, 1]], " and ", StyleBox["return", FontColor->RGBColor[0, 0, 1]], " keys on a Mac, simultaneously to evaluate a cell. In particular, that is \ how you evaluate a cell on a notebook computer without a number pad.)" }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{ Cell[TextData[{ "What ", StyleBox["Mathematica", FontSlant->"Italic"], " can do" }], "Section", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ StyleBox["Mathematica", FontSlant->"Italic"], " can perform many of the basic vector and matrix operations of Linear \ Algebra. These include dot and cross products, row reduction to Gauss-Jordan \ form, vector and matrix arithmetic, matrix multiplication and inversion, \ taking determinants, and finding eigenvalues and eigenvectors. Examples of \ these operations are in the cells below. Most of the Linear Algebra \ operations are available on the additional ", StyleBox["LinearAlgebra", FontColor->RGBColor[0, 0, 1]], " palette." }], "Text"] }, Closed]], Cell[CellGroupData[{ Cell["Vectors", "Section"], Cell[TextData[{ StyleBox["Mathematica", FontSlant->"Italic"], " stores vectors as lists. (See the Introduction to ", StyleBox["Mathematica", FontSlant->"Italic"], " Tutorial in this folder for more on lists.) For instance, the next cell \ defines two vectors v and w. ", StyleBox[" Evaluate it.", FontColor->RGBColor[1, 0, 1]] }], "Text"], Cell[BoxData[{ \(Clear[v, w]\), "\[IndentingNewLine]", \(\(v = {2, 1, 4, 5};\)\), "\[IndentingNewLine]", \(\(w = {\(-2\), 1, 3, 2};\)\)}], "Input"], Cell[TextData[{ "We can do vector addition using ", StyleBox["+", FontColor->RGBColor[1, 0, 0]], "." }], "Text"], Cell[BoxData[ \(v + w\)], "Input"], Cell[TextData[{ "Scalar multiplication is also easy. (Make sure to leave a space or use ", StyleBox["*", FontColor->RGBColor[1, 0, 0]], " between things you wish to multiply.)" }], "Text"], Cell[BoxData[ \(3\ v\)], "Input"], Cell[TextData[{ StyleBox["Mathematica", FontSlant->"Italic"], " will perform some list operations which are not Linear Algebra \ operations. For instance," }], "Text"], Cell[BoxData[ \(v*w\)], "Input"], Cell["\<\ simply multiplies the vectors v and w term-by-term, but is not a Linear \ Algebra operation.\ \>", "Text"], Cell["We can even have variables as entries in vectors.", "Text"], Cell[BoxData[ \({1, a, b} + {2, c, d}\)], "Input"] }, Closed]], Cell[CellGroupData[{ Cell["Dot, cross, and scalar triple products", "Section", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ "This cell produces the dot product of the vectors ", StyleBox["{2,1,-3}", FontColor->RGBColor[1, 0, 0]], " and ", StyleBox["{4,-2,1}", FontColor->RGBColor[1, 0, 0]], ", which ", StyleBox["Mathematica", FontSlant->"Italic"], " stores as lists." }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[BoxData[ \(Dot[{2, 1, \(-3\)}, {4, \(-2\), 1}]\)], "Input", AspectRatioFixed->True], Cell[TextData[{ "You can also do this for any two vectors in ", Cell[BoxData[ \(TraditionalForm\`R\^n\)]], " using a period ", StyleBox[". ", FontColor->RGBColor[1, 0, 0]], "." }], "Text"], Cell[BoxData[ \({2, 1, \(-3\)} . {4, \(-2\), 1}\)], "Input"], Cell[BoxData[ \({3, 4, \(-2\), 5} . {\(-2\), 4, 3, \(-1\)}\)], "Input"], Cell["\<\ This one gives the cross product of the original two vectors.\ \>", "Text", Evaluatable->False, AspectRatioFixed->True], Cell[BoxData[ \(Cross[{2, 1, \(-3\)}, {4, \(-2\), 1}]\)], "Input", AspectRatioFixed->True], Cell[TextData[{ "The function ", StyleBox["ScalarTripleProduct", FontColor->RGBColor[1, 0, 0]], " can be used to find the scalar triple product of these vectors together \ with the vector ", StyleBox["{2,-2,3}", FontColor->RGBColor[1, 0, 0]], ". You might remember that the absolute value of this number gives us the \ volume of the parallelepiped whose sides are formed from the three vectors. \ To use this function, you need to first load a package." }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[BoxData[ \(Needs["\"]\)], "Input", AspectRatioFixed->True], Cell[BoxData[ \(ScalarTripleProduct[{2, 1, \(-3\)}, {4, \(-2\), 1}, {2, \(-2\), 3}]\)], "Input", AspectRatioFixed->True], Cell["Of course you can simply use", "Text"], Cell[BoxData[ \({2, 1, \(-3\)} . Cross[{4, \(-2\), 1}, {2, \(-2\), 3}]\)], "Input"], Cell["\<\ to find the scalar triple product without having to load the package.\ \>", "Text"] }, Closed]], Cell[CellGroupData[{ Cell["Matrices", "Section"], Cell[TextData[{ "To ", StyleBox["Mathematica", FontSlant->"Italic"], ", a matrix is a list of lists. ", StyleBox["Be sure to evaluate the next cell defining a matrix m.", FontColor->RGBColor[1, 0, 1]] }], "Text"], Cell[BoxData[ \(m = {{1, 2, 3, 4}, {5, 6, 7, 8}, {9, 10, 11, 12}}\)], "Input"], Cell[TextData[{ "To get ", StyleBox["Mathematica", FontSlant->"Italic"], " to display it in our usual matrix form, we need ", StyleBox["MatrixForm[\[Placeholder]]", FontColor->RGBColor[1, 0, 0]], "." }], "Text"], Cell[BoxData[ \(MatrixForm[m]\)], "Input"], Cell[TextData[{ "Another way to enter a matrix into ", StyleBox["Mathematica", FontSlant->"Italic"], " is to use the ", StyleBox["Input...Create Table/Matrix/Palette", FontColor->RGBColor[0, 0, 1]], " menu item. You choose whether you are creating a table, matrix, or \ palette, and the number of rows and columns. You then fill in the boxes \ (either pressing the ", StyleBox["Tab", FontColor->RGBColor[1, 0, 0]], " key or using the mouse to move to empty boxes). The additional \ LinearAlgebra palette has buttons for 1 by 2, 2 by 1, and 2 by 2 matrices. \ Rows can be added to any matrix using ", StyleBox["Ctrl", FontColor->RGBColor[1, 0, 0]], "+", StyleBox["Enter ", FontColor->RGBColor[1, 0, 0]], "(simultaneous key press) and columns can be added by using ", StyleBox["Ctrl", FontColor->RGBColor[1, 0, 0]], "+", StyleBox[",", FontColor->RGBColor[1, 0, 0]], ". I used the menu item to create the matrix in the next cell. ", StyleBox["Be sure to evaluate it.", FontColor->RGBColor[1, 0, 1]] }], "Text"], Cell[BoxData[ RowBox[{"n", "=", RowBox[{"(", GridBox[{ {\(-2\), "1", "3", "5"}, {\(-3\), \(-4\), "2", "4"}, {\(-2\), \(-1\), "4", "3"} }], ")"}]}]], "Input"], Cell[TextData[{ "Matrix addition simply uses ", StyleBox["+", FontColor->RGBColor[1, 0, 0]], ". To get nicer output, we use ", StyleBox["MatrixForm[\[Placeholder]]", FontColor->RGBColor[1, 0, 0]], "." }], "Text"], Cell[BoxData[ \(MatrixForm[m + n]\)], "Input"], Cell["Again, we can easily multiply a matrix by a scalar.", "Text"], Cell[BoxData[ \(MatrixForm[3\ m]\)], "Input"], Cell[TextData[{ "The matrix product of m and n from above is not defined (since the number \ of columns of m is not equal to the number of rows of n). If we try to \ multiply them by using a space or ", StyleBox["*", FontColor->RGBColor[1, 0, 0]], ", ", StyleBox["Mathematica", FontSlant->"Italic"], " simply multiplies them entry by entry. This is not a Linear Algebra \ operation." }], "Text"], Cell[BoxData[ \(m*n\)], "Input"], Cell[TextData[{ "The n by n identity matrix is given by ", StyleBox["IdentityMatrix[n]", FontColor->RGBColor[1, 0, 0]], ". Here's the 3 by 3 one." }], "Text"], Cell[BoxData[ \(MatrixForm[IdentityMatrix[3]]\)], "Input"], Cell[TextData[{ "There is another way to get a matrix to appear in two-dimensional form. \ The second of the two cells below was obtained by simply copying the first \ one, and changing the cell type using the ", StyleBox["Cell...Convert To", FontColor->RGBColor[0, 0, 1]], " menu item and choosing ", StyleBox["TraditionalForm", FontColor->RGBColor[0, 0, 1]], ". Notice the difference between the brackets at the right side of the \ cells." }], "Text"], Cell[BoxData[ \({{1, 2, 3, 4}, {5, 6, 7, 8}, {9, 10, 11, 12}}\)], "Input"], Cell[BoxData[ FormBox[ RowBox[{"(", "\[NoBreak]", GridBox[{ {"1", "2", "3", "4"}, {"5", "6", "7", "8"}, {"9", "10", "11", "12"} }, ColumnAlignments->{Decimal}], "\[NoBreak]", ")"}], TraditionalForm]], "Input"], Cell[TextData[{ "You can even have ", StyleBox["TraditionalForm", FontColor->RGBColor[0, 0, 1]], " as the ", StyleBox["Default Output Format Type", FontColor->RGBColor[0, 0, 1]], " (in the ", StyleBox["Cell", FontColor->RGBColor[0, 0, 1]], " menu). This is not advised since ", StyleBox["TraditionalForm", FontColor->RGBColor[0, 0, 1]], " is not precise enough for ", StyleBox["Mathematica", FontSlant->"Italic"], ", and you might have problems if you use such output as input for later \ cells." }], "Text"] }, Closed]], Cell[CellGroupData[{ Cell["Matrix operations", "Section"], Cell[TextData[{ "To multiply two matrices, use a period ", StyleBox[".", FontColor->RGBColor[1, 0, 0]], "." }], "Text"], Cell[BoxData[ RowBox[{"MatrixForm", "[", RowBox[{ RowBox[{"(", GridBox[{ {"1", "2", "3", "4"}, {"3", \(-2\), \(-1\), "4"}, {"2", \(-5\), "1", "3"} }], ")"}], ".", RowBox[{"(", GridBox[{ {"1", "0"}, {"2", \(-2\)}, {\(-1\), "3"}, {"4", "2"} }], ")"}]}], "]"}]], "Input"], Cell[TextData[{ "The transpose of a matrix uses ", StyleBox["Transpose[\[Placeholder]]", FontColor->RGBColor[1, 0, 0]], "." }], "Text"], Cell[BoxData[ RowBox[{"MatrixForm", "[", RowBox[{"Transpose", "[", RowBox[{"(", GridBox[{ {"1", "2", "3", "4"}, {"3", \(-2\), \(-1\), "4"}, {"2", \(-5\), "1", "3"} }], ")"}], "]"}], "]"}]], "Input"], Cell[TextData[{ "To obtain the inverse of a square matrix (if it exists), use ", StyleBox["Inverse[\[Placeholder]]", FontColor->RGBColor[1, 0, 0]], ". (Why does the second inverse below not exist?)" }], "Text"], Cell[BoxData[ RowBox[{"MatrixForm", "[", RowBox[{"Inverse", "[", RowBox[{"(", GridBox[{ {"1", "2"}, {"3", "4"} }], ")"}], "]"}], "]"}]], "Input"], Cell[BoxData[ RowBox[{"MatrixForm", "[", RowBox[{"Inverse", "[", RowBox[{"(", GridBox[{ {"1", "2"}, {"2", "4"} }], ")"}], "]"}], "]"}]], "Input"], Cell[TextData[{ "To find the determinant of a square matrix, use ", StyleBox["Det[\[Placeholder]]", FontColor->RGBColor[1, 0, 0]], ". What do the following two determinants have to say about the \ invertibility of the matrices we saw above?" }], "Text"], Cell[BoxData[ RowBox[{"Det", "[", RowBox[{"(", GridBox[{ {"1", "2"}, {"3", "4"} }], ")"}], "]"}]], "Input"], Cell[BoxData[ RowBox[{"Det", "[", RowBox[{"(", GridBox[{ {"1", "2"}, {"2", "4"} }], ")"}], "]"}]], "Input"], Cell[TextData[{ "Finally, to raise a square matrix to a power, use ", StyleBox["MatrixPower[\[Placeholder],\[Placeholder]]", FontColor->RGBColor[1, 0, 0]], ". The first entry is the matrix, and the second the power to which you \ want to raise it." }], "Text"], Cell[BoxData[ RowBox[{"MatrixForm", "[", RowBox[{"MatrixPower", "[", RowBox[{ RowBox[{"(", GridBox[{ {"1", "2"}, {"3", "4"} }], ")"}], ",", "3"}], "]"}], "]"}]], "Input"] }, Closed]], Cell[CellGroupData[{ Cell["Row reduction", "Section"], Cell[TextData[{ StyleBox["Mathematica", FontSlant->"Italic"], " has a built-in command to row-reduce a matrix to Gauss-Jordan (reduced \ row-echelon) form. ", StyleBox["Evaluate the following cell.", FontColor->RGBColor[1, 0, 1]] }], "Text"], Cell[BoxData[ RowBox[{"RowReduce", "[", RowBox[{"(", GridBox[{ {"2", \(-2\), "3", "1"}, {"8", "3", "7", \(-4\)}, {"1", "4", \(-3\), "1"} }], ")"}], "]"}]], "Input"], Cell[TextData[{ "Notice that the output is in one-dimensional form. To get it in the \ two-dimensional matrix form, we need to use ", StyleBox["MatrixForm[\[Placeholder]]", FontColor->RGBColor[1, 0, 0]], "." }], "Text"], Cell[BoxData[ \(MatrixForm[%]\)], "Input"], Cell[TextData[{ "(", StyleBox["%", FontColor->RGBColor[1, 0, 0]], " is the way to refer to the last output from ", StyleBox["Mathematica", FontSlant->"Italic"], ".) You could also combine the two commands in one cell." }], "Text"], Cell[BoxData[ RowBox[{"MatrixForm", "[", RowBox[{"RowReduce", "[", RowBox[{"(", GridBox[{ {"2", \(-2\), "3", "1"}, {"8", "3", "7", \(-4\)}, {"1", "4", \(-3\), "1"} }], ")"}], "]"}], "]"}]], "Input"], Cell[TextData[{ "One word of warning - if you row reduce a matrix which contains symbols, \ ", StyleBox["Mathematica", FontSlant->"Italic"], " will assume that any expression containing a symbol is non-zero. This \ can sometimes lead to misleading answers. For instance, the following row \ reduces to the identity matrix, even though it should not if ad-bc = 0. (Why \ should it not in this case?)" }], "Text"], Cell[BoxData[ RowBox[{"MatrixForm", "[", RowBox[{"RowReduce", "[", RowBox[{"(", GridBox[{ {"a", "b"}, {"c", "d"} }], ")"}], "]"}], "]"}]], "Input"], Cell["It gets it right in this case, however.", "Text"], Cell[BoxData[ RowBox[{"MatrixForm", "[", RowBox[{"RowReduce", "[", RowBox[{"(", GridBox[{ {"a", "b"}, {"a", "b"} }], ")"}], "]"}], "]"}]], "Input"] }, Closed]], Cell[CellGroupData[{ Cell["Eigenvalues and Eigenvectors", "Section"], Cell[TextData[{ "An eigenvector for a matrix A is a vector v such A v = \[Lambda] v for \ some scalar \[Lambda]. The scalar \[Lambda] is an eigenvalue for the matrix. \ ", StyleBox["Mathematica", FontSlant->"Italic"], " can find these for us. To find the eigenvalues of a matrix, use ", StyleBox["Eigenvalues[\[Placeholder]]", FontColor->RGBColor[1, 0, 0]], "." }], "Text"], Cell[BoxData[ RowBox[{"Eigenvalues", "[", RowBox[{"(", GridBox[{ {"1", "0"}, {"1", \(-1\)} }], ")"}], "]"}]], "Input"], Cell[BoxData[ RowBox[{"Eigenvalues", "[", RowBox[{"(", GridBox[{ {\(-1\), \(-4\)}, {"1", \(-1\)} }], ")"}], "]"}]], "Input"], Cell[TextData[{ "(Remember that ", "\[ImaginaryI]", " = ", Cell[BoxData[ \(TraditionalForm\`\@\(-1\)\)]], ".)" }], "Text"], Cell[TextData[{ "We can also find a set of linearly independent eigenvectors for the \ matrices, using ", StyleBox["Eigenvectors[\[Placeholder]]", FontColor->RGBColor[1, 0, 0]], "." }], "Text"], Cell[BoxData[ RowBox[{"Eigenvectors", "[", RowBox[{"(", GridBox[{ {"1", "0"}, {"1", \(-1\)} }], ")"}], "]"}]], "Input"], Cell[BoxData[ RowBox[{"Eigenvectors", "[", RowBox[{"(", GridBox[{ {\(-1\), \(-4\)}, {"1", \(-1\)} }], ")"}], "]"}]], "Input"], Cell["\<\ Let's test which eigenvector of the first matrix corresponds with which \ eigenvalue.\ \>", "Text"], Cell[BoxData[ RowBox[{"MatrixForm", "[", RowBox[{ RowBox[{"(", GridBox[{ {"1", "0"}, {"1", \(-1\)} }], ")"}], ".", RowBox[{"(", GridBox[{ {"0"}, {"1"} }], ")"}]}], "]"}]], "Input"], Cell[TextData[{ "This shows that the vector {0,1} is an eigenvector of the matrix ", Cell[BoxData[ RowBox[{"(", GridBox[{ {"1", "0"}, {"1", \(-1\)} }], ")"}]]], " with eigenvalue -1, so the first eigenvalue corresponds to the first \ eigenvector." }], "Text"], Cell[TextData[{ "To get both eigenvalues and eigenvectors at once and to better see which \ eigenvalues are connected with which eigenvectors, it is a good idea to use \ ", StyleBox["Eigensystem[\[Placeholder]]", FontColor->RGBColor[1, 0, 0]], "." }], "Text"], Cell[BoxData[ RowBox[{"Eigensystem", "[", RowBox[{"(", GridBox[{ {"1", "0"}, {"1", \(-1\)} }], ")"}], "]"}]], "Input"], Cell[BoxData[ RowBox[{"Eigensystem", "[", RowBox[{"(", GridBox[{ {\(-1\), \(-4\)}, {"1", \(-1\)} }], ")"}], "]"}]], "Input"], Cell["\<\ This is a list of lists. The first list contains the eigenvalues, and the \ second the eigenvectors. The positions within the lists indicate which \ eigenvalue goes with which eigenvector.\ \>", "Text"], Cell["\<\ Finally, notice what can happen in cases where an eigenvalue is \"repeated.\"\ \ \>", "Text"], Cell[BoxData[ RowBox[{"Eigensystem", "[", RowBox[{"(", GridBox[{ {"2", "0"}, {"0", "2"} }], ")"}], "]"}]], "Input"], Cell[BoxData[ RowBox[{"Eigensystem", "[", RowBox[{"(", GridBox[{ {"2", "1"}, {"0", "2"} }], ")"}], "]"}]], "Input"], Cell[TextData[{ "In this last case, we only have a one dimensional eigenspace, so ", StyleBox["Mathematica", FontSlant->"Italic"], " pads the list of linearly independent eigenvectors with zero vectors in \ order to have as many as there are eigenvalues in the list of eigenvalues." }], "Text"] }, Closed]] }, Open ]] }, FrontEndVersion->"5.0 for Microsoft Windows", ScreenRectangle->{{0, 1280}, {0, 971}}, WindowToolbars->"EditBar", WindowSize->{559, 490}, WindowMargins->{{4, Automatic}, {Automatic, 0}} ] (******************************************************************* Cached data follows. 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