(*********************************************************************** Mathematica-Compatible Notebook This notebook can be used on any computer system with Mathematica 3.0, MathReader 3.0, or any compatible application. The data for the notebook starts with the line of 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[ 14687, 580]*) (*NotebookOutlinePosition[ 27545, 1051]*) (* CellTagsIndexPosition[ 27501, 1047]*) (*WindowFrame->Normal*) Notebook[{ Cell[CellGroupData[{ Cell["LISTS, VECTORS, LINEAR ALGEBRA", "Title", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{ Cell["Lists and elements", "Section", Evaluatable->False, AspectRatioFixed->True], Cell["\<\ A list (1-dimensional array) is an ordered sequence of expressions, \ separated by commas, and delimited by \"curly brackets\" {}.\ \>", "Text"], Cell[BoxData[ \({\ a, \ 17, \ Sqrt[9], \ \ yet\ another\ element, \ Pi}\)], "Input"], Cell["\<\ The elements of a list are indexed 1, 2, ..., n (not 0, 1, ..., (n-1) as in \ C and other languages). 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This is a most useful function, and worth learning in its various \ forms.\n\nThe simplest usage is to make a list of ", FontWeight->"Plain"], StyleBox["n", FontWeight->"Plain", FontSlant->"Italic"], StyleBox[" copies of an expression.", FontWeight->"Plain"] }], "Text", FontWeight->"Bold"], Cell[BoxData[ \(Table[\ \ 16 + x + 1, \ \ {5}\ ]\)], "Input"], Cell[TextData[{ StyleBox["\n", FontWeight->"Bold"], "A variation makes a list where dummy variable ", StyleBox["k", FontSlant->"Italic"], " runs from 1 to ", StyleBox["n", FontSlant->"Italic"], "." }], "Text"], Cell[BoxData[ \(Table[\ \ k^2, \ \ {k, \ 5}]\)], "Input"], Cell[TextData[{ StyleBox["\n", FontWeight->"Bold"], "The range of values for the dummy variable can be controlled more \ precisely:" }], "Text"], Cell[BoxData[{ \(Table[\ \ m^2, \ \ {m, \ 4, \ 7}]\), \(Table[\ \ m^2, \ \ {m, \ \ 5, \ \ 17, \ 3}]\)}], "Input"], Cell[TextData[{ StyleBox["\n", FontWeight->"Bold"], "A multidimensional array (nested lists) can be built by using several \ iteration variables." }], "Text"], Cell[BoxData[ \(Table[\ \ 2^i\ *\ 3^j, \ \ \ {i, 0, 3}, \ {j, 0, 3}]\)], "Input"], Cell[TextData[{ StyleBox["\n", FontWeight->"Bold"], "Any expression can be evaluated in a table." }], "Text"], Cell[BoxData[ \(\(Table[\ \ \ \ \ \ \(Plot[\ x^i, \ {x, 0, 2}];\), \ \ \ \ {i, 1, 3}]; \)\)], "Input"], Cell["\<\ 2-dimensional arrays can be displayed in \"list of lists\" form (which is the \ default), or more neatly using TableForm or MatrixForm:\ \>", "Text"], Cell[BoxData[{ \(Table[\ \ {k, \ N[Pi, k]}, \ \ \ \ {k, 15, 45, 10}]\ // \ TableForm\), \(Table[\ \ i/j, \ \ \ \ {i, 5}, \ {j, 5}]\ \ \ // \ MatrixForm\)}], "Input"] }, Closed]], Cell[CellGroupData[{ Cell["Applying a function to a list", "Section", Evaluatable->False, AspectRatioFixed->True], Cell["\<\ Basic arithmetic commands do the right thing with lists; for \ example;\ \>", "Text"], Cell[BoxData[{ \(list1\ \ = \ \ {a, b, c, d, e}; \nlist2\ \ = \ \ {1, 2, 3, 7, 8}; \n\n list1\ + \ 17\), \(list1\ + \ list2\), \(list1\ *\ list2\), \(Sqrt[\ \ list1\ ]\)}], "Input"], Cell[TextData[{ "Functions ", StyleBox["Plus", FontWeight->"Bold"], " (addition), ", StyleBox["Times", FontWeight->"Bold"], " (multiplication), and ", StyleBox["Sqrt", FontWeight->"Bold"], " are \"", StyleBox["listable", FontSlant->"Italic"], "\", which means they are applied to every element in a list which appears \ as a parameter. 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", StyleBox["Table", FontWeight->"Bold"], " is the most versatile." }], "Text"], Cell["\<\ IdentityMatrix[2] // MatrixForm DiagonalMatrix[ {a,b,c} ] // MatrixForm Table[ If[i<=j,1,0], {i,3}, {j,3}] // MatrixForm\ \>", "Input"], Cell[TextData[{ "\nThe functions ", StyleBox["Length", FontWeight->"Bold"], " and ", StyleBox["Dimensions", FontWeight->"Bold"], " can be used to find the size of a list or tensor." }], "Text"], Cell["\<\ MatrixForm[matr2] Length[matr2] Dimensions[matr2]\ \>", "Input"] }, Closed]], Cell[CellGroupData[{ Cell["Matrix functions", "Section", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ "All sorts of typical matrix operations are available with ", StyleBox["Mathematica.", FontSlant->"Italic"] }], "Text"], Cell["\<\ m = { {199,XX}, {e3,17} }; m // MatrixForm Transpose[m] // MatrixForm Det[m] Inverse[m]\ \>", "Input"], Cell["\<\ Some functions work quickly for numerical matrices, but take prohibitively \ long for non-numerical matrices other than the smallest and simplest. Before \ attempting to find eigenvectors of a large non-numerical matrix, for \ instance, be sure to save your work first; the computer may hang when all \ available RAM is used up...\ \>", "Text"], Cell["\<\ m = { {5.43, -0.96}, {0.0172, 3.801} }; m // MatrixForm MatrixExp[m] // MatrixForm Eigenvalues[m] Eigenvectors[m]\ \>", "Input"] }, Closed]], Cell[CellGroupData[{ Cell["Solving Linear Systems", "Section"], Cell[TextData[{ "A system of linear equations, ", StyleBox["m.x==b", FontFamily->"Courier", FontWeight->"Bold"], ", can be solved with ", StyleBox["LinearSolve", FontWeight->"Bold"], ". 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