{"id":48,"date":"2013-05-29T20:25:13","date_gmt":"2013-05-29T20:25:13","guid":{"rendered":"http:\/\/homepages.se.edu\/cmoretti\/?page_id=48"},"modified":"2019-10-28T15:42:41","modified_gmt":"2019-10-28T15:42:41","slug":"mathematica-notebooks-higher-algebra","status":"publish","type":"page","link":"https:\/\/www.se.edu\/cmoretti\/main\/mathematicalibrarymain\/mathematica-notebooks-higher-algebra\/","title":{"rendered":"Dr. Moretti&#8217;s Mathematica Notebooks &#8211; Linear and Abstract Algebra"},"content":{"rendered":"<h1 style=\"text-align: center\">Mathematica Notebooks for Linear &amp; Abstract Algebra<\/h1>\n<p><span style=\"text-decoration: underline\">Important Note:<\/span>\u00a0The links for the notebooks open a new window or tab with a Google Drive page &#8211; the current settings for our homepages won&#8217;t allow me to host mathematica notebooks locally.<\/p>\n<h2>Graphical Eigenvectors<\/h2>\n<p>This notebook gives a graphical demonstration of what it means to be an eigenvector. \u00a0In this case you can enter in a linear transformation in\u00a0<em>x<\/em> and\u00a0<em>y<\/em>. \u00a0You can then drag around the terminal point of a vector (in red) and the computer will plot the image of your vector under the transformation in green. \u00a0Your vector will be an eigenvector if the result vector is a scalar multiple of it \u00a0&#8211; that is, if it is in the same or opposite direction. \u00a0Of course you will only be able to do this if the eigenvalues for the linear transformation are real. \u00a0You can also choose to have the true eigenvectors or eigenvalues displayed for reference.<\/p>\n<p style=\"text-align: center\"><a href=\"https:\/\/www.se.edu\/cmoretti\/wp-content\/uploads\/sites\/81\/2013\/05\/graphicaleigenvectors2.png\"><img fetchpriority=\"high\" decoding=\"async\" class=\"aligncenter size-full wp-image-198\" alt=\"graphicaleigenvectors2\" src=\"https:\/\/www.se.edu\/cmoretti\/wp-content\/uploads\/sites\/81\/2013\/05\/graphicaleigenvectors2.png\" width=\"500\" height=\"645\" srcset=\"https:\/\/www.se.edu\/cmoretti\/wp-content\/uploads\/sites\/81\/2013\/05\/graphicaleigenvectors2.png 500w, https:\/\/www.se.edu\/cmoretti\/wp-content\/uploads\/sites\/81\/2013\/05\/graphicaleigenvectors2-233x300.png 233w\" sizes=\"(max-width: 500px) 100vw, 500px\" \/><\/a><\/p>\n<p style=\"text-align: center\"><a href=\"https:\/\/drive.google.com\/folderview?id=0BxSwUt0i0mImYlE0a2JnVW01WkU&amp;usp=sharing\" target=\"_blank\" rel=\"noopener noreferrer\">Download this or other Higher Algebra notebooks from Google Drive<\/a><\/p>\n<h2>A &#8220;home-brewed&#8221; solver for systems of linear equations<\/h2>\n<p>This notebook contains some code we wrote in our Technology 2 class to solve systems of linear equations (essentially duplicating a very small part of the functionality of Mathematica&#8217;s Solve command). \u00a0The code works by first checking if a system is linear in given variables, then putting the system in matrix form, performing Gauss-Jordan elimination on the resulting matrix, and then converting the reduced matrix to standard Mathematica notation. \u00a0Solving a linear system is a great example of both being able to understand and algorithm and then code it.<\/p>\n<p style=\"text-align: center\"><a href=\"https:\/\/www.se.edu\/cmoretti\/wp-content\/uploads\/sites\/81\/2013\/05\/linearsystems.jpg\"><img decoding=\"async\" class=\"aligncenter  wp-image-149\" style=\"border: 1px solid black\" alt=\"linearsystems\" src=\"https:\/\/www.se.edu\/cmoretti\/wp-content\/uploads\/sites\/81\/2013\/05\/linearsystems.jpg\" width=\"387\" height=\"196\" srcset=\"https:\/\/www.se.edu\/cmoretti\/wp-content\/uploads\/sites\/81\/2013\/05\/linearsystems.jpg 1548w, https:\/\/www.se.edu\/cmoretti\/wp-content\/uploads\/sites\/81\/2013\/05\/linearsystems-300x152.jpg 300w, https:\/\/www.se.edu\/cmoretti\/wp-content\/uploads\/sites\/81\/2013\/05\/linearsystems-768x389.jpg 768w, https:\/\/www.se.edu\/cmoretti\/wp-content\/uploads\/sites\/81\/2013\/05\/linearsystems-1024x519.jpg 1024w\" sizes=\"(max-width: 387px) 100vw, 387px\" \/><\/a><\/p>\n<p style=\"text-align: center\"><a href=\"https:\/\/drive.google.com\/folderview?id=0BxSwUt0i0mImYlE0a2JnVW01WkU&amp;usp=sharing\" target=\"_blank\" rel=\"noopener noreferrer\">Download this or other Higher Algebra notebooks from Google Drive<\/a><\/p>\n<h2>Cayley Tables<\/h2>\n<p>This notebook contains codes to let you quickly create &#8220;Cayley Tables&#8221; &#8211; operation tables that are commonly used to understand small groups and rings. \u00a0The commands CayleyTable and MakeRingTables allow you do to tables for either one or two operations respectively. \u00a0Both allow for the use of arbitrarily labels for the elements, and the MakeRingTables command requires you to put in labels for each of the tables so you can tell which is which. \u00a0The example below shows a quick construction of the Cayley table for the group GL(2,Z_2). \u00a0Once you have the Cayley table you can easily see that this is a group of 6 elements, and since the table isn&#8217;t symmetric about the main diagonal the group is non-Abelian, and therefore must be isomorphic to S_3. \u00a0The table can also be used to computer orders of elements, centralizers, etc.<\/p>\n<p style=\"text-align: center\"><a href=\"https:\/\/www.se.edu\/cmoretti\/wp-content\/uploads\/sites\/81\/2013\/05\/calyey2.jpg\"><img decoding=\"async\" class=\"aligncenter  wp-image-157\" style=\"border: 1px solid black\" alt=\"calyey2\" src=\"https:\/\/www.se.edu\/cmoretti\/wp-content\/uploads\/sites\/81\/2013\/05\/calyey2.jpg\" width=\"396\" height=\"238\" srcset=\"https:\/\/www.se.edu\/cmoretti\/wp-content\/uploads\/sites\/81\/2013\/05\/calyey2.jpg 1584w, https:\/\/www.se.edu\/cmoretti\/wp-content\/uploads\/sites\/81\/2013\/05\/calyey2-300x180.jpg 300w, https:\/\/www.se.edu\/cmoretti\/wp-content\/uploads\/sites\/81\/2013\/05\/calyey2-768x462.jpg 768w, https:\/\/www.se.edu\/cmoretti\/wp-content\/uploads\/sites\/81\/2013\/05\/calyey2-1024x615.jpg 1024w, https:\/\/www.se.edu\/cmoretti\/wp-content\/uploads\/sites\/81\/2013\/05\/calyey2-1568x942.jpg 1568w\" sizes=\"(max-width: 396px) 100vw, 396px\" \/><\/a><\/p>\n<p style=\"text-align: center\"><a href=\"https:\/\/drive.google.com\/folderview?id=0BxSwUt0i0mImYlE0a2JnVW01WkU&amp;usp=sharing\" target=\"_blank\" rel=\"noopener noreferrer\">Download this or other Higher Algebra notebooks from Google Drive<\/a><\/p>\n<h2>Modular Arithmetic<\/h2>\n<p>This notebook has a manipulation which lets you visualize operations in modular arithmetic (such as addition or multiplication modulo n) by assigning a color to each number and then creating the operation table. \u00a00 is always represented by black and 1 by white; this makes it easy to identify 0&#8217;s and 1&#8217;s and do investigations about concepts like units, zero-divisors, nilpotents, and so on. \u00a0I wrote a version of this for the <a href=\"http:\/\/demonstrations.wolfram.com\/ModularArithmetic\/\" target=\"_blank\" rel=\"noopener noreferrer\">Wolfram Demonstrations Project<\/a>.<\/p>\n<p style=\"text-align: center\"><a href=\"https:\/\/www.se.edu\/cmoretti\/wp-content\/uploads\/sites\/81\/2013\/05\/modular.jpg\"><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter  wp-image-147\" alt=\"modular\" src=\"https:\/\/www.se.edu\/cmoretti\/wp-content\/uploads\/sites\/81\/2013\/05\/modular.jpg\" width=\"313\" height=\"336\" srcset=\"https:\/\/www.se.edu\/cmoretti\/wp-content\/uploads\/sites\/81\/2013\/05\/modular.jpg 1252w, https:\/\/www.se.edu\/cmoretti\/wp-content\/uploads\/sites\/81\/2013\/05\/modular-279x300.jpg 279w, https:\/\/www.se.edu\/cmoretti\/wp-content\/uploads\/sites\/81\/2013\/05\/modular-768x824.jpg 768w, https:\/\/www.se.edu\/cmoretti\/wp-content\/uploads\/sites\/81\/2013\/05\/modular-954x1024.jpg 954w\" sizes=\"(max-width: 313px) 100vw, 313px\" \/><\/a><\/p>\n<p style=\"text-align: center\"><a href=\"https:\/\/drive.google.com\/folderview?id=0BxSwUt0i0mImYlE0a2JnVW01WkU&amp;usp=sharing\" target=\"_blank\" rel=\"noopener noreferrer\">Download this or other Higher Algebra notebooks from Google Drive<\/a><\/p>\n<p><a title=\"Mathematica Notebook Library\" href=\"http:\/\/homepages.se.edu\/cmoretti\/main\/mathematicalibrarymain\/\">Back to my Mathematica Library<\/a><\/p>\n<p><a title=\"Dr. Christopher Moretti\u2019s Homepage\" href=\"http:\/\/homepages.se.edu\/cmoretti\/\">Back to my homepage<\/a><\/p>\n<p><a title=\"Southeastern main page\" href=\"http:\/\/www.se.edu\">Back to the Southeastern Main Page<\/a>\t\t<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Mathematica Notebooks for Linear &amp; Abstract Algebra Important Note:\u00a0The links for the notebooks open a new window or tab with a Google Drive page &#8211; the current settings for our [&hellip;]<\/p>\n","protected":false},"author":24,"featured_media":0,"parent":19,"menu_order":0,"comment_status":"open","ping_status":"open","template":"page-templates\/page-with-sidebar.php","meta":{"footnotes":""},"class_list":["post-48","page","type-page","status-publish","hentry"],"acf":[],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v25.9 - https:\/\/yoast.com\/wordpress\/plugins\/seo\/ -->\n<title>Dr. Moretti&#039;s Mathematica Notebooks - Linear and Abstract Algebra -<\/title>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/www.se.edu\/cmoretti\/main\/mathematicalibrarymain\/mathematica-notebooks-higher-algebra\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Dr. Moretti&#039;s Mathematica Notebooks - Linear and Abstract Algebra -\" \/>\n<meta property=\"og:description\" content=\"Mathematica Notebooks for Linear &amp; 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