{"id":230,"date":"2024-06-15T17:20:50","date_gmt":"2024-06-15T22:20:50","guid":{"rendered":"https:\/\/mathtalker.org\/blog\/?page_id=230"},"modified":"2026-04-16T16:14:21","modified_gmt":"2026-04-16T21:14:21","slug":"applied-linear-algebra-class-notes","status":"publish","type":"page","link":"https:\/\/mathtalker.org\/blog\/applied-linear-algebra-class-notes\/","title":{"rendered":"Applied Linear Algebra"},"content":{"rendered":"<p><script type=\"application\/ld+json\"><br \/>\n{<br \/>\n  \"@context\": \"https:\/\/schema.org\",<br \/>\n  \"@type\": \"CollectionPage\",<br \/>\n  \"name\": \"Applied Linear Algebra Class Notes\",<br \/>\n  \"headline\": \"Applied Linear Algebra Class Notes\",<br \/>\n  \"description\": \"Free applied linear algebra class notes covering vector spaces, eigenvalues, orthogonality, and singular values, based on Olver and Shakiban.\",<br \/>\n  \"url\": \"https:\/\/mathtalker.org\/www\/classes\/appliedlinearalgebra.php\",<br \/>\n  \"author\": {<br \/>\n    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\"https:\/\/mathtalker.org\/www\/classes\/appliedlinearalgebra_pdfs\/04p4_Orthogonal_Projections_and_Subspaces_fredholm_alternative.pdf\",<br \/>\n      \"encodingFormat\": \"application\/pdf\"<br \/>\n    },<br \/>\n    {<br \/>\n      \"@type\": \"CreativeWork\",<br \/>\n      \"name\": \"5.2 Minimization of Quadratic Functions\",<br \/>\n      \"url\": \"https:\/\/mathtalker.org\/www\/classes\/appliedlinearalgebra_pdfs\/05p2_Minimization_of_Quadratic_Functions.pdf\",<br \/>\n      \"encodingFormat\": \"application\/pdf\"<br \/>\n    },<br \/>\n    {<br \/>\n      \"@type\": \"CreativeWork\",<br \/>\n      \"name\": \"7.1 Linear Functions\",<br \/>\n      \"url\": \"https:\/\/mathtalker.org\/www\/classes\/appliedlinearalgebra_pdfs\/07p1_Linear_Functions.pdf\",<br \/>\n      \"encodingFormat\": \"application\/pdf\"<br \/>\n    },<br \/>\n    {<br \/>\n      \"@type\": \"CreativeWork\",<br \/>\n      \"name\": \"7.2 Linear Transformations\",<br \/>\n      \"url\": \"https:\/\/mathtalker.org\/www\/classes\/appliedlinearalgebra_pdfs\/07p2_Linear_Transformations.pdf\",<br \/>\n      \"encodingFormat\": \"application\/pdf\"<br \/>\n    },<br \/>\n    {<br \/>\n      \"@type\": \"CreativeWork\",<br \/>\n      \"name\": \"8.2 Eigenvalues and Eigenvectors\",<br \/>\n      \"url\": \"https:\/\/mathtalker.org\/www\/classes\/appliedlinearalgebra_pdfs\/08p2_Eigenvalues_and_Eigenvectors.pdf\",<br \/>\n      \"encodingFormat\": \"application\/pdf\"<br \/>\n    },<br \/>\n    {<br \/>\n      \"@type\": \"CreativeWork\",<br \/>\n      \"name\": \"8.3 Eigenvector Bases\",<br \/>\n      \"url\": \"https:\/\/mathtalker.org\/www\/classes\/appliedlinearalgebra_pdfs\/08p3_Eigenvector_Bases.pdf\",<br \/>\n      \"encodingFormat\": \"application\/pdf\"<br \/>\n    },<br \/>\n    {<br \/>\n      \"@type\": \"CreativeWork\",<br \/>\n      \"name\": \"8.5 Eigenvalues of Symmetric Matrices\",<br \/>\n      \"url\": \"https:\/\/mathtalker.org\/www\/classes\/appliedlinearalgebra_pdfs\/08p5_Eigenvalues_of_Symmetric_Matrices.pdf\",<br \/>\n      \"encodingFormat\": \"application\/pdf\"<br \/>\n    },<br \/>\n    {<br \/>\n      \"@type\": \"CreativeWork\",<br \/>\n      \"name\": \"8.7 Singular Values\",<br \/>\n      \"url\": \"https:\/\/mathtalker.org\/www\/classes\/appliedlinearalgebra_pdfs\/08p7_Singular_Values.pdf\",<br \/>\n      \"encodingFormat\": \"application\/pdf\"<br \/>\n    }<br \/>\n  ]<br \/>\n}<br \/>\n<\/script><\/p>\n<h2>Applied Linear Algebra Class Notes<\/h2>\n<p><strong>Keywords:<\/strong> Applied Linear Algebra notes, Olver and Shakiban, eigenvalues, Gram\u2013Schmidt, orthogonal projections, singular values, vector spaces.<\/p>\n<p>As an instructor of applied linear algebra, I prepared these class notes as a review of the chapters in<br \/>\n<a href=\"https:\/\/www-users.cse.umn.edu\/~olver\/ala.html\" target=\"_blank\" rel=\"noopener noreferrer\"><br \/>\n&#8220;Applied Linear Algebra, 2nd Edition&#8221; by Olver and Shakiban<\/a>. They include example problems which have been worked out in some detail. I hope you find these useful.<\/p>\n<style>\n.notes-table {<br \/>\n    width: 100%;<br \/>\n    border-collapse: collapse;<br \/>\n    margin-top: 20px;<br \/>\n}<\/p>\n<p>.notes-table th,<br \/>\n.notes-table td {<br \/>\n    border: 1px solid #999;<br \/>\n    padding: 10px;<br \/>\n    text-align: left;<br \/>\n    vertical-align: top;<br \/>\n}<\/p>\n<p>.notes-table th {<br \/>\n    background-color: #f2f2f2;<br \/>\n}<\/p>\n<p>.notes-table tr:nth-child(even) {<br \/>\n    background-color: #fafafa;<br \/>\n}<br \/>\n<\/style>\n<table class=\"notes-table\">\n<thead>\n<tr>\n<th>Class Notes<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td><a href=\"https:\/\/mathtalker.org\/www\/classes\/appliedlinearalgebra_pdfs\/01p1_1p3_Linear_Systems_Matrices_Vectors_Gaussian_Elimination.pdf\" target=\"_blank\" rel=\"noopener noreferrer\">1.1\u20131.3 Linear Systems, Matrices, Vectors, Gaussian Elimination<\/a><\/td>\n<\/tr>\n<tr>\n<td><a href=\"https:\/\/mathtalker.org\/www\/classes\/appliedlinearalgebra_pdfs\/01p4_1p5_Pivoting_Permutations_Inverse_Matrices.pdf\" target=\"_blank\" rel=\"noopener noreferrer\">1.4\u20131.5 Pivoting, Permutations, Inverse Matrices<\/a><\/td>\n<\/tr>\n<tr>\n<td><a href=\"https:\/\/mathtalker.org\/www\/classes\/appliedlinearalgebra_pdfs\/01p6_Transposes_and_Symmetric_Matrices.pdf\" target=\"_blank\" rel=\"noopener noreferrer\">1.6 Transposes and Symmetric Matrices<\/a><\/td>\n<\/tr>\n<tr>\n<td><a href=\"https:\/\/mathtalker.org\/www\/classes\/appliedlinearalgebra_pdfs\/01p8_General_Linear_Systems.pdf\" target=\"_blank\" rel=\"noopener noreferrer\">1.8 General Linear Systems<\/a><\/td>\n<\/tr>\n<tr>\n<td><a href=\"https:\/\/mathtalker.org\/www\/classes\/appliedlinearalgebra_pdfs\/01p9_Determinants.pdf\" target=\"_blank\" rel=\"noopener noreferrer\">1.9 Determinants<\/a><\/td>\n<\/tr>\n<tr>\n<td><a href=\"https:\/\/mathtalker.org\/www\/classes\/appliedlinearalgebra_pdfs\/02p1_Real_Vector_Spaces.pdf\" target=\"_blank\" rel=\"noopener noreferrer\">2.1 Real Vector Spaces<\/a><\/td>\n<\/tr>\n<tr>\n<td><a href=\"https:\/\/mathtalker.org\/www\/classes\/appliedlinearalgebra_pdfs\/02p2_Subspaces.pdf\" target=\"_blank\" rel=\"noopener noreferrer\">2.2 Subspaces<\/a><\/td>\n<\/tr>\n<tr>\n<td><a href=\"https:\/\/mathtalker.org\/www\/classes\/appliedlinearalgebra_pdfs\/02p3_Span_and_Linear_Independence.pdf\" target=\"_blank\" rel=\"noopener noreferrer\">2.3 Span and Linear Independence<\/a><\/td>\n<\/tr>\n<tr>\n<td><a href=\"https:\/\/mathtalker.org\/www\/classes\/appliedlinearalgebra_pdfs\/02p4_Basis_and_Dimension.pdf\" target=\"_blank\" rel=\"noopener noreferrer\">2.4 Basis and Dimension<\/a><\/td>\n<\/tr>\n<tr>\n<td><a href=\"https:\/\/mathtalker.org\/www\/classes\/appliedlinearalgebra_pdfs\/02p5_Fundamental_Matrix_Subspaces.pdf\" target=\"_blank\" rel=\"noopener noreferrer\">2.5 Fundamental Matrix Subspaces<\/a><\/td>\n<\/tr>\n<tr>\n<td><a href=\"https:\/\/mathtalker.org\/www\/classes\/appliedlinearalgebra_pdfs\/02p6_Graphs_and_Digraphs.pdf\" target=\"_blank\" rel=\"noopener noreferrer\">2.6 Graphs and Digraphs<\/a><\/td>\n<\/tr>\n<tr>\n<td><a href=\"https:\/\/mathtalker.org\/www\/classes\/appliedlinearalgebra_pdfs\/03p1_Inner_Products.pdf\" target=\"_blank\" rel=\"noopener noreferrer\">3.1 Inner Products<\/a><\/td>\n<\/tr>\n<tr>\n<td><a href=\"https:\/\/mathtalker.org\/www\/classes\/appliedlinearalgebra_pdfs\/03p2_Cauchy_Schwarz_and_Triangle_Inequalities.pdf\" target=\"_blank\" rel=\"noopener noreferrer\">3.2 Cauchy\u2013Schwarz and Triangle Inequalities<\/a><\/td>\n<\/tr>\n<tr>\n<td><a href=\"https:\/\/mathtalker.org\/www\/classes\/appliedlinearalgebra_pdfs\/03p3_Norms.pdf\" target=\"_blank\" rel=\"noopener noreferrer\">3.3 Norms<\/a><\/td>\n<\/tr>\n<tr>\n<td><a href=\"https:\/\/mathtalker.org\/www\/classes\/appliedlinearalgebra_pdfs\/03p4_Positive_Definite_Matrices.pdf\" target=\"_blank\" rel=\"noopener noreferrer\">3.4 Positive Definite Matrices<\/a><\/td>\n<\/tr>\n<tr>\n<td><a href=\"https:\/\/mathtalker.org\/www\/classes\/appliedlinearalgebra_pdfs\/03p5_Completing_the_square.pdf\" target=\"_blank\" rel=\"noopener noreferrer\">3.5 Completing the Square<\/a><\/td>\n<\/tr>\n<tr>\n<td><a href=\"https:\/\/mathtalker.org\/www\/classes\/appliedlinearalgebra_pdfs\/04p1_Orthogonal_and_Orthonormal_Bases.pdf\" target=\"_blank\" rel=\"noopener noreferrer\">4.1 Orthogonal and Orthonormal Bases<\/a><\/td>\n<\/tr>\n<tr>\n<td><a href=\"https:\/\/mathtalker.org\/www\/classes\/appliedlinearalgebra_pdfs\/04p2_Gram_Schmidt_Process.pdf\" target=\"_blank\" rel=\"noopener noreferrer\">4.2 Gram\u2013Schmidt Process<\/a><\/td>\n<\/tr>\n<tr>\n<td><a href=\"https:\/\/mathtalker.org\/www\/classes\/appliedlinearalgebra_pdfs\/04p3_Orthogonal_Matrices.pdf\" target=\"_blank\" rel=\"noopener noreferrer\">4.3 Orthogonal Matrices<\/a><\/td>\n<\/tr>\n<tr>\n<td><a href=\"https:\/\/mathtalker.org\/www\/classes\/appliedlinearalgebra_pdfs\/04p4_Orthogonal_Projections_and_Subspaces_fredholm_alternative.pdf\" target=\"_blank\" rel=\"noopener noreferrer\">4.4 Orthogonal Projections and Subspaces; Fredholm Alternative<\/a><\/td>\n<\/tr>\n<tr>\n<td><a href=\"https:\/\/mathtalker.org\/www\/classes\/appliedlinearalgebra_pdfs\/05p2_Minimization_of_Quadratic_Functions.pdf\" target=\"_blank\" rel=\"noopener noreferrer\">5.2 Minimization of Quadratic Functions<\/a><\/td>\n<\/tr>\n<tr>\n<td><a href=\"https:\/\/mathtalker.org\/www\/classes\/appliedlinearalgebra_pdfs\/07p1_Linear_Functions.pdf\" target=\"_blank\" rel=\"noopener noreferrer\">7.1 Linear Functions<\/a><\/td>\n<\/tr>\n<tr>\n<td><a href=\"https:\/\/mathtalker.org\/www\/classes\/appliedlinearalgebra_pdfs\/07p2_Linear_Transformations.pdf\" target=\"_blank\" rel=\"noopener noreferrer\">7.2 Linear Transformations<\/a><\/td>\n<\/tr>\n<tr>\n<td><a href=\"https:\/\/mathtalker.org\/www\/classes\/appliedlinearalgebra_pdfs\/08p2_Eigenvalues_and_Eigenvectors.pdf\" target=\"_blank\" rel=\"noopener noreferrer\">8.2 Eigenvalues and Eigenvectors<\/a><\/td>\n<\/tr>\n<tr>\n<td><a href=\"https:\/\/mathtalker.org\/www\/classes\/appliedlinearalgebra_pdfs\/08p3_Eigenvector_Bases.pdf\" target=\"_blank\" rel=\"noopener noreferrer\">8.3 Eigenvector Bases<\/a><\/td>\n<\/tr>\n<tr>\n<td><a href=\"https:\/\/mathtalker.org\/www\/classes\/appliedlinearalgebra_pdfs\/08p5_Eigenvalues_of_Symmetric_Matrices.pdf\" target=\"_blank\" rel=\"noopener noreferrer\">8.5 Eigenvalues of Symmetric Matrices<\/a><\/td>\n<\/tr>\n<tr>\n<td><a href=\"https:\/\/mathtalker.org\/www\/classes\/appliedlinearalgebra_pdfs\/08p7_Singular_Values.pdf\" target=\"_blank\" rel=\"noopener noreferrer\">8.7 Singular Values<\/a><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n","protected":false},"excerpt":{"rendered":"<p>Applied Linear Algebra Class Notes Keywords: Applied Linear Algebra notes, Olver and Shakiban, eigenvalues, Gram\u2013Schmidt, orthogonal projections, singular values, vector spaces. As an instructor of applied linear algebra, I prepared these class notes as a review of the chapters in &hellip; <a href=\"https:\/\/mathtalker.org\/blog\/applied-linear-algebra-class-notes\/\">Continue reading <span class=\"meta-nav\">&rarr;<\/span><\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"parent":0,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"","meta":{"jetpack_post_was_ever_published":false,"footnotes":""},"class_list":["post-230","page","type-page","status-publish","hentry"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v27.5 - https:\/\/yoast.com\/product\/yoast-seo-wordpress\/ -->\n<title>Applied Linear Algebra - MathTalker<\/title>\n<meta name=\"description\" content=\"Free applied linear algebra class notes covering vector spaces, eigenvalues, orthogonality, and singular values, based on Olver and Shakiban.\" \/>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/mathtalker.org\/blog\/applied-linear-algebra-class-notes\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Applied Linear Algebra - 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