# row reduction Published articles for row reduction. This is one page of public article previews, not the complete archive. Follow Next page to continue. Summaries are not the original full articles. ## Linear Programming and the Simplex Algorithm DevFeed: [Linear Programming and the Simplex Algorithm]() Original publisher: [Read original article]() Published: 2014-12-01T10:00:59Z Content type: tutorial Language: en Sources: [Jeremy Kun]() Topics: [Algorithm](), [Programming]() Tags: [algorithm](), [code](), [exponential-time-algorithms](), [integer-programming](), [linear-algebra](), [linear-programming](), [lp-relaxation](), [optimization](), [programming](), [row-reduction](), [simplex-algorithm]() ### AI overview This tutorial explains how to implement the simplex algorithm for solving linear programs. It introduces standard form and shows how slack variables convert inequality constraints into equality constraints. ### Source excerpt In the last post in this series we saw some simple examples of linear programs, derived the concept of a dual linear program, and saw the duality theorem and the complementary slackness conditions which give a rough sketch of the stopping criterion for an algorithm. This time we'll go ahead and write this algorithm for solving linear programs, and next time we'll apply the algorithm to an industry-strength version of the nutrition problem we saw last time. ## Fixing Bugs in "Computing Homology" DevFeed: [Fixing Bugs in "Computing Homology"]() Original publisher: [Read original article]() Published: 2014-01-23T23:05:16Z Content type: tutorial Language: en Sources: [Jeremy Kun]() Topics: [Computing](), [Mathematics](), [Code](), [function](), [Matrix]() Tags: [bugs](), [code](), [computing](), [function](), [homology](), [linear-algebra](), [mathematics](), [matrices](), [persistent-homology](), [programming](), [python](), [quotients](), [row-reduction](), [test]() ### AI overview This article corrects bugs in code for computing homology. It identifies an indexing error, explains a mathematical mistake in simultaneous row and column reduction, and shows how further row reduction is needed to obtain the correct rank for a triangulation of the Möbius band. ### Source excerpt A few awesome readers have posted comments in Computing Homology to the effect of, "Your code is not quite correct!" And they're right! Despite the almost year since that post's publication, I haven't bothered to test it for more complicated simplicial complexes, or even the basic edge cases! When I posted it the mathematics just felt so solid to me that it had to be right (the irony is rich, I know). ## Computing Homology DevFeed: [Computing Homology]() Original publisher: [Read original article]() Published: 2013-04-10T16:38:25Z Content type: tutorial Language: en Sources: [Jeremy Kun]() Topics: [Computing](), [Code](), [Algorithm](), [Collections](), [.NET 11 Preview 7]() Tags: [algebraic-topology](), [algorithm](), [code](), [collections](), [computing](), [homology](), [linear-maps](), [matrices](), [matrix](), [ordering](), [row-reduction](), [structure]() ### AI overview This tutorial explains an algorithmic approach to computing homology for a simplicial complex. It represents chain groups and boundary homomorphisms as matrices, uses row reduction, and works through an extended example for computing H₁. ### Source excerpt Update: the mistakes made in the code posted here are fixed and explained in a subsequent post (one minor code bug was fixed here, and a less minor conceptual bug is fixed in the linked post). In our last post in this series on topology, we defined the homology group. Specifically, we built up a topological space as a simplicial complex (a mess of triangles glued together), we defined an algebraic way to represent collections of simplices called chains as vectors in a vector space, we defined the boundary homomorphism $ \partial_k$ as a linear map on chains, and finally defined the homology groups as the quotient vector spaces ## Row Reduction Over A Field DevFeed: [Row Reduction Over A Field]() Original publisher: [Read original article]() Published: 2011-12-30T15:36:28Z Content type: tutorial Language: en Sources: [Jeremy Kun]() Topics: [Computing](), [Optimization]() Tags: [eigenvalues](), [eigenvectors](), [field](), [linear-algebra](), [linear-maps](), [mathematics](), [matrices](), [optimization](), [programming](), [python](), [row-reduction]() ### AI overview This tutorial introduces row reduction over a field through matrices representing linear maps between finite-dimensional vector spaces. It explains row equivalence and describes how suitable matrix forms help determine kernels, images, dimensions, eigenvalues, and eigenvectors, with applications to persistent homology and optimization problems. ### Source excerpt We're quite eager to get to applications of algebraic topology to things like machine learning (in particular, persistent homology). Even though there's a massive amount of theory behind it (and we do plan to cover some of the theory), a lot of the actual computations boil down to working with matrices. Of course, this means we're in the land of linear algebra; for a refresher on the terminology, see our primers on linear algebra.