# linear maps Published articles for linear maps. This is one page of public article previews, not the complete archive. Follow Next page to continue. Summaries are not the original full articles. ## Concrete Examples of Quantum Gates DevFeed: [Concrete Examples of Quantum Gates]() Original publisher: [Read original article]() Published: 2016-01-11T14:05:22Z Content type: tutorial Language: en Sources: [Jeremy Kun]() Topics: [circuit]() Tags: [circuit](), [circuits](), [linear-maps](), [mathematics](), [matrices](), [python](), [quantum](), [quantum-computing]() ### AI overview A quantum-computing tutorial explains how local quantum gates affect the full state vector of a multi-qubit register, using a three-qubit swap-gate example. ### Source excerpt So far in this series we've seen a lot of motivation and defined basic ideas of what a quantum circuit is. But on rereading my posts, I think we would all benefit from some concreteness. "Local" operations So by now we've understood that quantum circuits consist of a sequence of gates $ A_1, \dots, A_k$, where each $ A_i$ is an 8-by-8 matrix that operates "locally" on some choice of three (or fewer) qubits. ## 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. ## Linear Algebra--A Primer DevFeed: [Linear Algebra--A Primer]() Original publisher: [Read original article]() Published: 2011-06-19T18:39:40Z Content type: tutorial Language: en Sources: [Jeremy Kun]() Topics: [Mathematics](), [Matrix](), [Graphs]() Tags: [adjacency-matrix](), [algebra](), [eigenvalues](), [eigenvectors](), [graph](), [history](), [linear-algebra](), [linear-independence](), [linear-maps](), [mathematics](), [matrices](), [matrix](), [primer](), [transformation](), [vector](), [vector-spaces]() ### AI overview This primer introduces linear algebra through its historical origins in solving systems of linear equations. It explains determinants, matrices, vector algebra, and linear transformations, and shows how matrices can model graphs and compute path counts. ### Source excerpt Story Time Linear algebra was founded around the same time as Calculus (think Leibniz, circa 1700) solely for the purpose of solving general systems of linear equations. The coefficients of a system were written in a grid form, with rows corresponding to equations and columns to the unknown variables. Using a computational tool called the determinant (an awkward, but computable formula involving only the coefficients of the equations in a system), researchers were able to solve these systems, opening a world of information about the positions of celestial bodies and large-scale measurements (of geodesic arcs) on the surface of the earth.