# vector spaces Published articles for vector spaces. This is one page of public article previews, not the complete archive. Follow Next page to continue. Summaries are not the original full articles. ## (Finite) Fields -- A Primer DevFeed: [(Finite) Fields -- A Primer]() Original publisher: [Read original article]() Published: 2014-02-26T10:00:01Z Content type: tutorial Language: en Sources: [Jeremy Kun]() Topics: [Mathematics](), [Math and Logic]() Tags: [complex-numbers](), [cryptography](), [euclidean-domains](), [field](), [field-characteristic](), [finite-fields](), [groups](), [ideals](), [mathematics](), [monoids](), [operations](), [polynomial-ring](), [vector-spaces]() ### AI overview This primer introduces fields as commutative rings with 0 and 1 in which every nonzero element has a multiplicative inverse. It defines the field axioms, places fields within related algebraic structures, and raises the question of finite fields. ### Source excerpt So far on this blog we've given some introductory notes on a few kinds of algebraic structures in mathematics (most notably groups and rings, but also monoids). Fields are the next natural step in the progression. If the reader is comfortable with rings, then a field is extremely simple to describe: they're just commutative rings with 0 and 1, where every nonzero element has a multiplicative inverse. We'll give a list of all of the properties that go into this "simple" definition in a moment, but an even more simple way to describe a field is as a place where "arithmetic makes sense. ## Homology Theory -- A Primer DevFeed: [Homology Theory -- A Primer]() Original publisher: [Read original article]() Published: 2013-04-03T20:07:46Z Content type: tutorial Language: en Sources: [Jeremy Kun]() Topics: [Computing]() Tags: [abelian-groups](), [algebraic-topology](), [fundamental-group](), [groups](), [homology](), [homotopy](), [linear-algebra](), [mathematics](), [matrices](), [primer](), [row-reduction](), [series](), [simplicial-complex](), [topological-invariant](), [topology](), [tor](), [vector-spaces]() ### AI overview A primer on homology theory that introduces homology groups as computable algebraic invariants of topological spaces. It emphasizes linear algebra, row reduction, vector spaces, and the mathematical background needed to implement related programs. ### Source excerpt This series on topology has been long and hard, but we're are quickly approaching the topics where we can actually write programs. For this and the next post on homology, the most important background we will need is a solid foundation in linear algebra, specifically in row-reducing matrices (and the interpretation of row-reduction as a change of basis of a linear operator). Last time we engaged in a whirlwind tour of the fundamental group and homotopy theory. ## The Discrete Fourier Transform -- A Primer DevFeed: [The Discrete Fourier Transform -- A Primer]() Original publisher: [Read original article]() Published: 2012-06-23T14:13:53Z Content type: tutorial Language: en Sources: [Jeremy Kun]() Topics: [Programming]() Tags: [analysis](), [course](), [fourier-analysis](), [fourier-transform](), [function](), [mathematics](), [primer](), [vector-spaces]() ### AI overview This primer explains the intuitive connections between continuous and discrete Fourier transforms. It covers discrete approximations of functions and transforms, the transition between discrete representations, and the role of sampling as motivation. ### Source excerpt So here we are. We have finally made it to a place where we can transition with confidence from the classical continuous Fourier transform to the discrete version, which is the foundation for applications of Fourier analysis to programming. Indeed, we are quite close to unfurling the might of the Fast Fourier Transform algorithm, which efficiently computes the discrete Fourier transform. But because of its focus on algorithmic techniques, we will save it for a main content post and instead focus here on the intuitive connections between the discrete and continuous realms. ## Inner Product Spaces--A Primer DevFeed: [Inner Product Spaces--A Primer]() Original publisher: [Read original article]() Published: 2011-07-25T00:29:00Z Content type: tutorial Language: en Sources: [Jeremy Kun]() Topics: [Mathematics](), [Math and Logic](), [Computing]() Tags: [big-o-notation](), [eigenvectors](), [inner-product](), [mathematics](), [matrices](), [orthogonality](), [symmetry](), [vector-spaces](), [vectors]() ### AI overview This primer explains why vector spaces can be extended with a dot-product-like operation and defines inner products through conjugate symmetry, linearity, additivity, and positive definiteness. It introduces inner product spaces and discusses their connection to linear functionals. ### Source excerpt Vector spaces alone are not enough to do a lot of the interesting things we'd like them to do. Since a vector space is a generalization of Euclidean space, it is natural for us to investigate more specific types of vector spaces which are more akin to Euclidean space. In particular, we want to include the notion of a dot product. By admitting additional structure to a vector space, we may perform more computations, and hopefully get more interesting results. ## 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.