# linear algebra Published articles for linear algebra. This is one page of public article previews, not the complete archive. Follow Next page to continue. Summaries are not the original full articles. ## The Single Most Undervalued Fact of Linear Algebra DevFeed: [The Single Most Undervalued Fact of Linear Algebra]() Original publisher: [Read original article]() Author: Tivadar Danka Published: 2026-08-24T09:47:15Z Content type: opinion Language: en Sources: [The Palindrome]() Topics: [Graphs]() Tags: [animation](), [arts](), [audio](), [graphs](), [linear](), [linear-algebra](), [matrices](), [video]() ### AI overview The author presents matrices and graphs as related representations and describes remastering an earlier piece into a video with improved animation and audio recording. ### Source excerpt Matrices are graphs and graphs are matrices ## Bicyclic Matrix-Matrix Multiplication in Fully Homomorphic Encryption DevFeed: [Bicyclic Matrix-Matrix Multiplication in Fully Homomorphic Encryption]() Original publisher: [Read original article]() Published: 2025-11-17T16:41:28Z Content type: tutorial Language: en Sources: [Jeremy Kun]() Topics: [FHE](), [Encryption](), [Algorithm](), [Code](), [GitHub]() Tags: [algorithm](), [circuit](), [code](), [cryptography](), [encryption](), [fhe](), [github](), [homomorphic-encryption](), [linear-algebra](), [lwe](), [mathematics](), [programming](), [rlwe]() ### AI overview This article explains the bicyclic method for matrix-matrix multiplication in fully homomorphic encryption. It introduces the method's packing scheme, relates it to Halevi-Shoup diagonal packing, and discusses properties including multiplicative depth, layout invariance, and rotation complexity. The implementation is provided in a GitHub repository in a file named bicyclic.py. ### Source excerpt In an earlier article, I covered the basic technique for performing matrix-vector multiplication in fully homomorphic encryption (FHE), known as the Halevi-Shoup diagonal method. This article covers a more recent method for matrix-matrix multiplication known as the bicyclic method. The code implementing this method is in the same GitHub repository as the previous article, and the bicyclic method is in a file called bicyclic.py. The previous article linked above covers the general concepts behind "FHE packing," which I will assume as background knowledge for this article: ## My Graduate Career in Math DevFeed: [My Graduate Career in Math]() Original publisher: [Read original article]() Published: 2025-05-12T18:35:57Z Content type: opinion Language: en Sources: [Jeremy Kun]() Topics: [Mathematics](), [Computer science](), [graph theory]() Tags: [computer-science](), [education](), [essay](), [essays](), [game-theory](), [graph-theory](), [group-theory](), [linear-algebra](), [math](), [mathematics](), [retrospective](), [topology](), [university]() ### AI overview An autobiographical essay about the author's transition from computer science to mathematics at Cal Poly, including university coursework, study abroad in Budapest, and reflections on the intellectual environment and an early group theory project. ### Source excerpt Editor's note: This essay was originally published on Medium on 2016-03-05. I have made minor edits in this republishing and added a few small retrospective notes. 2010-2011 (Year 0) I had just switched my major at Cal Poly State University from computer science to math. I wanted to double major but California was in a budget crisis and a few weeks before I tried submitting my double-major request the Provost for the CSU system put a blanket ban on double majors. ## Packing Matrix-Vector Multiplication in Fully Homomorphic Encryption DevFeed: [Packing Matrix-Vector Multiplication in Fully Homomorphic Encryption]() Original publisher: [Read original article]() Published: 2024-09-07T04:18:09Z Content type: tutorial Language: en Sources: [Jeremy Kun]() Topics: [homomorphic encryption](), [FHE](), [Encryption](), [layout](), [parallel](), [Python]() Tags: [arithmetic](), [code](), [cryptography](), [data](), [encryption](), [fhe](), [github-repository](), [homomorphic-encryption](), [layout](), [linear-algebra](), [lwe](), [mathematics](), [packing](), [parallel](), [programming](), [python](), [rlwe](), [simd](), [strategies]() ### AI overview This article explains packing for SIMD-style fully homomorphic encryption. It describes how to arrange plaintext data in RLWE ciphertexts so matrix-vector multiplication requires fewer alignment multiplications and rotations, then introduces two basic packing techniques and a computational model. ### Source excerpt In my recent overview of homomorphic encryption, I underemphasized the importance of data layout when working with arithmetic (SIMD-style) homomorphic encryption schemes. In the FHE world, the name given to data layout strategies is called "packing," because it revolves around putting multiple plaintext data into RLWE ciphertexts in carefully-chosen ways that mesh well with the operations you'd like to perform. By "mesh well" I mean it reduces the number of extra multiplications and rotations required merely to align data elements properly, rather than doing the actual computation you care about. ## A High-Level Technical Overview of Fully Homomorphic Encryption DevFeed: [A High-Level Technical Overview of Fully Homomorphic Encryption]() Original publisher: [Read original article]() Published: 2024-05-04T15:30:00Z Content type: tutorial Language: en Sources: [Jeremy Kun]() Topics: [FHE](), [homomorphic encryption](), [Encryption](), [Cryptography](), [toolchain](), [Compiler](), [Post-quantum cryptography]() Tags: [compiler](), [cryptography](), [encryption](), [fhe](), [google](), [homomorphic-encryption](), [linear-algebra](), [lwe](), [mathematics](), [post-quantum-cryptography](), [programming](), [rlwe](), [technical]() ### AI overview A technical, high-level survey of fully homomorphic encryption (FHE), explaining how programs can operate on encrypted data without decrypting it. The article discusses the field's current limitations, key techniques, and its relationship to the HEIR compiler toolchain. ### Source excerpt About two years ago, I switched teams at Google to focus on fully homomorphic encryption (abbreviated FHE, or sometimes HE). Since then I've got to work on a lot of interesting projects, learning along the way about post-quantum cryptography, compiler design, and the ins and outs of fully homomorphic encryption. If you've heard about FHE and you're a software person, you've probably heard two things: it lets you run programs directly on encrypted data without ever decrypting it; and it's still too slow to be useful for anything. ## Sample Extraction from RLWE to LWE DevFeed: [Sample Extraction from RLWE to LWE]() Original publisher: [Read original article]() Published: 2023-02-27T08:00:00Z Content type: tutorial Language: en Sources: [Jeremy Kun]() Topics: [FHE](), [Cryptography](), [Encryption]() Tags: [cryptography](), [encryption](), [fhe](), [homomorphic-encryption](), [linear-algebra](), [lwe](), [mathematics](), [programming](), [rlwe](), [scheme]() ### AI overview A tutorial deriving sample extraction, an FHE technique that partially converts a ciphertext from the Ring Learning With Errors (RLWE) scheme to the Learning With Errors (LWE) scheme. It introduces the relevant LWE and RLWE encryption constructions and explains the role of error terms. ### Source excerpt In this article I'll derive a trick used in FHE called sample extraction. In brief, it allows one to partially convert a ciphertext in the Ring Learning With Errors (RLWE) scheme to the Learning With Errors (LWE) scheme. Here are some other articles I've written about other FHE building blocks, though they are not prerequisites for this article. Modulus Switching in LWE Key Switching in LWE The Gadget Decomposition in FHE Negacyclic Polynomial Multiplication Estimating the Security of Ring Learning With Errors LWE and RLWE The first two articles in the list above define the Learning With Errors problem (LWE). ## The Gadget Decomposition in FHE DevFeed: [The Gadget Decomposition in FHE]() Original publisher: [Read original article]() Published: 2021-12-11T13:57:25Z Content type: tutorial Language: en Sources: [Jeremy Kun]() Topics: [FHE](), [homomorphic encryption](), [Cryptography](), [Computing](), [data](), [Code](), [GitHub]() Tags: [bootstrapping](), [core](), [encryption](), [fhe](), [gadget-decomposition](), [group-theory](), [homomorphic-encryption](), [learning-with-errors](), [linear-algebra](), [lwe](), [mathematics](), [matrix](), [number-theory](), [operations](), [programming](), [python]() ### AI overview A tutorial on gadget decomposition in fully homomorphic encryption (FHE). It explains how GSW and related schemes use random noise, how homomorphic operations increase that noise, why bootstrapping is needed, and how gadget decomposition helps limit noise growth. ### Source excerpt Lately I've been studying Fully Homomorphic Encryption, which is the miraculous ability to perform arbitrary computations on encrypted data without learning any information about the underlying message. It's the most comprehensive private computing solution that can exist (and it does exist!). The first FHE scheme by Craig Gentry was based on ideal lattices and was considered very complex (I never took the time to learn how it worked). Some later schemes (GSW = Gentry-Sahai-Waters) are based on matrix multiplication, and are conceptually much simpler. ## Regression and Linear Combinations DevFeed: [Regression and Linear Combinations]() Original publisher: [Read original article]() Published: 2021-03-29T09:00:00Z Content type: tutorial Language: en Sources: [Jeremy Kun]() Topics: [linear-regression](), [math](), [Optimization](), [Algorithms, Complexity]() Tags: [data](), [gradient-descent](), [kernelization](), [linear-algebra](), [linear-combination](), [linear-regression](), [mathematics](), [optimization](), [programming](), [regression]() ### AI overview The article explains why linear combinations matter to programmers, using linear regression as a practical example. It describes representing inputs and weights as vectors, incorporating an intercept into the input vector, and formulating regression as a least-squares optimization problem. The supplied excerpt then begins introducing basis functions for modeling nonlinearity. ### Source excerpt Recently I've been helping out with a linear algebra course organized by Tai-Danae Bradley and Jack Hidary, and one of the questions that came up a few times was, "why should programmers care about the concept of a linear combination?" For those who don't know, given vectors $ v_1, \dots, v_n$, a linear combination of the vectors is a choice of some coefficients $ a_i$ with which to weight the vectors in a sum $ v = \sum_{i=1}^n a_i v_i$. ## A parlor trick for SET DevFeed: [A parlor trick for SET]() Original publisher: [Read original article]() Published: 2018-03-25T09:51:34Z Content type: article Language: en Sources: [Jeremy Kun]() Topics: [math](), [color]() Tags: [color](), [finite-fields](), [game](), [invariant](), [linear-algebra](), [math](), [mathematics](), [set](), [technical]() ### AI overview This mathematical article presents a parlor trick for the card game SET. By modeling cards as vectors in the finite vector space F₃⁴, it explains why the remaining visible board uniquely determines the final undealt card when all previously claimed sets were valid. ### Source excerpt Tai-Danae Bradley is one of the hosts of PBS Infinite Series, a delightful series of vignettes into fun parts of math. The video below is about the same of SET, a favorite among mathematicians. Specifically, Tai-Danae explains how SET cards lie in (using more technical jargon) a vector space over a finite field, and that valid sets correspond to lines. If you don't immediately know how this would work, watch the video. ## The Inner Product as a Decision Rule DevFeed: [The Inner Product as a Decision Rule]() Original publisher: [Read original article]() Published: 2017-05-22T08:00:00Z Content type: tutorial Language: en Sources: [Jeremy Kun]() Topics: [math](), [Code]() Tags: [arithmetic](), [inner-product](), [javascript](), [linear-algebra](), [machine-learning](), [math](), [mathematics](), [programming](), [projection](), [support-vector-machines](), [vectors]() ### AI overview This article explains how the standard inner product, or dot product, acts as a geometric decision rule. It shows how the sign of the product determines which side of a line through the origin a vector lies on, while zero indicates that the vector lies on the line. The explanation connects this behavior to vector projection. ### Source excerpt The standard inner product of two vectors has some nice geometric properties. Given two vectors $ x, y \in \mathbb{R}^n$, where by $ x_i$ I mean the $ i$-th coordinate of $ x$, the standard inner product (which I will interchangeably call the dot product) is defined by the formula $$\displaystyle \langle x, y \rangle = x_1 y_1 + \dots + x_n y_n$$ This formula, simple as it is, produces a lot of interesting geometry. ## Singular Value Decomposition Part 1: Perspectives on Linear Algebra DevFeed: [Singular Value Decomposition Part 1: Perspectives on Linear Algebra]() Original publisher: [Read original article]() Published: 2016-04-18T09:00:40Z Content type: article Language: en Sources: [Jeremy Kun]() Topics: [Data analysis](), [Computer science](), [Statistics](), [Optimization]() Tags: [algorithms](), [data-analysis](), [linear-algebra](), [mathematics](), [matrix](), [optimization](), [regression](), [singular-value-decomposition](), [statistics]() ### AI overview This first post in a two-part series motivates and introduces singular value decomposition (SVD). It explains how matrices can represent both linear transformations and organized data, and outlines the role of SVD in applications such as regression, prediction, and approximate optimization solutions. ### Source excerpt The singular value decomposition (SVD) of a matrix is a fundamental tool in computer science, data analysis, and statistics. It's used for all kinds of applications from regression to prediction, to finding approximate solutions to optimization problems. In this series of two posts we'll motivate, define, compute, and use the singular value decomposition to analyze some data. (Jump to the second post) I want to spend the first post entirely on motivation and background. ## Tensorphobia and the Outer Product DevFeed: [Tensorphobia and the Outer Product]() Original publisher: [Read original article]() Published: 2016-03-28T09:00:46Z Content type: article Language: en Sources: [Jeremy Kun]() Topics: [math](), [Programming]() Tags: [linear-algebra](), [math](), [mathematics](), [matrices](), [matrix](), [primer](), [tensors]() ### AI overview An explanation of the outer product of vectors, connecting modern tensor concepts with practical linear algebra. The article develops why the construction of two vectors as a linear map should be understood as natural or canonical. ### Source excerpt Variations on a theme Back in 2014 I wrote a post called How to Conquer Tensorphobia that should end up on Math $ \cap$ Programming's "greatest hits" album. One aspect of tensors I neglected to discuss was the connection between the modern views of tensors and the practical views of linear algebra. I feel I need to write this because every year or two I forget why it makes sense. ## The Čech Complex and the Vietoris-Rips Complex DevFeed: [The Čech Complex and the Vietoris-Rips Complex]() Original publisher: [Read original article]() Published: 2015-08-06T09:00:00Z Content type: article Language: en Sources: [Jeremy Kun]() Topics: [data](), [datasets](), [Point cloud](), [Computing]() Tags: [approximation](), [cech-complex](), [complex](), [computational-topology](), [data-mining](), [data-science](), [dataset](), [homology](), [linear-algebra](), [mathematics](), [persistent-homology](), [point](), [points](), [simplicial-complex](), [vietoris-rips-complex]() ### AI overview This article introduces computational topology for analyzing the shape of data. It explains how point clouds can be converted into simplicial complexes so homology and persistent homology can identify qualitative features such as connected components and holes, with some resistance to noise. ### Source excerpt It's about time we got back to computational topology. Previously in this series we endured a lightning tour of the fundamental group and homology, then we saw how to compute the homology of a simplicial complex using linear algebra. What we really want to do is talk about the inherent shape of data. Homology allows us to compute some qualitative features of a given shape, i.e., find and count the number of connected components or a given shape, or the number of "2-dimensional holes" it has. ## Hamming's Code DevFeed: [Hamming's Code]() Original publisher: [Read original article]() Published: 2015-03-02T09:00:00Z Content type: tutorial Language: en Sources: [Jeremy Kun]() Topics: [Code](), [Encoding](), [digital]() Tags: [coding-theory](), [compression](), [computing](), [encoding](), [error](), [errors](), [finite-fields](), [hamming](), [hamming-code](), [hypercube](), [information-theory](), [linear-algebra](), [linear-codes](), [np-hard](), [reed-solomon-codes](), [solved](), [transmission]() ### AI overview This tutorial introduces Hamming codes as efficiently computable encoding schemes for detecting and correcting errors caused by noise during digital transmission. It defines a code as a subset of binary strings and relates codewords to an injective encoding function. ### Source excerpt Or how to detect and correct errors Last time we made a quick tour through the main theorems of Claude Shannon, which essentially solved the following two problems about communicating over a digital channel. What is the best encoding for information when you are guaranteed that your communication channel is error free? Are there any encoding schemes that can recover from random noise introduced during transmission? The answers to these questions were purely mathematical theorems, of course. ## Multiple Qubits and the Quantum Circuit DevFeed: [Multiple Qubits and the Quantum Circuit]() Original publisher: [Read original article]() Published: 2015-01-26T09:00:00Z Content type: article Language: en Sources: [Jeremy Kun]() Topics: [circuit]() Tags: [2](), [bits](), [circuits](), [entanglement](), [linear-algebra](), [multiple](), [physics](), [quantum](), [quantum-computing](), [tensor-product](), [tensors]() ### AI overview This article explains why the tensor product is the natural mathematical representation of the joint state of multiple qubits. It also introduces basic quantum gates and the definition of a quantum circuit. ### Source excerpt Last time we left off with the tantalizing question: how do you do a quantum "AND" operation on two qubits? In this post we'll see why the tensor product is the natural mathematical way to represent the joint state of multiple qubits. Then we'll define some basic quantum gates, and present the definition of a quantum circuit. Working with Multiple Qubits In a classical system, if you have two bits with values $ b_1, b_2$, then the "joint state" of the two bits is given by the concatenated string $ b_1b_2$. ## The Quantum Bit DevFeed: [The Quantum Bit]() Original publisher: [Read original article]() Published: 2014-12-15T10:00:52Z Content type: tutorial Language: en Sources: [Jeremy Kun]() Topics: [Quantum Computing](), [Computing](), [circuit]() Tags: [bits](), [circuit](), [circuits](), [complex-numbers](), [computing](), [linear-algebra](), [mathematics](), [measurement](), [quantum](), [quantum-computing](), [quantum-mechanics](), [qubit](), [unitary-matrices]() ### AI overview An introduction to quantum computing that extends classical circuit concepts to qubits. It defines a qubit as a unit vector in the complex plane of two dimensions and explains why extracting information from qubits differs from reading classical bits. ### Source excerpt The best place to start our journey through quantum computing is to recall how classical computing works and try to extend it. Since our final quantum computing model will be a circuit model, we should informally discuss circuits first. A circuit has three parts: the "inputs," which are bits (either zero or one); the "gates," which represent the lowest-level computations we perform on bits; and the "wires," which connect the outputs of gates to the inputs of other gates. ## 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). ## A Reminder of Lagrange Multipliers for Optimization Problems DevFeed: [A Reminder of Lagrange Multipliers for Optimization Problems]() Original publisher: [Read original article]() Published: 2013-11-30T09:00:50Z Content type: tutorial Language: en Sources: [Jeremy Kun]() Topics: [Optimization](), [function](), [Variable]() Tags: [convex-functions](), [function](), [gradient-descent](), [lagrange](), [lagrange-multipliers](), [linear-algebra](), [optimization](), [problems](), [variable](), [vectors]() ### AI overview A tutorial-style reminder of Lagrange multipliers for optimization problems. It reviews gradients, partial derivatives, dot products, and directional change for multivariable functions. ### Source excerpt For a while I've been meaning to do some more advanced posts on optimization problems of all flavors. One technique that comes up over and over again is Lagrange multipliers, so this post is going to be a leisurely reminder of that technique. I often forget how to do these basic calculus-type things, so it's good practice. We will assume something about the reader's knowledge, but it's a short list: know how to operate with vectors and the dot product, know how to take a partial derivative, and know that in single-variable calculus the local maxima and minima of a differentiable function $ f(x)$ occur when the derivative $ f'(x)$ vanishes. ## 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. ## K-Nearest-Neighbors and Handwritten Digit Classification DevFeed: [K-Nearest-Neighbors and Handwritten Digit Classification]() Original publisher: [Read original article]() Published: 2012-08-26T12:19:43Z Content type: tutorial Language: en Sources: [Jeremy Kun]() Topics: [Machine Learning & Artificial Intelligence](), [Data Science](), [Math and Logic]() Tags: [dimension](), [linear-algebra](), [machine-learning](), [mathematics](), [metric](), [process](), [programming](), [supervised-learning]() ### AI overview This tutorial introduces supervised classification in machine learning and explains how labeled data, training algorithms, models, and classification algorithms work together. It then examines the assumption that data points in the same class are close under an appropriate metric. ### Source excerpt The Recipe for Classification One important task in machine learning is to classify data into one of a fixed number of classes. For instance, one might want to discriminate between useful email and unsolicited spam. Or one might wish to determine the species of a beetle based on its physical attributes, such as weight, color, and mandible length. These "attributes" are often called "features" in the world of machine learning, and they often correspond to dimensions when interpreted in the framework of linear algebra. ## Generalized Functions -- A Primer DevFeed: [Generalized Functions -- A Primer]() Original publisher: [Read original article]() Published: 2012-06-06T20:57:05Z Content type: tutorial Language: en Sources: [Jeremy Kun]() Topics: [Mathematics](), [Math and Logic](), [abstraction](), [Programming]() Tags: [abstraction](), [fourier-analysis](), [fourier-transform](), [functional-analysis](), [generalized-functions](), [linear-algebra](), [mathematics](), [primer](), [programming](), [theory]() ### AI overview This primer develops a more rigorous mathematical framework for Fourier transforms. It motivates the search for a class of functions that remains well behaved under the Fourier transform and its inverse, while avoiding divergent integrals. ### Source excerpt Last time we investigated the naive (which I'll henceforth call "classical") notion of the Fourier transform and its inverse. While the development wasn't quite rigorous, we nevertheless discovered elegant formulas and interesting properties that proved useful in at least solving differential equations. Of course, we wouldn't be following this trail of mathematics if it didn't result in some worthwhile applications to programming. While we'll get there eventually, this primer will take us deeper down the rabbit hole of abstraction. ## Double Angle Trigonometric Formulas DevFeed: [Double Angle Trigonometric Formulas]() Original publisher: [Read original article]() Published: 2012-05-19T23:28:53Z Content type: tutorial Language: en Sources: [Jeremy Kun]() Topics: [Mathematics](), [Math and Logic]() Tags: [cos](), [double-angle-identities](), [geometric-transformations](), [linear-algebra](), [mathematics](), [matrices](), [matrix](), [rotation](), [trigonometry](), [vector]() ### AI overview This tutorial derives the double-angle sine and cosine identities by representing planar rotations with a 2x2 matrix, squaring that matrix, and comparing it with a rotation by 2θ. ### Source excerpt Problem: Derive the double angle identities $$\sin(2\theta) = 2\sin(\theta)\cos(\theta)\\\ \cos(2\theta) = \cos^2(\theta) - \sin^2(\theta)$$ Solution: Recall from linear algebra how one rotates a point in the plane. The matrix of rotation (derived by seeing where $ (1,0)$ and $ (0,1)$ go under a rotation by $ \theta$, and writing those coordinates in the columns) is $$A = \begin{pmatrix} \cos(\theta) & -\sin(\theta) \\\ \sin(\theta) & \cos(\theta) \end{pmatrix}$$ Next, note that to rotate a point twice by $ \theta$, we simply multiply the point (as a vector) by $ A$ twice. ## 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. [Next page]()