Matrix Algebra Basics || Matrix Algebra for Beginners

In mathematics, a matrix is a rectangular array or table of numbers, symbols, or expressions, arranged in rows and columns.

This course is about basics of matrix algebra.

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Matrix Algebra Basics || Matrix Algebra for Beginners
Python  Library

Python Library

1657400640

Synapse: Matrix Homeserver Written in Python 3/Twisted

Introduction

Matrix is an ambitious new ecosystem for open federated Instant Messaging and VoIP. The basics you need to know to get up and running are:

  • Everything in Matrix happens in a room. Rooms are distributed and do not exist on any single server. Rooms can be located using convenience aliases like #matrix:matrix.org or #test:localhost:8448.
  • Matrix user IDs look like @matthew:matrix.org (although in the future you will normally refer to yourself and others using a third party identifier (3PID): email address, phone number, etc rather than manipulating Matrix user IDs)

The overall architecture is:

client <----> homeserver <=====================> homeserver <----> client
       https://somewhere.org/_matrix      https://elsewhere.net/_matrix

#matrix:matrix.org is the official support room for Matrix, and can be accessed by any client from https://matrix.org/docs/projects/try-matrix-now.html or via IRC bridge at irc://irc.libera.chat/matrix.

Synapse is currently in rapid development, but as of version 0.5 we believe it is sufficiently stable to be run as an internet-facing service for real usage!

About Matrix

Matrix specifies a set of pragmatic RESTful HTTP JSON APIs as an open standard, which handle:

  • Creating and managing fully distributed chat rooms with no single points of control or failure
  • Eventually-consistent cryptographically secure synchronisation of room state across a global open network of federated servers and services
  • Sending and receiving extensible messages in a room with (optional) end-to-end encryption
  • Inviting, joining, leaving, kicking, banning room members
  • Managing user accounts (registration, login, logout)
  • Using 3rd Party IDs (3PIDs) such as email addresses, phone numbers, Facebook accounts to authenticate, identify and discover users on Matrix.
  • Placing 1:1 VoIP and Video calls

These APIs are intended to be implemented on a wide range of servers, services and clients, letting developers build messaging and VoIP functionality on top of the entirely open Matrix ecosystem rather than using closed or proprietary solutions. The hope is for Matrix to act as the building blocks for a new generation of fully open and interoperable messaging and VoIP apps for the internet.

Synapse is a Matrix "homeserver" implementation developed by the matrix.org core team, written in Python 3/Twisted.

In Matrix, every user runs one or more Matrix clients, which connect through to a Matrix homeserver. The homeserver stores all their personal chat history and user account information - much as a mail client connects through to an IMAP/SMTP server. Just like email, you can either run your own Matrix homeserver and control and own your own communications and history or use one hosted by someone else (e.g. matrix.org) - there is no single point of control or mandatory service provider in Matrix, unlike WhatsApp, Facebook, Hangouts, etc.

We'd like to invite you to join #matrix:matrix.org (via https://matrix.org/docs/projects/try-matrix-now.html), run a homeserver, take a look at the Matrix spec, and experiment with the APIs and Client SDKs.

Thanks for using Matrix!

Support

For support installing or managing Synapse, please join #synapse:matrix.org (from a matrix.org account if necessary) and ask questions there. We do not use GitHub issues for support requests, only for bug reports and feature requests.

Synapse's documentation is nicely rendered on GitHub Pages, with its source available in docs.

Synapse Installation

Connecting to Synapse from a client

The easiest way to try out your new Synapse installation is by connecting to it from a web client.

Unless you are running a test instance of Synapse on your local machine, in general, you will need to enable TLS support before you can successfully connect from a client: see TLS certificates.

An easy way to get started is to login or register via Element at https://app.element.io/#/login or https://app.element.io/#/register respectively. You will need to change the server you are logging into from matrix.org and instead specify a Homeserver URL of https://<server_name>:8448 (or just https://<server_name> if you are using a reverse proxy). If you prefer to use another client, refer to our client breakdown.

If all goes well you should at least be able to log in, create a room, and start sending messages.

Registering a new user from a client

By default, registration of new users via Matrix clients is disabled. To enable it, specify enable_registration: true in homeserver.yaml. (It is then recommended to also set up CAPTCHA - see docs/CAPTCHA_SETUP.md.)

Once enable_registration is set to true, it is possible to register a user via a Matrix client.

Your new user name will be formed partly from the server_name, and partly from a localpart you specify when you create the account. Your name will take the form of:

@localpart:my.domain.name

(pronounced "at localpart on my dot domain dot name").

As when logging in, you will need to specify a "Custom server". Specify your desired localpart in the 'User name' box.

Security note

Matrix serves raw, user-supplied data in some APIs -- specifically the content repository endpoints.

Whilst we make a reasonable effort to mitigate against XSS attacks (for instance, by using CSP), a Matrix homeserver should not be hosted on a domain hosting other web applications. This especially applies to sharing the domain with Matrix web clients and other sensitive applications like webmail. See https://developer.github.com/changes/2014-04-25-user-content-security for more information.

Ideally, the homeserver should not simply be on a different subdomain, but on a completely different registered domain (also known as top-level site or eTLD+1). This is because some attacks are still possible as long as the two applications share the same registered domain.

To illustrate this with an example, if your Element Web or other sensitive web application is hosted on A.example1.com, you should ideally host Synapse on example2.com. Some amount of protection is offered by hosting on B.example1.com instead, so this is also acceptable in some scenarios. However, you should not host your Synapse on A.example1.com.

Note that all of the above refers exclusively to the domain used in Synapse's public_baseurl setting. In particular, it has no bearing on the domain mentioned in MXIDs hosted on that server.

Following this advice ensures that even if an XSS is found in Synapse, the impact to other applications will be minimal.

Upgrading an existing Synapse

The instructions for upgrading synapse are in the upgrade notes. Please check these instructions as upgrading may require extra steps for some versions of synapse.

Using a reverse proxy with Synapse

It is recommended to put a reverse proxy such as nginx, Apache, Caddy, HAProxy or relayd in front of Synapse. One advantage of doing so is that it means that you can expose the default https port (443) to Matrix clients without needing to run Synapse with root privileges.

For information on configuring one, see docs/reverse_proxy.md.

Identity Servers

Identity servers have the job of mapping email addresses and other 3rd Party IDs (3PIDs) to Matrix user IDs, as well as verifying the ownership of 3PIDs before creating that mapping.

They are not where accounts or credentials are stored - these live on home servers. Identity Servers are just for mapping 3rd party IDs to matrix IDs.

This process is very security-sensitive, as there is obvious risk of spam if it is too easy to sign up for Matrix accounts or harvest 3PID data. In the longer term, we hope to create a decentralised system to manage it (matrix-doc #712), but in the meantime, the role of managing trusted identity in the Matrix ecosystem is farmed out to a cluster of known trusted ecosystem partners, who run 'Matrix Identity Servers' such as Sydent, whose role is purely to authenticate and track 3PID logins and publish end-user public keys.

You can host your own copy of Sydent, but this will prevent you reaching other users in the Matrix ecosystem via their email address, and prevent them finding you. We therefore recommend that you use one of the centralised identity servers at https://matrix.org or https://vector.im for now.

To reiterate: the Identity server will only be used if you choose to associate an email address with your account, or send an invite to another user via their email address.

Password reset

Users can reset their password through their client. Alternatively, a server admin can reset a users password using the admin API or by directly editing the database as shown below.

First calculate the hash of the new password:

$ ~/synapse/env/bin/hash_password
Password:
Confirm password:
$2a$12$xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx

Then update the users table in the database:

UPDATE users SET password_hash='$2a$12$xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx'
    WHERE name='@test:test.com';

Synapse Development

The best place to get started is our guide for contributors. This is part of our larger documentation, which includes information for synapse developers as well as synapse administrators.

Developers might be particularly interested in:

Alongside all that, join our developer community on Matrix: #synapse-dev:matrix.org, featuring real humans!

Quick start

Before setting up a development environment for synapse, make sure you have the system dependencies (such as the python header files) installed - see Platform-specific prerequisites.

To check out a synapse for development, clone the git repo into a working directory of your choice:

git clone https://github.com/matrix-org/synapse.git
cd synapse

Synapse has a number of external dependencies. We maintain a fixed development environment using Poetry. First, install poetry. We recommend:

pip install --user pipx
pipx install poetry

as described here. (See poetry's installation docs for other installation methods.) Then ask poetry to create a virtual environment from the project and install Synapse's dependencies:

poetry install --extras "all test"

This will run a process of downloading and installing all the needed dependencies into a virtual env.

We recommend using the demo which starts 3 federated instances running on ports 8080 - 8082:

poetry run ./demo/start.sh

(to stop, you can use poetry run ./demo/stop.sh)

See the demo documentation for more information.

If you just want to start a single instance of the app and run it directly:

# Create the homeserver.yaml config once
poetry run synapse_homeserver \
  --server-name my.domain.name \
  --config-path homeserver.yaml \
  --generate-config \
  --report-stats=[yes|no]

# Start the app
poetry run synapse_homeserver --config-path homeserver.yaml

Running the unit tests

After getting up and running, you may wish to run Synapse's unit tests to check that everything is installed correctly:

poetry run trial tests

This should end with a 'PASSED' result (note that exact numbers will differ):

Ran 1337 tests in 716.064s

PASSED (skips=15, successes=1322)

For more tips on running the unit tests, like running a specific test or to see the logging output, see the CONTRIBUTING doc.

Running the Integration Tests

Synapse is accompanied by SyTest, a Matrix homeserver integration testing suite, which uses HTTP requests to access the API as a Matrix client would. It is able to run Synapse directly from the source tree, so installation of the server is not required.

Testing with SyTest is recommended for verifying that changes related to the Client-Server API are functioning correctly. See the SyTest installation instructions for details.

Platform dependencies

Synapse uses a number of platform dependencies such as Python and PostgreSQL, and aims to follow supported upstream versions. See the docs/deprecation_policy.md document for more details.

Troubleshooting

Need help? Join our community support room on Matrix: #synapse:matrix.org

Running out of File Handles

If synapse runs out of file handles, it typically fails badly - live-locking at 100% CPU, and/or failing to accept new TCP connections (blocking the connecting client). Matrix currently can legitimately use a lot of file handles, thanks to busy rooms like #matrix:matrix.org containing hundreds of participating servers. The first time a server talks in a room it will try to connect simultaneously to all participating servers, which could exhaust the available file descriptors between DNS queries & HTTPS sockets, especially if DNS is slow to respond. (We need to improve the routing algorithm used to be better than full mesh, but as of March 2019 this hasn't happened yet).

If you hit this failure mode, we recommend increasing the maximum number of open file handles to be at least 4096 (assuming a default of 1024 or 256). This is typically done by editing /etc/security/limits.conf

Separately, Synapse may leak file handles if inbound HTTP requests get stuck during processing - e.g. blocked behind a lock or talking to a remote server etc. This is best diagnosed by matching up the 'Received request' and 'Processed request' log lines and looking for any 'Processed request' lines which take more than a few seconds to execute. Please let us know at #synapse:matrix.org if you see this failure mode so we can help debug it, however.

Help!! Synapse is slow and eats all my RAM/CPU!

First, ensure you are running the latest version of Synapse, using Python 3 with a PostgreSQL database.

Synapse's architecture is quite RAM hungry currently - we deliberately cache a lot of recent room data and metadata in RAM in order to speed up common requests. We'll improve this in the future, but for now the easiest way to either reduce the RAM usage (at the risk of slowing things down) is to set the almost-undocumented SYNAPSE_CACHE_FACTOR environment variable. The default is 0.5, which can be decreased to reduce RAM usage in memory constrained enviroments, or increased if performance starts to degrade.

However, degraded performance due to a low cache factor, common on machines with slow disks, often leads to explosions in memory use due backlogged requests. In this case, reducing the cache factor will make things worse. Instead, try increasing it drastically. 2.0 is a good starting value.

Using libjemalloc can also yield a significant improvement in overall memory use, and especially in terms of giving back RAM to the OS. To use it, the library must simply be put in the LD_PRELOAD environment variable when launching Synapse. On Debian, this can be done by installing the libjemalloc1 package and adding this line to /etc/default/matrix-synapse:

LD_PRELOAD=/usr/lib/x86_64-linux-gnu/libjemalloc.so.1

This can make a significant difference on Python 2.7 - it's unclear how much of an improvement it provides on Python 3.x.

If you're encountering high CPU use by the Synapse process itself, you may be affected by a bug with presence tracking that leads to a massive excess of outgoing federation requests (see discussion). If metrics indicate that your server is also issuing far more outgoing federation requests than can be accounted for by your users' activity, this is a likely cause. The misbehavior can be worked around by setting the following in the Synapse config file:

presence:
    enabled: false

People can't accept room invitations from me

The typical failure mode here is that you send an invitation to someone to join a room or direct chat, but when they go to accept it, they get an error (typically along the lines of "Invalid signature"). They might see something like the following in their logs:

2019-09-11 19:32:04,271 - synapse.federation.transport.server - 288 - WARNING - GET-11752 - authenticate_request failed: 401: Invalid signature for server <server> with key ed25519:a_EqML: Unable to verify signature for <server>

This is normally caused by a misconfiguration in your reverse-proxy. See docs/reverse_proxy.md and double-check that your settings are correct.

Download Details:
Author: matrix-org
Source Code: https://github.com/matrix-org/synapse
License: Apache-2.0 license

#python

Algebra for Beginners | Basics of Algebra

Algebra is one of the broad parts of mathematics, together with number theory, geometry and analysis. In its most general form, algebra is the study of mathematical symbols and the rules for manipulating these symbols; it is a unifying thread of almost all of mathematics.

Table of Contents


  • Welcome to Algebra (0:00)
  • Numbers (natural, integer, rational, real, complex) (3:35)
  • Associative property of addition and multiplication (10:04)
  • Commutative property of addition and multiplication (11:52)
  • Cancelling fractions (15:02)
  • Multiplying fractions (21:35)
  • Subtraction (25:34)
  • Factoring a cubic polynomial (32:40)

#data-analysis #data-science #algebra #developer

Bongani  Ngema

Bongani Ngema

1645518900

Algebra: Algebraic Structures

algebra

means completeness and balancing, from the Arabic word الجبر 

Algebra OnQuaternionsAndOctonions

Table Of Contents

Features

Installation

With npm do

npm install algebra

or use a CDN adding this to your HTML page

<script src="https://unpkg.com/algebra/dist/algebra.min.js"></script>

Quick start

This is a 60 seconds tutorial to get your hands dirty with algebra.

NOTA BENE Imagine all code examples below as written in some REPL where expected output is documented as a comment.

All code in the examples below should be contained into a single file, like test/quickStart.js.

First of all, import algebra package.

const algebra = require('algebra')

Scalars

Use the Real numbers as scalars.

const R = algebra.Real

Every operator is implemented both as a static function and as an object method.

Static operators return raw data, while class methods return object instances.

Use static addition operator to add three numbers.

R.add(1, 2) // 3

Create two real number objects: x = 2, y = -2

// x will be overwritten, see below
let x = new R(2)
const y = new R(-2)

The value r is the result of x multiplied by y.

// 2 * (-2) = -4
const r = x.mul(y)

r // Scalar { data: -4 }

// x and y are not changed
x.data // 2
y.data // -2

Raw numbers are coerced, operators can be chained when it makes sense. Of course you can reassign x, for example, x value will be 0.1: x -> x + 3 -> x * 2 -> x ^-1

// ((2 + 3) * 2)^(-1) = 0.1
x = x.add(3).mul(2).inv()

x // Scalar { data: 0.1 }

Comparison operators equal and notEqual are available, but they cannot be chained.

x.equal(0.1) // true
x.notEqual(Math.PI) // true

You can also play with Complexes.

const C = algebra.Complex

let z1 = new C([1, 2])
const z2 = new C([3, 4])

z1 = z1.mul(z2)

z1 // Scalar { data: [-5, 10] }

z1 = z1.conj().mul([2, 0])

z1.data // [-10, -20]

Vectors

Create vector space of dimension 2 over Reals.

const R2 = algebra.VectorSpace(R)(2)

Create two vectors and add them.

let v1 = new R2([0, 1])
const v2 = new R2([1, -2])

// v1 -> v1 + v2 -> [0, 1] + [1, -2] = [1, -1]
v1 = v1.add(v2)

v1 // Vector { data: [1, -1] }

Matrices

Create space of matrices 3 x 2 over Reals.

const R3x2 = algebra.MatrixSpace(R)(3, 2)

Create a matrix.

//       | 1 1 |
//  m1 = | 0 1 |
//       | 1 0 |
//
const m1 = new R3x2([1, 1,
                     0, 1,
                     1, 0])

Multiply m1 by v1, the result is a vector v3 with dimension 3. In fact we are multiplying a 3 x 2 matrix by a 2 dimensional vector, but v1 is traited as a column vector so it is like a 2 x 1 matrix.

Then, following the row by column multiplication law we have

//  3 x 2  by  2 x 1  which gives a   3 x 1
//      ↑      ↑
//      +------+----→  by removing the middle indices.
//
//                   | 1 1 |
//    v3 = m1 * v1 = | 0 1 | * [1 , -1] = [0, -1, 1]
//                   | 1 0 |

const v3 = m1.mul(v1)

v3.data // [0, -1, 1]

Let's try with two square matrices 2 x 2.

const R2x2 = algebra.MatrixSpace(R)(2, 2)

let m2 = new R2x2([1, 0,
                   0, 2])

const m3 = new R2x2([0, -1,
                     1,  0])

m2 = m2.mul(m3)

m2 // Matrix { data: [0, -1, 2, 0] }

Since m2 is a square matrix we can calculate its determinant.

m2.determinant // Scalar { data: 2 }

API

About operators

All operators can be implemented as static methods and as object methods. In both cases, operands are coerced to raw data. As an example, consider addition of vectors in a Cartesian Plane.

const R2 = algebra.R2

const vector1 = new R2([1, 2])
const vector2 = new R2([3, 4])

The following static methods, give the same result: [4, 6].

R2.addition(vector1, [3, 4])
R2.addition([1, 2], vector2)
R2.addition(vector1, vector2)

The following object methods, give the same result: a vector instance with data [4, 6].

const vector3 = vector1.addition([3, 4])
const vector4 = vector1.addition(vector2)

R2.equal(vector3, vector4) // true

Operators can be chained when it makes sense.

vector1.addition(vector1).equality([2, 4]) // true

Objects are immutable

vector1.data // still [1, 2]

Scalar

The scalars are the building blocks, they are the elements you can use to create vectors and matrices. They are the underneath set enriched with a ring structure which consists of two binary operators that generalize the arithmetic operations of addition and multiplication. A ring that has the commutativity property is called abelian (in honour to Abel) or also a field.

Ok, let's make a simple example. Real numbers, with common addition and multiplication are a scalar field.

The good new is that you can create any scalar field as long as you provide a set with two internal operations and related neutral elements that satisfy the ring axioms.

We are going to create a scalar field using BigInt elements to implement something similar to a Rational Number. The idea is to use a couple of numbers, the first one is the numerator and the second one the denominator.

Arguments we need are the same as algebra-ring. Let's start by unities; every element is a couple of numbers, the numerator and the denominator, hence unitites are:

  • zero: [ BigInt(0), BigInt(1) ]
  • one: [ BigInt(1), BigInt(1) ]

We need a function that computes the Great Common Divisor.

function greatCommonDivisor (a, b) {
  if (b === BigInt(0)) {
    return a
  } else {
    return greatCommonDivisor(b, a % b)
  }
}

So now we can normalize a rational number, by removing the common divisors of numerator and denominator. Also, denominator should be positive.

function normalizeRational ([numerator, denominator]) {
  const divisor = greatCommonDivisor(numerator, denominator)

  const sign = denominator > 0 ? 1 : -1

  return denominator > 0 ? (
    [numerator / divisor, denominator / divisor]
  ) : (
    [-numerator / divisor, -denominator / divisor]
  )
}
const Rational = algebra.Scalar({
  zero: [BigInt(0), BigInt(1)],
  one: [BigInt(1), BigInt(1)],
  equality: ([n1, d1], [n2, d2]) => (n1 * d2 === n2 * d1),
  contains: ([n, d]) => (typeof n === 'bigint' && typeof d === 'bigint'),
  addition: ([n1, d1], [n2, d2]) => normalizeRational([n1 * d2 + n2 * d1, d1 * d2]),
  negation: ([n, d]) => ([-n, d]),
  multiplication: ([n1, d1], [n2, d2]) => normalizeRational([n1 * n2, d1 * d2]),
  inversion: ([n, d]) => ([d, n])
})

So far so good, algebra dependencies will do some checks under the hood and will complain if something looks wrong.

Let's create few rational numbers.

const bigHalf = new Rational([BigInt(1), BigInt(2)])
const bigTwo = new Rational([BigInt(2), BigInt(1)])

Scalar.one

Is the neutral element for multiplication operator.

Rational.one // [1n, 1n]

Scalar.zero

Is the neutral element for addition operator.

Rational.zero // [0n, 1n]

scalar.data

The data attribute holds the raw data underneath our scalar instance.

bigHalf.data // [1n, 2n]

Scalar.contains(scalar)

Checks a given argument is contained in the scalar field that was defined.

Rational.contains(bigHalf) // true
Rational.contains([BigInt(1), BigInt(2)]) // true

scalar1.belongsTo(Scalar)

This is a class method that checks a scalar instance is contained in the given scalar field.

bigHalf.belongsTo(Rational) // true

Scalar.equality(scalar1, scalar2)

Rational.equality(bigHalf, [BigInt(5), BigInt(10)]) // true
Rational.equality(bigTwo, [BigInt(-4), BigInt(-2)]) // true

scalar1.equals(scalar2)

bigHalf.equals([BigInt(2), BigInt(4)])

Scalar.disequality(scalar1, scalar2)

Rational.disequality(bigHalf, bigTwo) // true

scalar1.disequality(scalar2)

bigHalf.disequality(bigTwo) // true

Scalar.addition(scalar1, scalar2)

Rational.addition(bigHalf, bigTwo) // [5n , 2n]

scalar1.addition(scalar2)

bigHalf.addition(bigTwo) // Scalar { data: [5n, 2n] }

Scalar.subtraction(scalar1, scalar2)

Rational.subtraction(bigTwo, bigHalf) // [3n , 2n]

scalar1.subtraction(scalar2)

bigTwo.multiplication(bigHalf) // Scalar { data: [1n, 1n] }

Scalar.multiplication(scalar1, scalar2)

Rational.multiplication(bigHalf, bigTwo) // [1n, 1n]

scalar1.multiplication(scalar2)

bigHalf.multiplication(bigTwo) // Scalar { data: [1n, 1n] }

Scalar.division(scalar1, scalar2)

Rational.division(bigTwo, bigHalf) // [1n, 4n]

scalar1.division(scalar2)

bigHalf.division(bigTwo) // Scalar { data: [1n, 4n] }

Scalar.negation(scalar)

Rational.negation(bigTwo) // [-2n, 1n]

scalar.negation()

bigTwo.negation() // Scalar { data: [-2n, 1n] }

Scalar.inversion(scalar)

Rational.inversion(bigTwo) // [1n, 2n]

scalar.inversion()

bigTwo.inversion() // Scalar { data: [1n, 2n] }

Real

Inherits everything from Scalar. Implements algebra of real numbers.

const Real = algebra.Real

Real.addition(1, 2) // 3

const pi = new Real(Math.PI)
const twoPi = pi.mul(2)

Real.subtraction(twoPi, 2 * Math.PI) // 0

Composition Algebra

A composition algebra is one of ℝ, ℂ, ℍ, O: Real, Complex, Quaternion, Octonion. A generic function is provided to iterate the Cayley-Dickson construction over any field.

CompositionAlgebra(field[, num])

  • num can be 1, 2, 4 or 8

Let's use for example the algebra.Boole which implements Boolean Algebra by exporting an object with all the stuff needed by algebra-ring npm package.

const CompositionAlgebra = algebra.CompositionAlgebra

const Boole = algebra.Boole

const Bit = CompositionAlgebra(Boole)

Bit.contains(false) // true
Bit.contains(4) // false

const bit = new Bit(true)
Bit.addition(false).data // true

Not so exciting, let's build something more interesting. Let's pass a second parameter, that is used to build a Composition algebra over the given field. It is something experimental also for me, right now I am writing this but I still do not know how it will behave. My idea (idea feliz) is that

A byte is an octonion of bits

Maybe we can discover some new byte operator, taken from octonion rich algebra structure. Create an octonion algebra over the binary field, a.k.a Z2 and create the eight units.

// n must be a power of two
const Byte = CompositionAlgebra(Boole, 8)

// Use a single char const for better indentation.
const t = true
const f = false

const byte1 = new Byte([t, f, f, f, f, f, f, f])
const byte2 = new Byte([f, t, f, f, f, f, f, f])
const byte3 = new Byte([f, f, t, f, f, f, f, f])
const byte4 = new Byte([f, f, f, t, f, f, f, f])
const byte5 = new Byte([f, f, f, f, t, f, f, f])
const byte6 = new Byte([f, f, f, f, f, t, f, f])
const byte7 = new Byte([f, f, f, f, f, f, t, f])
const byte8 = new Byte([f, f, f, f, f, f, f, t])

The first one corresponds to one, while the rest are immaginary units. Every imaginary unit multiplied by itself gives -1, but since the underlying field is homomorphic to Z2, -1 corresponds to 1.

byte1.mul(byte1).data // [t, f, f, f, f, f, f, f]
byte2.mul(byte2).data // [t, f, f, f, f, f, f, f]
byte3.mul(byte3).data // [t, f, f, f, f, f, f, f]
byte4.mul(byte4).data // [t, f, f, f, f, f, f, f]
byte5.mul(byte5).data // [t, f, f, f, f, f, f, f]
byte6.mul(byte6).data // [t, f, f, f, f, f, f, f]
byte7.mul(byte7).data // [t, f, f, f, f, f, f, f]
byte8.mul(byte8).data // [t, f, f, f, f, f, f, f]

You can play around with this structure.

const max = byte1.add(byte2).add(byte3).add(byte4)
                 .add(byte5).add(byte6).add(byte7).add(byte8)

max.data // [t, t, t, t, t, t, t, t]

Complex

Inherits everything from Scalar.

It is said the Gauss brain is uncommonly big and folded, much more than the Einstein brain (both are conserved and studied). Gauss was one of the biggest mathematicians and discovered many important results in many mathematic areas. One of its biggest intuitions, in my opinion, was to realize that the Complex number field is geometrically a plane. The Complex numbers are an extension on the Real numbers, they have a real part and an imaginary part. The imaginary numbers, as named by Descartes later, were discovered by italian mathematicians Cardano, Bombelli among others as a trick to solve third order equations.

Complex numbers are a goldmine for mathematics, they are incredibly rich of deepest beauty: just as a divulgative example, take a look to the Mandelbrot set, but please trust me, this is nothing compared to the divine nature of Complex numbers.

Mandelbrot Set{:.responsive}

The first thing I noticed when I started to study the Complex numbers is conjugation. Every Complex number has its conjugate, that is its simmetric counterparte respect to the Real numbers line.

const Complex = algebra.Complex

const complex1 = new Complex([1, 2])

complex1.conjugation() // Complex { data: [1, -2] }

Quaternion

Inherits everything from Scalar.

Quaternions are not commutative, usually if you invert the operands in a multiplication you get the same number in absolute value but with the sign inverted.

const Quaternion = algebra.Quaternion

const j = new Quaternion([0, 1, 0, 0])
const k = new Quaternion([0, 0, 1, 0])

// j * k = - k * j
j.mul(k).equal(k.mul(j).neg()) // true

Octonion

Inherits everything from Scalar.

Octonions are not associative, this is getting hard: a * (b * c) could be equal to the negation of (a * b) * c.

const Octonion = algebra.Octonion

const a = new Octonion([0, 1, 0, 0, 0, 0, 0, 0])
const b = new Octonion([0, 0, 0, 0, 0, 1, 0, 0])
const c = new Octonion([0, 0, 0, 1, 0, 0, 0, 0])

// a * ( b * c )
const abc1 = a.mul(b.mul(c)) // Octonion { data: [0, 0, 0, 0, 0, 0, 0, -1] }

// (a * b) * c
const abc2 = a.mul(b).mul(c) // Octonion { data: [0, 0, 0, 0, 0, 0, 0, 1] }

Octonion.equality(Octonion.negation(abc1), abc2)

Common spaces

R

The real line.

It is in alias of Real.

const R = algebra.R

R2

The Cartesian Plane.

const R2 = algebra.R2

It is in alias of VectorSpace(Real)(2).

R3

The real space.

const R3 = algebra.R3

It is in alias of VectorSpace(Real)(3).

R2x2

Real square matrices of rank 2.

const R2x2 = algebra.R2x2

It is in alias of MatrixSpace(Real)(2).

C

The complex numbers.

It is in alias of Complex.

const C = algebra.C

C2x2

Complex square matrices of rank 2.

const C2x2 = algebra.C2x2

It is in alias of MatrixSpace(Complex)(2).

H

Usually it is used the H in honour of Sir Hamilton.

It is in alias of Quaternion.

const H = algebra.H

Vector

A Vector extends the concept of number, since it is defined as a tuple of numbers. For example, the Cartesian plane is a set where every point has two coordinates, the famous (x, y) that is in fact a vector of dimension 2. A Scalar itself can be identified with a vector of dimension 1.

We have already seen an implementation of the plain: R2.

If you want to find the position of an airplain, you need latitute, longitude but also altitude, hence three coordinates. That is a 3-ple, a tuple with three numbers, a vector of dimension 3.

An implementation of the vector space of dimension 3 over reals is given by R3.

VectorSpace(Scalar)(dimension)

Vector dimension

Strictly speaking, dimension of a Vector is the number of its elements.

Vector.dimension

It is a static class attribute.

R2.dimension // 2
R3.dimension // 3

vector.dimension

It is also defined as a static instance attribute.

const vector = new R2([1, 1])

vector.dimension // 2

Vector norm

The norm, at the end, is the square of the vector length: the good old Pythagorean theorem. It is usually defined as the sum of the squares of the coordinates. Anyway, it must be a function that, given an element, returns a positive real number. For example in Complex numbers it is defined as the multiplication of an element and its conjugate: it works as a well defined norm. It is a really important property since it shapes a metric space. In the Euclidean topology it gives us the common sense of space, but it is also important in other spaces, like a functional space. In fact a norm gives us a distance defined as its square root, thus it defines a metric space and hence a topology: a lot of good stuff.

Vector.norm()

Is a static operator that returns the square of the lenght of the vector.

R2.norm([3, 4]).data // 25

vector.norm

This implements a static attribute that returns the square of the length of the vector instance.

const vector = new R2([1, 2])

vector.norm.data // 5

Vector addition

Vector.addition(vector1, vector2)

R2.addition([2, 1], [1, 2]) // [3, 3]

vector1.addition(vector2)

const vector1 = new R2([2, 1])
const vector2 = new R2([2, 2])

const vector3 = vector1.addition(vector2)

vector3 // Vector { data: [4, 3] }

Vector cross product

It is defined only in dimension three. See Cross product on wikipedia.

Vector.crossProduct(vector1, vector2)

R3.crossProduct([3, -3, 1], [4, 9, 2]) // [-15, 2, 39]

vector1.crossProduct(vector2)

const vector1 = new R3([3, -3, 1])
const vector2 = new R3([4, 9, 2])

const vector3 = vector1.crossProduct(vector2)

vector3 // Vector { data: [-15, 2, 39] }

Matrix

MatrixSpace(Scalar)(numRows[, numCols])

Matrix.isSquare

Matrix.numCols

Matrix.numRows

Matrix multiplication

Matrix.multiplication(matrix1, matrix2)

matrix1.multiplication(matrix2)

Matrix inversion

It is defined only for square matrices which determinant is not zero.

Matrix.inversion(matrix)

matrix.inversion

Matrix determinant

It is defined only for square matrices.

Matrix.determinant(matrix)

matrix.determinant

Matrix adjoint

Matrix.adjoint(matrix1)

matrix.adjoint

Author: Fibo
Source Code: https://github.com/fibo/algebra 
License: MIT License

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