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Research draft

linear map

vr.tr.linear-map · XCT.QLT

Let an agent handle linear maps by definition, representation as matrices, properties and applications.

Thing Registry Cross-cutting context

Research draft, second pass

A second pass drafted this model: the structure a model of this thing needs, and what is known about it in the world. The line under this one says how the second half was obtained - researched against sources, or recalled without web access, in which case nothing here was read anywhere and every claim is a lead to verify. Unreviewed either way.

written by Claude from model knowledge without web access - no source was read, every claim is a lead to verify

Researched by: Claude

Purpose and description

Let an agent handle linear maps by definition, representation as matrices, properties and applications.

A function between vector spaces that preserves addition and scalar multiplication, such as rotations, projections and matrix transformations, and in analysis operators such as integral transforms and the Fourier transform.

What it is for: Linear algebra, physics, engineering, computer graphics and data science.

It can be test whether a map is linear; represent linear maps as matrices; compute kernels, images and eigenvalues; apply transformations in graphics and signals.

Distinguishing features

Preserves linear structure

Represented by matrices in finite dimensions

Has kernel and image

Central to linear algebra

What it looks like

Written as functions or matrices.

How it is recognised

T(u + v) = T(u) + T(v) and T(cu) = cT(u)

Matrix representations

Affine maps add a translation and are not linear

Related models

is a kind of - category

function and additive map

is represented by - representation

matrix

has - properties

eigenvalue and kernel

is used in - applications

computer graphics and quantum mechanics

In practice

Families and kinds

linear transformations of finite-dimensional spaces

linear operators

integral transforms

projections and reflections

Householder transformations

Standards and regulation

ISO 80000-2 notation

Failure modes and hazards

Confusing linear and affine maps

Basis-dependent errors

Also called

quantum Fourier transformHouseholder transformationintegral transformHecke operatorHodge star operatorpseudo-differential operatorlinear operatoraffine tensorunitary transformationlinear functionaltranspose of a linear maplinear endomorphismWeingarten operatororthogonal transformationDelta operatorpseudoreflectionsqueeze mappinglinear isomorphismvectorizationsine and cosine transformscontinuous wavelet transformJacobi transformoscillatory integral operatorintegral operatorLinear canonical transformationfractional derivativeshape operatoralgebraic operatorEigenoperatordiscontinuous linear operatorantisymmetrizercomposition operatordensely defined operatormultiplication operatorpositive linear functionalsingular integral operator on a closed curveclosed operator

Where this came from

wikidata · CC0 1.0

Source repository Registry entry point

Drafted structure

Bundle to layer to finding to question, as the second pass will find it: 4 bundles · 8 layers · 8 findings · 16 questions.

Definition Is it linear.

Tests decide.

Test

Linearity.

Test

Linearity test.

  1. Does the map preserve addition and scalar multiplication? boundary
  2. Can a counterexample be given? action

Spaces

Domain and codomain.

Spaces

Vector spaces.

  1. Between which vector spaces does it map? definition
  2. What are their dimensions? measurement
Representation Matrices.

Bases matter.

Matrix

In a basis.

Matrix

Matrix representation.

  1. What is the matrix of the map in this basis? measurement
  2. How does it change with a change of basis? definition

Invariants

Rank and determinant.

Invariants

Invariants.

  1. What are its rank, determinant and trace? measurement
  2. What do they tell us? definition
Structure Kernel and eigenvalues.

Structure reveals behaviour.

Kernel

Kernel and image.

Kernel

Kernel and image.

  1. What are the kernel and image? definition
  2. Is the map injective or surjective? boundary

Eigen

Eigenvalues.

Eigen

Eigenvalues.

  1. What are its eigenvalues and eigenvectors? measurement
  2. Is it diagonalisable? boundary
Applications Uses.

Linear maps are everywhere.

Graphics

Transformations.

Graphics

Graphics.

  1. How are rotations and scalings used in computer graphics? definition
  2. Why are homogeneous coordinates used for translations? definition

Signals

Transforms.

Signals

Signal processing.

  1. How is the Fourier transform a linear map? definition
  2. What does that imply? definition

What the second pass must settle

  • Should linear operators be a separate entry?
  • How should matrix representations be linked?
  • How should infinite-dimensional cases be handled?
Draft as JSON