Linear map¶

A function between vector spaces that preserves vector addition and scalar multiplication, equivalently preserving every finite linear combination.

Version: v2 · 2026-09-19 · History
Domain-specific #: 5342
Domain group: Formal Sciences
Origin domain: Mathematics
Subdomains: Linear Algebra, Vector Space Morphisms → Mathematics

Core Idea¶

A map T:V→W is linear when T(u+v)=T(u)+T(v) and T(av)=aT(v) for all vectors and scalars. Preservation of the two vector-space operations makes T determined by its values on a basis and representable by matrix multiplication after bases are chosen. The abstraction is therefore identified by a declared carrier, a transformation or constraint over that carrier, and an invariant that tells an analyst whether the named structure is genuinely present.

How would you explain it like I'm…

The Fair Arrow Machine

Imagine a machine that changes arrows. A Linear map is a fair machine: if you join two arrows tip to tail and then feed them in, you get the same result as feeding each arrow in alone and then joining what comes out. And if you make an arrow twice as long first, the result also comes out twice as long.

Add-and-Stretch Rule

Vectors are things you can add together and stretch by numbers, like arrows. A Linear map is a rule that turns vectors into other vectors while keeping those two operations working. Adding first and then mapping gives the same answer as mapping first and then adding. Stretching first and then mapping gives the same answer as mapping first and then stretching. Because of this, if you know what the map does to a few building-block arrows, you know what it does to every arrow.

Structure-Preserving Vector Map

A Linear map T from a vector space V to a vector space W satisfies T(u + v) = T(u) + T(v) and T(av) = aT(v) for all vectors u, v and scalars a. Together these mean T preserves every linear combination. As a result, T is completely determined by what it does to a basis of V, and once bases are chosen, T can be written as a matrix that acts by multiplication. The matrix depends on the bases, but the map itself doesn't. Some properties are basis-free, such as its kernel (the vectors sent to zero) and its image (everything it reaches). Something like f(x) = x + 1 is not linear in this sense, because it fails T(0) = 0.

A Linear map is a function T: V → W between vector spaces over the same field satisfying additivity, T(u + v) = T(u) + T(v), and homogeneity, T(av) = aT(v). Equivalently, it preserves all linear combinations, which makes it the structure-preserving morphism of vector spaces. Fixing a basis of V shows that T is determined by the images of the basis vectors. Fixing bases of both V and W gives a matrix representation in which applying T becomes matrix multiplication. The intrinsic content is basis-independent: kernel, image, and the relations between them, such as rank–nullity in finite dimensions. To check linearity, you must specify the spaces and the scalar field and verify both laws. Affine maps with a nonzero translation, and maps that respect only addition over a different field, are neighboring objects, not instances.

Scope of Application¶

Linear map belongs to linear algebra and is useful where the analyst can specify vector spaces over a common field, a function between them, vector addition, scalar multiplication, kernels, images, and optional bases or matrices, then evaluate the map preserves all linear combinations under the same scalar field. The scope is broad within that domain but bounded by the need for the map preserves all linear combinations under the same scalar field. The entry records a descriptive analytical identity; practical use requires the governing domain's evidence, standards, and safety obligations.

Clarity¶

The abstraction clarifies a crowded vocabulary by making the map preserves all linear combinations under the same scalar field the center of the account. A claim should name the carrier, the governing operation or relation, the applicable assumptions, and the recognition test. A bare label is insufficient because the name Linear map can be used for a formal identity, an implementation, or a neighboring result unless carrier and convention are stated.

Manages Complexity¶

Without the abstraction, an analyst must reason directly over many local details: the carrier roles, admissibility assumptions, competing conventions, derived invariants, boundary cases, and proof or validation obligations specific to Linear map. Linear map compresses them into the roles in the structural signature. That compression permits comparison across instances without erasing the variables that determine validity. It also exposes which details may be varied safely and which are constitutive.

Abstract Reasoning¶

Identify the carrier. State what the elements, states, objects, or observations are: vector spaces over a common field, a function between them, vector addition, scalar multiplication, kernels, images, and optional bases or matrices. Reject examples whose alleged carrier belongs to a different problem.
Lock the constitutive rule. Express the map preserves all linear combinations under the same scalar field independently of one notation or implementation.

Knowledge Transfer¶

Knowledge transfers strongly among subfields of linear algebra because they reuse vector spaces over a common field, a function between them, vector addition, scalar multiplication, kernels, images, and optional bases or matrices, Preservation of the two vector-space operations makes T determined by its values on a basis and representable by matrix multiplication after bases are chosen., and type the carrier, state every parameter and convention in the definition, test that the map preserves all linear combinations under the same scalar field, compare the nearest accepted identity, and report counterexamples, uncertainty, and limiting cases.

Relationships to Other Abstractions¶

Current abstraction Linear map Domain-specific

Parents (1) — outgoing typed relations

Linear map is a kind of Linearity Prime

It is the canonical morphism preserving linear structure.

Children (3) — abstractions linked here by typed relations

Discrete Chebyshev Transform Domain-specific is a kind of Linear map

A fixed-convention discrete Chebyshev transform is a linear map between finite nodal-value and polynomial-coefficient spaces.
Elasticity tensor Domain-specific is a kind of Linear map

A small-strain elasticity tensor is a linear map from symmetric strain to symmetric stress, with an elastic constitutive differentia.
Invariant Subspace Domain-specific presupposes Linear map

Invariant Subspace presupposes Linear map because invariance is defined by a specified linear operator mapping the subspace into itself.

Hierarchy path (1) — routes to 1 parentless root

Linear map → is a kind of Linearity

Neighborhood in Abstraction Space¶

Linear map sits in a crowded region of the domain-specific corpus (6^th percentile for distinctiveness): several abstractions share nearly its structure, so a description that fits it tends to fit its neighbors too.

Family — Vector Spaces & Linear Structures (17 abstractions)

Nearest neighbors

Computed from structural-signature embeddings · 2026-10-25