Linear map¶

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

Version: v1 · 2026-09-08 · History
Domain-specific #: 5342
Origin domain: linear algebra
Subdomain: vector space morphisms

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.

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. 2. 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) — more general patterns this builds on

Linear map is a kind of Linearity Prime

The proposed strict upward parent is prime:linearity.

Hierarchy path (1) — routes to 1 parentless root

Linear map → Linearity

Neighborhood in Abstraction Space¶

Linear map sits in a crowded region of the domain-specific corpus (10^th percentile for distinctiveness): several abstractions share nearly its structure, so a description that fits it tends to fit its neighbors too.

Family — Matrix Structure & Linear Maps (48 abstractions)

Nearest neighbors

Computed from structural-signature embeddings · 2026-09-08