Linear map between real vector spaces

Mathematical statement

A real linear map T:V→W satisfies T(alpha*u+beta*v)=alpha*T(u)+beta*T(v) for all u,v and real alpha,beta. A finite-dimensional matrix formula supplies a universal coordinate certificate; a nonzero affine translation fails linearity.

Representative formula (exact TeX)

T(v)

Assumptions

• Exercises explicitly give a matrix or matrix-plus-bias formula from R^n to R^m.
• Matrix distributivity supplies the universal certificate; finitely many sampled inputs do not prove an arbitrary black-box map linear.

Symbol scope

• T — real linear map
• V — source space
• W — target space

Intuition, definition, and misconceptions

One identity encompasses both additivity and real homogeneity.

Matrix multiplication gives a linear map in finite-dimensional coordinates.

The affine formula Ax+b qualifies as linear exactly when b=0.

An arbitrary affine map, a black box passing only a few samples, or an additive function without real homogeneity need not be real-linear.

Adopted derivation guide

The original MKO is retained. The guide has its own append-only revision and verification boundaries, not a new Lean proof.

Prerequisites

• Real vector space — Both source and target are real vector spaces.
• Function mapping — T must be an explicit total map.
• Linear combinations and span — A linear map preserves finite linear combinations.

Verification boundary

The coordinate-matrix certificate covers only the explicit finite-dimensional formula; it does not certify arbitrary black boxes or make separate boundedness, continuity, injectivity or surjectivity claims.

not_formalized

This does not mean the content is false; it means no replayable computational or formal evidence is registered yet.

Next obligation

Add bounded linear operators on normed spaces.

Same-source structured data

This page and the interactive view use the same object and existing language edition; this adds no proof or human review.