Real vector space

Mathematical statement

A real vector space V has vector addition, a zero vector, additive inverses and real scalar multiplication, satisfying associativity, commutativity, distributivity, the unit-scalar law and scalar-multiplication compatibility. Exercises here check only exact axiom instances in fixed-dimensional real coordinate spaces.

Representative formula (exact TeX)

V(R)

Assumptions

• Exercises use equal fixed dimensions, integer coordinate vectors and integer scalars as special cases of real values.
• Successful coordinate instances do not fully prove an arbitrary candidate structure is a vector space.

Symbol scope

• V — real vector space
• R — real scalar field

Intuition, definition, and misconceptions

Vector addition is defined only between matching dimensions and adds corresponding coordinates.

Scalar multiplication scales each component. The zero vector and additive inverses support the additive-group structure.

Distributivity can be recomputed exactly for specified vectors and scalars; general axioms still quantify over all relevant elements.

Do not directly add vectors of different dimensions or present a few examples as a proof of general axioms.

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

• Plane vectors in a fixed basis — Coordinate vectors supply a concrete model.
• Real number — Scalars come from the real numbers.
• Arithmetic operations — Coordinate addition and scalar multiplication are componentwise.

Verification boundary

Only a fixed R^n coordinate slice is checked; no arbitrary abstract structure is claimed to be a proved vector space.

not_formalized

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

Next obligation

Add linear combinations, spans, independence, bases and dimension.

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.