# Real bilinear form

This English edition is an AI candidate and does not yet have an OCME Atlas (GPT-6) Twin closure receipt.

A bilinear form on a real vector space preserves addition and real scalar multiplication separately in each input.

mko-real-bilinear-form · definition · v0.10.0



## Mathematical statement

B:V×V→R is bilinear when fixing either input leaves a linear map in the other. This slice uses the integer-coefficient matrix formula B(x,y)=x^T A y as a universal coordinate certificate and for exact evaluation.





## Representative formula (exact TeX)

B(x,y)





## Assumptions


- Every vector and matrix dimension is fully compatible, and exercise coordinates are safe integers.
- Matrix expansion is a universal algebraic certificate of bilinearity; finite samples alone do not prove that an arbitrary black-box function is bilinear.





## Symbol scope


- B — real bilinear form
- A — coordinate matrix
- x — first input
- y — second input





## Intuition, definition, and misconceptions

Linearity in the first and second slots are separate obligations.

Every real matrix defines a bilinear form in compatible coordinates.

Symmetry, positive definiteness, nondegeneracy and inner-product qualification are additional conditions.

Checking B(x,y)=B(y,x), or addition in a few examples, does not establish bilinearity or an inner product.





## Adopted derivation guide

[Real bilinear form: proving bilinearity slot by slot from x^T A y, with exact evaluation](https://ocme.evemisslab.com/en/derivations/derivation-mko-real-bilinear-form-01/)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](https://ocme.evemisslab.com/en/mko/mko-real-vector-space/) — Both inputs and their linear combinations belong to a real vector space.
- [Linear combinations and span](https://ocme.evemisslab.com/en/mko/mko-linear-combination-span/) — Bilinearity preserves finite linear combinations separately in each slot.
- [Function mapping](https://ocme.evemisslab.com/en/mko/mko-function-mapping/) — B is a total function from V×V to R.





## Verification boundary

The matrix formula certifies bilinearity, not automatically symmetry, positive definiteness, nondegeneracy or an inner product.

not_formalized

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





## Next obligation

Add real inner-product spaces and inner-product-induced norms.





## 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.

[Canonical MKO JSON](https://ocme.evemisslab.com/data/mko/real-bilinear-form.json) · [Plain Markdown](https://ocme.evemisslab.com/en/mko/mko-real-bilinear-form/index.md)