Real bilinear form

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

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 inputs and their linear combinations belong to a real vector space.
• Linear combinations and span — Bilinearity preserves finite linear combinations separately in each slot.
• 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.