---
tag: 0P799D6AMN
kind: definition
name: Bilinear form
msc: [15A63]
defines: 0SQZMXB6GP
uses: [05EGHEQZAW]
sources:
  - url: https://en.wikipedia.org/wiki/Bilinear_form
    locator: lead section
---

## Definition

Let $F$ be a field and let $V$ be a [[vector space|000AYPA9AJ]] over $F$.
A *bilinear form* on $V$ is a map $B \colon V \times V \to F$ such that, for all $u, v, w \in V$ and $a \in F$:

1. $B(u + v, w) = B(u, w) + B(v, w)$ and $B(au, w) = a B(u, w)$.
2. $B(u, v + w) = B(u, v) + B(u, w)$ and $B(u, aw) = a B(u, w)$.

## Remarks

The [[dot product|0S52TNNTH9]] on $\R^n$ is a bilinear form.
For each $n \times n$ matrix $M$ over $F$, the map $B(x, y) = x^T M y$ is a bilinear form on $F^n$.
The bilinear forms on $V$ form a vector space over $F$; if $\dim V = n$, it has dimension $n^2$.
More generally, a bilinear map $V \times W \to F$ uses two different spaces.

## Explanations

### L1 human

A bilinear form takes two vectors and gives a number.
It is linear in the first vector and also linear in the second vector.
The dot product is the main example, but bilinear forms need not be symmetric or positive.