Bilinear Forms — Mathematics Knowledge System

Overview

A bilinear form is a function of two vector arguments that is linear in each argument separately. Bilinear forms generalize the dot product and are the foundation for quadratic forms, inner products, and the geometry of vector spaces. They appear throughout differential geometry, physics, and numerical analysis.

Intuition

Think of a bilinear form B(u, v) as a 'pairing' between two vectors. Fix one argument and you get an ordinary linear function of the other. The dot product is the prototypical example: u · v is linear in u for fixed v, and linear in v for fixed u. Bilinear forms let you encode geometry — lengths, angles, and volumes — through a single algebraic structure that can be represented by a matrix.

Formal Definition

Definition

Let V be a vector space over a field F. A bilinear form on V is a map B : V × V → F satisfying linearity in each slot:

B(\alpha u + \beta w,\, v) = \alpha B(u,v) + \beta B(w,v)

Linearity in first argument

B(u,\, \alpha v + \beta w) = \alpha B(u,v) + \beta B(u,w)

Linearity in second argument

B(u,v) = u^\top M v, \quad M_{ij} = B(e_i, e_j)

In a basis {e_i}, B is represented by the matrix M whose (i,j) entry is B(e_i, e_j).

Matrix representation

Notation

Notation	Meaning
$B(u,v)$	Bilinear form evaluated on vectors u and v
$M$	Matrix of the bilinear form in a given basis
$[B]$	Matrix representation of B

Properties

Symmetric

B \text{ is symmetric if } B(u,v) = B(v,u) \text{ for all } u,v. \text{ Equivalently, } M = M^\top.

Skew-symmetric (alternating)

B \text{ is skew-symmetric if } B(u,v) = -B(v,u). \text{ Equivalently, } M = -M^\top.

Non-degenerate

B \text{ is non-degenerate if } B(u,v)=0 \text{ for all } v \implies u=0. \text{ Equivalently, } \det(M) \neq 0.

Change of basis

\text{Under a change of basis } P, \text{ the matrix transforms as } M' = P^\top M P.

Worked Examples

1
Compute M_{ij} = B(e_i, e_j). We have e_1 = (1,0), e_2 = (0,1).
$M_{11} = B(e_1,e_1) = 2(1)(1)+0+0-0 = 2$
2
Compute the off-diagonal and (2,2) entries.
$M_{12} = B(e_1,e_2) = 0+1+0-0 = 1, \quad M_{21} = B(e_2,e_1) = 0+0+1-0 = 1, \quad M_{22} = B(e_2,e_2) = 0+0+0-1 = -1$
3
Assemble and check symmetry.
$M = \begin{pmatrix}2 & 1 \\ 1 & -1\end{pmatrix}, \quad M^\top = M, \text{ so } B \text{ is symmetric.}$

✓ Answer

M = [[2,1],[1,-1]], and B is symmetric.

Practice Problems

Mediumfree response

Find the matrix of B(u,v) = 3u_1v_1 - 2u_2v_2 + u_1v_2 + u_2v_1 on R^2 in the standard basis. Is B symmetric?

Mediumfree response

A bilinear form on R^2 has matrix M = [[1,2],[3,4]]. Decompose M into its symmetric and skew-symmetric parts and interpret each.

Common Mistakes

Common Mistake

Confusing a bilinear form with a linear map.

A bilinear form takes TWO vectors and returns a scalar; a linear map takes one vector and returns another vector.

Common Mistake

Thinking all bilinear forms are symmetric.

Bilinear forms can be symmetric, skew-symmetric, or neither; any matrix M represents a bilinear form via B(u,v)=u^T M v.

Quiz

Summary

A bilinear form B: V×V→F is linear in each argument separately.
In a basis, B is represented by a matrix M with M_{ij} = B(e_i, e_j), giving B(u,v) = u^T M v.
B is symmetric if M = M^T, skew-symmetric if M = -M^T, non-degenerate if det(M) ≠ 0.
Under a change of basis P, the matrix transforms as M' = P^T M P (congruence).
Any matrix M splits as M = (M+M^T)/2 + (M-M^T)/2 into symmetric and skew-symmetric parts.

References

WebsiteWikipedia — Bilinear form