Bilinear Form - Maps To The Dual Space

Maps To The Dual Space

Every bilinear form B on V defines a pair of linear maps from V to its dual space V*. Define B₁, B₂: V → V* by

B₁(v)(w) = B(v, w)

B₂(v)(w) = B(w, v)

This is often denoted as

B₁(v) = B(v, ⋅)

B₂(v) = B(⋅, v)

where the ( ⋅ ) indicates the slot into which the argument for the resulting linear functional is to be placed.

If either of B₁ or B₂ is an isomorphism, then both are, and the bilinear form B is said to be nondegenerate. This can only occur if V is finite-dimensional since V* has higher dimension than V otherwise, and in finite dimension isomorphism (of equal dimensional vector spaces) is equivalent to injective (and also equivalent to surjective). More concretely, for a finite-dimensional vector space, non-degenerate means that every non-zero element pairs non-trivially with some other element:

for all implies that x = 0 and

for all implies that y = 0.

The corresponding notion for a module over a ring is that a bilinear form is unimodular if is an isomorphism. Given a finite dimensional module over a commutative ring, the pairing may be injective (hence "nondegenerate" in the above sense) but not unimodular. For example, over the integers, the pairing is nondegenerate but not unimodular, as the induced map from V = Z to V* = Z is multiplication by 2.

If V is finite-dimensional then one can identify V with its double dual V**. One can then show that B₂ is the transpose of the linear map B₁ (if V is infinite-dimensional then B₂ is the transpose of B₁ restricted to the image of V in V**). Given B one can define the transpose of B to be the bilinear form given by

B*(v, w) = B(w, v).

The left radical and right radical of the form B are the kernels of B₂ and B₁ respectively; they are the vectors orthogonal to the whole space on the left and on the right.

If V is finite-dimensional then the rank of B₁ is equal to the rank of B₂. If this number is equal to dim(V) then B₁ and B₂ are linear isomorphisms from V to V*. In this case B is nondegenerate. By the rank-nullity theorem, this is equivalent to the condition that the left and equivalently right radicals be trivial. In fact, for finite dimensional spaces, this is often taken as the definition of nondegeneracy:

Definition: B is nondegenerate if and only if B(v, w) = 0 for all w implies v = 0.

Given any linear map A : V → V* one can obtain a bilinear form B on V via

B(v, w) = A(v)(w).

This form will be nondegenerate if and only if A is an isomorphism.

If V is finite-dimensional then, relative to some basis for V, a bilinear form is degenerate if and only if the determinant of the associated matrix is zero. Likewise, a nondegenerate form is one for which the determinant of the associated matrix is non-zero (the matrix is non-singular). These statements are independent of the chosen basis. For a module over a ring, a unimodular form is one for which the determinant of the associate matrix is a unit (for example 1), hence the term; note that a form whose matrix is non-zero but not a unit will be nondegenerate but not unimodular, for example over the integers.