##  [Degenerate Bilinear Form](/degenerate-bilinear-form) 

  ##  [Degenerate Bilinear Form](https://puremath.quantumdictionary.io/degenerate-bilinear-form-0) 

  

 [![Pure Mathematics Dictionary](/sites/default/files/styles/large/public/2026-01/Pure%20Mathematics.png.webp?itok=5pZnFQ59)](/topic-specific-dictionaries/mathematics-logic/pure-mathematics)

- Mathematics & Logic -

**Pure Mathematics Dictionary**

 







 

 

 

 



 

 

 

 

Definition

A bilinear form B: V × V → F on a vector space V over a field F that has a nontrivial kernel: there exists a nonzero v in V such that B(v,w)=0 for every w in V. Equivalently the linear map V → V* induced by v ↦ B(v,−) is not injective.

 

 

 

 

 





 

 



 ##  [Degenerate Bilinear Form](https://algebra.quantumdictionary.io/degenerate-bilinear-form-1) 

  

 [![Algebra](/sites/default/files/styles/large/public/2026-01/Algebra.png.webp?itok=3pHxBnUF)](/topic-specific-dictionaries/pure-mathematics/algebra)

- Pure Mathematics -

**Algebra Dictionary**

 







 

 

 

 



 

 

 

 

Definition

A bilinear form B: V×V→K on a vector space (or module) whose radical {v∈V | B(v,w)=0 for all w∈V} is nontrivial. Equivalently its Gram matrix (with respect to some basis) is singular, so B fails to induce an isomorphism V→V* or a nondegenerate pairing on V.