[Tensors](Tensors%20and%20Multilinear%20Algebra%20(index).md#Tensors) of degree $(0,2)$ or $(2,0)$ are referred to as _bilinear forms._ A bilinear form, $B$, is a [bilinear map](Bilinear%20map.md) on a [vector space](Vector%20spaces.md) $\mathcal{V}$ of a number field, $\mathbb{F}$, which is the two input map $B:(.,.)\in\mathcal{V}\times\mathcal{V}\rightarrow (.,.)\in\mathbb{F}.$ # Properties ## Bilinearity Consider a set of inputs $v,v',v''\in\mathcal{V}$ and a constant $c\in\mathbb{F}.$ The principal property of bilinear forms is in the name: bilinearity. This means that it is [linear](Linear%20map.md) for both inputs. Thus, 1. $B(v+v',v'')=B(v,v'')+B(v',v'')$ and $B(cv,v')=cB(v,v').$ 2. $B(v,v'+v'')=B(v,v')+B(v,v'')$ and $B(v,cv')=cB(v,v').$ ## Symmetry properties of bilinear forms A [bilinear form](Bilinear%20form.md), $B(v',v),$ is _symmetric_ if $B(v',v)=B(v,v')$ and _antisymmetric_ if $B(v',v)=-B(v,v').$ #MathematicalFoundations/Algebra/AbstractAlgebra/LinearAlgebra/MultiLinearAlgebra/Tensors