IEVref: 102-03-16 ID: Language: en Status: Standard Term: bilinear form Synonym1: Synonym2: Synonym3: Symbol: Definition: function f that attributes a scalar $f\left(U\text{,}\text{\hspace{0.17em}}V\right)$ to any pair of vectors U and V in a given vector space, with the following properties: $f\left(\alpha \text{\hspace{0.17em}}U,\text{\hspace{0.17em}}V\right)=\alpha \text{\hspace{0.17em}}f\left(U,\text{\hspace{0.17em}}V\right)$ and $f\left(U,\text{\hspace{0.17em}}\beta \text{\hspace{0.17em}}V\right)=\beta \text{\hspace{0.17em}}f\left(U,\text{\hspace{0.17em}}V\right)$ where α and β are scalars, $f\left(U+V,\text{\hspace{0.17em}}W\right)=f\left(U,\text{\hspace{0.17em}}W\right)+f\left(V,\text{\hspace{0.17em}}W\right)$ and $f\left(W,\text{\hspace{0.17em}}U+V\right)=f\left(W,\text{\hspace{0.17em}}U\right)+f\left(W,\text{\hspace{0.17em}}V\right)$ for any vector W existing in the same vector spaceNote 1 to entry: A bilinear form over an n-dimensional vector space can be represented by a square matrix $\left({k}_{ij}\right)$ and the scalar is $f\left(U\text{,}\text{\hspace{0.17em}}V\right)=\sum _{ij}{k}_{ij}{U}_{i}{V}_{j}$. Note 2 to entry: The bilinear forms over a given n-dimensional vector space constitute an ${n}^{2}$-dimensional vector space. Note 3 to entry: The concept of bilinear form extends to "linear form" in the case of one vector and to "multilinear form" (or m-linear form) in the case of an ordered set of m vectors. Publication date: 2008-08 Source: Replaces: Internal notes: 2017-02-20: Editorial revisions in accordance with the information provided in C00019 (IEV 102) - evaluation. JGO CO remarks: TC/SC remarks: VT remarks: Domain1: Domain2: Domain3: Domain4: Domain5: