• $f(\alpha U,V)=\alpha f(U,V)$ and $f(U,\beta V)=\beta f(U,V)$ where α and β are scalars,
• $f(U+V,W)=f(U,W)+f(V,W)$ and $f(W,U+V)=f(W,U)+f(W,V)$ for any vector W existing in the same vector space

Note 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 $f(U\text{,}V)={\displaystyle \sum _{ij}{k}_{ij}}{U}_i{V}_j$.

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.