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IEVref: | 102-03-16 | ID: | |

Language: | en | Status: Standard | |

Term: | bilinear form | ||

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Definition: | function f that attributes a scalar $f(U\text{,}\text{\hspace{0.17em}}V)$ to any pair of vectors and U in a given vector space, with the following properties: V- $f(\alpha \text{\hspace{0.17em}}U,\text{\hspace{0.17em}}V)=\alpha \text{\hspace{0.17em}}f(U,\text{\hspace{0.17em}}V)$ and $f(U,\text{\hspace{0.17em}}\beta \text{\hspace{0.17em}}V)=\beta \text{\hspace{0.17em}}f(U,\text{\hspace{0.17em}}V)$ where
*α*and*β*are scalars, - $f(U+V,\text{\hspace{0.17em}}W)=f(U,\text{\hspace{0.17em}}W)+f(V,\text{\hspace{0.17em}}W)$ and $f(W,\text{\hspace{0.17em}}U+V)=f(W,\text{\hspace{0.17em}}U)+f(W,\text{\hspace{0.17em}}V)$ for any vector
existing in the same vector space*W*
Note 1 to entry: A bilinear form over an Note 2 to entry: The bilinear forms over a given Note 3 to entry: The concept of bilinear form extends to "linear form" in the case of one vector and to "multilinear form" (or | ||

Publication date: | 2008-08 | ||

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Internal notes: | 2017-02-20: Editorial revisions in accordance with the information provided in C00019 (IEV 102) - evaluation. JGO | ||

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- $f(\alpha \text{\hspace{0.17em}}U,\text{\hspace{0.17em}}V)=\alpha \text{\hspace{0.17em}}f(U,\text{\hspace{0.17em}}V)$ and $f(U,\text{\hspace{0.17em}}\beta \text{\hspace{0.17em}}V)=\beta \text{\hspace{0.17em}}f(U,\text{\hspace{0.17em}}V)$ where
*α*and*β*are scalars, - $f(U+V,\text{\hspace{0.17em}}W)=f(U,\text{\hspace{0.17em}}W)+f(V,\text{\hspace{0.17em}}W)$ and $f(W,\text{\hspace{0.17em}}U+V)=f(W,\text{\hspace{0.17em}}U)+f(W,\text{\hspace{0.17em}}V)$ for any vector
existing in the same vector space*W*

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 $f(U\text{,}\text{\hspace{0.17em}}V)={\displaystyle \sum _{ij}{k}_{ij}}{U}_{i}{V}_{j}$.

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.