# Definition A bilinear form $\braket{\cdot|\cdot}$ is a [[Quadratic Forms on Vector Spaces|quadratic form]] on a [[vector space]] $V$ $ \begin{align} \braket{\cdot|\cdot}: &V\times V \rightarrow \mathbb{F} , \\ &(\pmb{x}, \pmb{y}) \mapsto \braket{\pmb{x}|\pmb{y}} \end{align} $ where $\mathbb{F}$ is the field over which $V$ is defined. It has the following properties 1. *Linearity in its second argument:* $\braket{\pmb{z}|c\pmb{x}+\pmb{y}} = c\braket{\pmb{z}|\pmb{x}} + \braket{\pmb{z}|\pmb{y}}$ 2. *Linearity in its first argument:* $\braket{c\pmb{x}+\pmb{y}|\pmb{z}} = c \braket{\pmb{x}|\pmb{z}} + \braket{\pmb{y}|\pmb{z}}$ # More General Definition More generally, a bilinear form $B$ can take arguments from two distinct vector spaces $E$ and $F$ $ B: E \times F \rightarrow \mathbb{F} $ Let $\mathcal{B}_E = \{e_i\}$ be a [[basis]] for $E$ and $\mathcal{B}_F = \{f_i\}$ a basis for $F$, then the [[matrix]] representation of $B$ relative to these two bases is given by: $ [B]_{ij} = B(e_i, f_j) $ such that: $ B(\pmb{v}, \pmb{w}) = B(v^ie_i, w^jf_j) = v^iw_j B(e_i,f_j) = v^i[B]_{ij} w^j $ ([[Einstein Summation Convention|Einstein summation]] is implied). # Associated Linear Map For every bilinear form $B$ described as above, we can associate a [[linear map]] $B^\flat$ as follows: $ B^\flat : E \rightarrow F^* $ where $F^*$ is the [[vector space]] [[Dual Space|dual]] to $F$. This map is defined such that (via [[Currying]]): $ B^\flat(\pmb{v})(\pmb{w}) = B(\pmb{v}, \pmb{w}) $ Note that $B^\flat$ has [[Components#Linear Maps|components]] $ \begin{align} B^\flat(e_i) &= B^\flat_{ki} f^k\\ B^\flat(e_i)(f_j) &= B^\flat_{ki} f^k f_j = B^\flat_{ji} = B(e_i, f_j) = B_{ij} \end{align} $ So we have: $ B_{ji}^\flat = B_{ij}, $ i.e., they are each others' transpose. %% $ \begin{align} B^\sharp(e_j) &= B^\sharp_{kj}f^k\\ B^\sharp(e_j)(f_i) &= B^\sharp_{kj}f^kf_i = B^\sharp_{ij} = B(e_i, f_j) = B_{ij} \end{align} $ %%