Bilinear Forms

Bilinear forms provide a different and more general way of looking at the relationship between a vector space and its dual.

Definition. A map $\phi: V \times W \to \mathbb{F}$ is a bilinear form if it is linear in both arguments, i.e. if $\phi(v, -): W \to \mathbb{F} \in W^\ast$ for all $v \in V$ and $\phi(-, w): V \to \mathbb{F} \in V^\ast$ for all $w \in W$.

Definition. Let $(e_1, …, e_m)$ be a basis for $V$ and $(f_1, …, f_n)$ be a basis for $W$ and $\phi: V \times W \to \mathbb{F}$ a bilinear form. Then the $m \times n$ matrix $A$ representing $\phi$ with respect to $(e_1, …, e_m)$ and $(f_1, …, f_n)$ is given by $A_{ij} = \phi(e_i, f_j)$.

Consider vectors $v = \sum \lambda_i e_i$ and $w = \sum \mu_j f_j$ then

\[\phi(v, w) = \phi \left( \sum_i \lambda_i e_i, \sum_j \mu_j f_j \right) = \sum_i \sum_j \lambda_i \mu_j \phi(e_i, f_j)\]

Therefore, if $A$ is the matrix representing $\phi$ then

\[\phi(v, w) = \begin{pmatrix} \lambda_1 & \cdots & \lambda_m \end{pmatrix} A \begin{pmatrix} \mu_1 \\ \vdots \\ \mu_n \end{pmatrix}\]

Definition. A bilinear form $\phi: V \times W \to \mathbb{F}$ induces linear maps $\phi_L: V \to W^\ast$ and $\phi_R: W \to V^\ast$ defined by

\[\phi_L(v)(w) = \phi(v, w) = \phi_R(w)(v)\]

for $v \in V$ and $w \in W$.

Base on the above, we can see that $\phi: V \times V^\ast \to \mathbb{F}, \phi(v, f) = f(v)$ is a bilinear form with $\phi_L: V \to V^{\ast\ast} = \text{ev}$ and $\phi_R: V^\ast \to V^\ast = \iota_{V^\ast}$.