# Definitions Let $V$ and $W$ be [[Vector spaces|vector spaces]] over a field $K$. > [!NOTE] Definition \[MA266\] > A bilinear map on $V$ and $W$ is a map $\tau: V \times W \rightarrow K$ such that > (i) $\tau\left(\alpha_1 \mathbf{v}_1+\alpha_2 \mathbf{v}_2, \mathbf{w}\right)=\alpha_1 \tau\left(\mathbf{v}_1, \mathbf{w}\right)+\alpha_2 \tau\left(\mathbf{v}_2, \mathbf{w}\right)$; and > (ii) $\tau\left(\mathbf{v}, \alpha_1 \mathbf{w}_1+\alpha_2 \mathbf{w}_2\right)=\alpha_1 \tau\left(\mathbf{v}, \mathbf{w}_1\right)+\alpha_2 \tau\left(\mathbf{v}, \mathbf{w}_2\right)$ > for all $\mathbf{v}, \mathbf{v}_1, \mathbf{v}_2 \in V, \mathbf{w}, \mathbf{w}_1, \mathbf{w}_2 \in W$, and $\alpha_1, \alpha_2 \in K$. **Terminology**: - If $V=W$, $\tau$ is called **bilinear form**. - The **rank** of $\tau$, is defined as the [[Rank of Linear Map|rank]] of any of its [[Matrix representations of bilinear map|matrix representations]] (equivalent representations $A,B$ of $\tau$ satisfy $B=P^{T}AQ$) - The **right radical** of $\tau$ is the kernel of $A$ and the **left radical** of $\tau$ is kernel of $A^T$ for any $A$ matrix representation of $\tau$ wrt given bases. - We say that a **bilinear form**, on say $V$, is **non-degenerate** if its matrix representation is invertible or equivalently, its left and right radicals are zero. Note $v\in \text{Ker}(A)$ iff $Av=0\iff w^{T}Av = 0 \iff \tau(w,v)=0, \forall w\in W$ and similarly $v\in \text{Ker}(A)$ iff $\tau(v,w)=0, \forall w\in V$. # Properties ###### Bilinear form We say that $\tau$ is **symmetric** if $\tau(v,w)=\tau(w,v)$ for all $v\in V$ and **antisymmetric** iff $\tau(v,v)=0$ for all $v\in V$. Note that this equivalent to $\tau(v,w)=-\tau(w,v)$ (prove by taking $\tau(v+w,v+w)$) except when $K$ is field such that $1_{K}+1_{K}=2 =0_{K}$ (such as in $\mathbb{Z}/2\mathbb{Z}$). See [[Normal form of bilinear form on vector space whose characteristic is not two]].