## Bilinear Functions

A bilinear function on a vector space
$V$
is a function
$f$

$\ \colon V \times V \rightarrow \mathbb{R}$
linear in both arguments.
The dot product
$f( \mathbf{v}_1, \mathbf{v}_2)= \mathbf{v}_1 \cdot \mathbf{v}_2$
is a bilinear function.
$f( k\mathbf{v}_1, \mathbf{v}_2)= (k \mathbf{v}_1) \cdot \mathbf{v}_2= k( \mathbf{v}_1 \cdot \mathbf{v}_2)$

$f(\mathbf{v}_1, k\mathbf{v}_2)= \mathbf{v}_1 \cdot (k \mathbf{v}_2)= k( \mathbf{v}_1 \cdot \mathbf{v}_2)$

$f( \mathbf{v}_1 + \mathbf{w}_1, \mathbf{v}_2)= ( \mathbf{v}_1 + \mathbf{w}_1) \cdot \mathbf{v}_2= \mathbf{v}_1 \cdot \mathbf{v}_2+ \mathbf{w}_1 \cdot \mathbf{v}_2$

$f( \mathbf{v}_1, \mathbf{v}_2++ \mathbf{w}_2)= \mathbf{v}_1 \cdot (\mathbf{v}_2+ \mathbf{w}_2)= \mathbf{v}_1 \cdot \mathbf{v}_2+ \mathbf{v}_1 \cdot \mathbf{w}_2$

The function that takes one of a pair of vectors and takes the dot product of that vector with itself is not a bilinear function.
$f( k\mathbf{v}_1, \mathbf{v}_2)=(k \mathbf{v}_1) \cdot (k\mathbf{v}_1)=k^2 f(\mathbf{v}_1, \mathbf{v}_2)$