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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.
\[f( k\mathbf{v}_1, \mathbf{v}_2)=(k \mathbf{v}_1) \cdot (k\mathbf{v}_1)=k^2 f(\mathbf{v}_1, \mathbf{v}_2)\]

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