Bilinear map

{wiki}

= Bilinear product
{synonym}

<Linear map> of two variables.

More formally, given 3 <vector spaces> X, Y, Z over a single <field (mathematics)>, a bilinear map is a function from:
$$
f : X \times Y \to Z
$$
that is linear on the first two arguments from X and Y, i.e.:
$$
f(a_1\vec{x_1} + a_2\vec{x_2}, \vec{y}) = a_1f(\vec{x_1}, \vec{y}) + a_2f(\vec{x_2}, \vec{y})
$$
Note that the definition only makes sense if all three vector spaces are over the same field, because linearity can mix up each of them.

The most important example by far is the <dot product> from $\R^n \times \R^n \to \R$, which is more specifically also a <symmetric bilinear form>.
