Definition:Bilinear Mapping

Definition
Let $\struct {R, +_R, \times_R}$ be a commutative ring.

Let $\struct {A_1, +_1, \circ_1}_R, \struct {A_2, +_2, \circ_2}_R, \struct {A_3, +_3, \circ_3}_R$ be $R$-modules.

Let $\oplus: A_1 \times A_2 \to A_3$ be a binary operator with the property that: $\forall \tuple {a_1, a_2} \in A_1 \times A_2$:
 * $a_1 \mapsto a_1 \oplus a_2$ is a linear transformation from $A_1$ to $A_3$
 * $a_2 \mapsto a_1 \oplus a_2$ is a linear transformation from $A_2$ to $A_3$

Then $\oplus$ is a bilinear mapping.

That is, $\forall a, b \in R, \forall x, y \in A_2, z \in A_3$:


 * $\paren {\paren {a \circ_1 x} +_1 \paren {y \circ_1 b} } \oplus z = \paren {a \circ_3 \paren {x \oplus z} } +_3 \paren {\paren {y \oplus z} \circ_3 b}$

and for all $z \in A_1, x,y \in A_2$:
 * $z \oplus \paren {\paren {a \circ_2 x} +_2 \paren {y \circ_2 b} } = \paren {a \circ_3 \paren {z \oplus x} } +_3 \paren {\paren {z \oplus y} \circ_3 b}$

Equivalently, this can be expressed:


 * $\paren {x +_1 y} \oplus z = \paren {x \oplus z} +_3 \paren {y \oplus z}$
 * $z \oplus \paren {x +_2 y} = \paren {z \oplus x} +_3 \paren {z \oplus y}$


 * $\paren {a \circ_1 x} \oplus z = a \circ_3 \paren {x \oplus z}$
 * $z \oplus \paren {y \circ_2 b} = \paren {z \oplus y} \circ_3 b$

If $\struct {A, +, \circ}_R = A_1 = A_2 = A_3$, the notation simplifies considerably:


 * $\paren {\paren {a \circ x} + \paren {b \circ y} } \oplus z = \paren {a \circ \paren {x \oplus z} } + \paren {b \circ \paren {y \oplus z} }$
 * $z \oplus \paren {\paren {a \circ x} + \paren {y \circ b} } = \paren {a \circ \paren {z \oplus x} } + \paren {\paren {z \oplus y} \circ b}$

or equivalently, more easily digested:


 * $\paren {x + y} \oplus z = \paren {x \oplus z} + \paren {y \oplus z}$
 * $z \oplus \paren {x + y} = \paren {z \oplus x} + \paren {z \oplus y}$


 * $\paren {a \circ x} \oplus z = a \circ \paren {x \oplus z}$
 * $z \oplus \paren {y \circ b} = \paren {z \oplus y} \circ b$

Also see

 * Definition:Bilinear Form