# Definition:Symmetric Mapping

This page is about symmetric mappings in the context of linear algebra.. For other uses, see Definition:Symmetry.

## Definition

Let $\R$ be the field of real numbers.

Let $\F$ be a subfield of $\R$.

Let $V$ be a vector space over $\F$

Let $\left \langle {\cdot, \cdot} \right \rangle : V \times V \to \mathbb F$ be a mapping.

Then $\left \langle {\cdot, \cdot} \right \rangle : V \times V \to \mathbb F$ is symmetric if and only if:

$\forall x, y \in V: \quad \left \langle {x, y} \right \rangle = \left \langle {y, x} \right \rangle$

## Linguistic Note

This property as a noun is referred to as symmetry.