# Definition:Symmetric Mapping

## Definition

### Mapping Theory

Let $n \in \N$ be a natural number.

Let $S^n$ be an $n$-dimensional cartesian space on a set $S$.

Let $f: S^n \to T$ be a mapping from $S^n$ to a set $T$.

Then $f$ is a symmetric mapping if and only if:

$\map f {x_1, x_2, \dotsc, x_n} = \map f {x_{\map \pi 1}, x_{\map \pi 2}, \dotsc x_{\map \pi n} }$

for all permutations $\pi$ on $\set {1, 2, \dotsc n}$.

That is, a symmetric mapping is a mapping defined on a cartesian space whose values are preserved under permutation of its arguments.

### Linear Algebra

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:

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