Definition:Symmetric Mapping
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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 $\innerprod \cdot \cdot: V \times V \to \mathbb F$ be a mapping.
Then $\innerprod \cdot \cdot: V \times V \to \mathbb F$ is symmetric if and only if:
- $\forall x, y \in V: \innerprod x y = \innerprod y x$