Category:Translation Mappings

From ProofWiki
Jump to navigation Jump to search

This category contains results about Translation Mappings.
Definitions specific to this category can be found in Definitions/Translation Mappings.

In an Abelian Group

Let $\struct {G, +}$ be an abelian group.

Let $g \in G$.

Then translation by $g$ is the mapping $\tau_g: G \to G$ defined by:

$\forall h \in G: \map {\tau_g} h = h + \paren {-g}$

where $-g$ is the inverse of $g$ with respect to $+$ in $G$.

In a Euclidean Space

A translation $\tau_\mathbf x$ is an isometry on the real Euclidean space $\Gamma = \R^n$ defined as:

$\forall \mathbf y \in \R^n: \map {\tau_\mathbf x} {\mathbf y} = \mathbf y - \mathbf x$

where $\mathbf x$ is a vector in $\R^n$.

In an Affine Space

Let $\EE$ and $\FF$ be affine spaces.

Let $\TT: \EE \to \FF$ be affine transformations.

Then $\TT$ is a translation if and only if the tangent map $\vec \TT$ is the identity on the tangent space $\vec \EE$.

In a Vector Space

Let $K$ be a field.

Let $X$ be a vector space over $K$.

Let $x \in X$.

We define the translation mapping $\tau_x : X \to X$ by:

$\map {\tau_x} y = y - x$

for each $y \in X$.