Definition:Translation Mapping/Vector Space

Definition

Let $K$ be a field.

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

Let $x \in X$.

The translation mapping $\tau_x : X \to X$ is defined as:

$\forall y \in X: \map {\tau_x} y = y - x$

where $y - x$ denotes vector subtraction.

This is often defined separately when the vector space in question is 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 $\R^n$.

The map $\tau_x$ may also be called the translation (by $x$) operator.

It is easy to confuse the mappings $\tau_{\mathbf x}$ and $\tau_{-\mathbf x}$, and the choice made here is arbitrary.

The map $\tau_{\mathbf x}$ can be understood (appealing to our planar $\R^2$ intuition) as translating the coordinate axes by $\mathbf x$.