Isometry

linear-algebra

Definition

Isometry

Let $V$ and $W$ be inner product spaces over the same field $F$.

A linear map $T:V\to W$ is isometric if it preserves the inner product:

$$⟨Tx,Ty⟩=⟨x,y⟩\forall x,y\in V.$$

Equivalently, $T$ preserves the norm: $\Vert Tx\Vert =\Vert x\Vert$ for all $x\in V$.