In linear algebra, an orthogonal transformation is a linear transformation T : V → V on a real inner product space V, that preserves the inner product.
That is, for each pair u, v of elements of V, we have:⟨u, v⟩ = ⟨Tu, Tv⟩
Since the lengths of vectors and the angles between them are defined through the inner product, orthogonal transformations preserve lengths of vectors and angles between them.