Proving that Isometries Preserving Norms in Euclidean Spaces Are Linear

Isometries in Euclidean spaces with their usual norms play a significant role in understanding and applying various properties in fields such as geometry, linear algebra, and even physics. One common misconception is that isometries in general do not preserve norms. However, under specific conditions, isometries can indeed preserve norms. This article will explore the conditions in which an isometry preserves norms and whether such an isometry must be a linear transformation.

Definition and Conditions for Isometries

An isometry in the context of normed vector spaces is a function f: mathbb{R}^n rightarrow mathbb{R}^n that preserves distances between points. Formally, f is an isometry if for all mathbf{u}, mathbf{v} in mathbb{R}^n, |f(mathbf{u}) - f(mathbf{v})| |mathbf{u} - mathbf{v}|.

Norm Preservation and Linear Isometries

If an isometry f satisfies f(mathbf{0}) mathbf{0}, then it is easy to show that |f(mathbf{u})| |mathbf{u}| for all mathbf{u} in mathbb{R}^n. This means that such an isometry f will preserve the norm of any vector. Furthermore, if f is a linear isometry, it will preserve the norm of any vector due to its linearity. The presence of a translation, i.e., if f(mathbf{0}) eq mathbf{0}, would prevent f from preserving the norm, as f can be decomposed into a translation and an isometry that fixes the origin.

Translation and Preservation of Norms

Consider the case where f(mathbf{a}) eq mathbf{0}. In this situation, f can be written as f t_{mathbf{a}} circ g, where t_{mathbf{a}} is a translation by the vector mathbf{a} and g is an isometry that fixes the origin. Since translations do not preserve norms, an isometry f that does not fix the origin cannot preserve norms.

Preservation of the Dot Product and Linear Isometries

Isometries that fix the origin in mathbb{R}^n also preserve the dot product. This is because the norm is defined as |mathbf{x}| sqrt{mathbf{x} cdot mathbf{x}}. Using the given condition, the isometry f must satisfy |f(mathbf{u}) - f(mathbf{v})| |mathbf{u} - mathbf{v}|, which can be squared to obtain (f(mathbf{u}) - f(mathbf{v})) cdot (f(mathbf{u}) - f(mathbf{v})) (mathbf{u} - mathbf{v}) cdot (mathbf{u} - mathbf{v}). Expanding and simplifying using the fact that f preserves norms results in f(mathbf{u}) cdot f(mathbf{v}) mathbf{u} cdot mathbf{v}.

Conclusion

To conclude, isometries that fix the origin in Euclidean spaces and preserve the dot product are linear. This is due to the fact that if an isometry preserves the dot product, it must preserve the norms as well. Therefore, in the specific context of mathbb{R}^n with the usual norm, an isometry that fixes the origin is linear.