Final answer:
An isometry is a function that preserves distances in a space. By isometry definition, for any point in the target space, we can find a preimage in the domain, proving surjectivity. This concept is related to the principle that distances remain constant under coordinate rotations.
Step-by-step explanation:
The question requires proving that an isometry is surjective, which means every point in the target space is mapped to by some point in the domain. The definition of isometry in this context is a function f from Rn to Rn that preserves distances, thus for all points u and v in Rn, the equation |f(u)-f(v)| = |u-v| holds.
To prove isometry is surjective, consider two points P and Q in Rn. The distance preserved by the isometry implies that if there is a point Q, there must also be a point P such that the distance between P and Q is the same in both the domain and the target. By the definition of isometry, for any point Q in the target, we can construct such a point P in the domain, hence every point in the target space has a preimage, which makes the function surjective.
This principle of invariance under transformations, such as rotations, is analogous to how the distance between points remains the same under rotations of the coordinate system. For instance, when a coordinate system is rotated, the coordinates (x,y) may change to (x', y'), but the magnitude of any vector, and consequently the distances between points, remain unchanged.