What is one-to-one structure-preserving correspondence?

Question

I know what a one-one correspondence is but when I was reading this How is free vector a same thing as translation? and it mentions a special type of one-one correspondence as structure-preserving 1-1 correspondence. Can someone explain more about this topic and the different types of one-one correspondence?

In the same text it is written that $T_v(x)=|v|^2v+x$ is not a structure preserving one-one correspondence. So, is it the case that in such cases the geometry of the object changes?

Answer 1

In general a structure preserving correspondence is a function which preserves the structure of the objects you're studying (not very illuminating is it?).

For example consider to vector spaces $V$ and $W$ over $\mathbb{R}$, with a function $T: V \rightarrow W$. Then we say $T$ preserves the vector space structure if:

1. $T(v_1 + v_2) = T(v_1) + T(v_2)$ for all $v_1,v_2 \in V$
2. $T(\alpha v) = \alpha T(v)$ for all $v \in V$ and $\alpha \in \mathbb{R}$.

Such a correspondence is called a linear transformation, and we say it preserves structure because it doesn't matter if we add vectors or multiply vectors by scalars before or after applying the function.

Similarly the correspondence you're confused about is structure preserving because $T_{v+w} = T_v \circ T_w$. So the translation corresponding to $v+w$ is the same as the translation corresponding to $v$ applied after the translation corresponding to $w$. Ie it doesn't matter if we calculate $T_{v+w}$ or $T_v$ and $T_w$ and then compose the translations, the answer is the same.

Edit: As an exercise can you find why $T_v(x) = |v|^2v + x$ is not structure preserving? (Ie, find two vectors $v,w$ such that $T_{v + w} \neq T_v \circ T_w$)