Show that if $\gamma$ is any isometry on $\mathbb{R}^n$, then so is $a\gamma(\dfrac{1}{a})$

Question

Take $v \in \mathbb{R}$ and denote translation over $v$ as $\tau v$. Let a ∈ $\mathbb{R}$ with $a \neq 0$.

a) Verify that $a \tau_v \dfrac{1}{a} $ is again a translation

b) Show that if $\gamma$ is any isometry on $\mathbb{R}^n$, then so is $a\gamma(\dfrac{1}{a})$

I'm stuck at both of these questions, can somebody help?

Answer 1

These both answer easily by applying the definitions of things.

$\tau_v:x\mapsto x+v$ is a translation, and $\rho_a:x\mapsto ax$ would be a dilation.

So $a\tau_v\frac1a$ can be thought of as three things:

• 1) $x\mapsto x/a$,
• 2) $x/a\mapsto x/a+v$
• 3) $x/a+v\mapsto a(x/a)+av=x+av$

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Next, what is an isometry? Here it's a transformation that preserves the Euclidean metric, rather than some other metric. What properties does this metric satisfy?