I'm working on an exercise in the text and am proving the following statement:
Suppose $T: \mathbb{R}^{n} \to \mathbb{R}^{n}$ is a linear transform. Prove that T is an isometry if and only if $T(v) \cdot T(w) = v \cdot w$
I'm currently working to prove the reverse statement and would like some clarification on a step, and potentially some guidance if my approach or any of my calculations are incorrect. I reference $2$ parts of the following proposition in my proof thus far
Proposition: $\forall u,v,w \in \mathbb{R}^{n}$ and $a \in \mathbb{R}$ we have
- $v \cdot v \geq 0$
- $v \cdot w = w \cdot v$
- $(av + u) \cdot w = a(v \cdot w) + (u \cdot w)$
The proof (thus far)
Assume $T$ is an isometry. This means $\lvert T(v) - T(w)\rvert = \lvert v-w \rvert$. We have $$\lvert T(v)-T(w) \rvert = \sqrt{[T(v)-T(w)] \cdot [T(v)-T(w)]} $$ By part $3$ of the proposition we have $$\sqrt{T(v) \cdot T(v) - T(v) \cdot T(w) - T(w) \cdot T(v) + T(w) \cdot T(w)}$$ Now by part $2$ of the proposition we obtain $$\sqrt{T(v) \cdot T(v) - 2T(v) \cdot T(w) + T(w) \cdot T(w)}$$ Which is $$\lvert T(v) \rvert + \sqrt{-2T(v) \cdot T(w)} + \lvert T(w) \rvert = \lvert T(v) \rvert + i \sqrt{2T(v) \cdot T(w)} + \lvert T(w) \rvert$$
But, since $T$ is an isometry we have $\lvert T(v) - T(w) \rvert = \lvert v-w \rvert$ and by a similar calculation for $\lvert v-w \rvert$ we have: $$\lvert T(v) \rvert + i \sqrt{2T(v) \cdot T(w)} + \lvert T(w) \rvert = \lvert v \rvert + i \sqrt{2v \cdot w} + \lvert w \rvert$$
Now at this part I'm a little confused. What I would like to do is claim that $\lvert T(v) \rvert = \lvert v \rvert$ and the same for the vector $w$ to get $$i \sqrt{2T(v) \cdot T(w)} = i \sqrt{2v \cdot w}$$ Where then I could cancel the $i$'s and the square roots (then the $2$'s) and show that the assumption that $T$ is an isometry directly implies that inner products are preserved. But I don't know if I can claim that$\lvert T(v) \rvert = \lvert v \rvert$ since isometries preserve distances and they don't say anything directly about preserving the length of vectors.