How do I prove that a map preserves the angle if and only if it's the scalar multiple of an isometry.

I get the "if" direction by using definition of isometry.

How do I show the other direction, i.e. that $$\frac{<u, v>}{\left\lVert u\right\rVert\left\lVert v\right\rVert} = \frac{<L(u), L(v)>}{\left\lVert L(u)\right\rVert\left\lVert L(v)\right\rVert}$$ implies scalar multiple of isometry?

  • $\begingroup$ Hint: let $v=u$. $\endgroup$ – S. Dewar Jan 17 at 21:53

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