Why does $T(x)=2x$ keep angles but is not an isometry?

Question

I would like to know why a defined operator $T$ on $V$, as following: $T(\alpha)=2\alpha$ preserves angles but is not an isometry?

Answer 1

Well I assume by the tags you used that $V$ is supposed to be an inner product space (otherwise we can't really talk about angles). Then $$\frac{\langle Tx, Ty \rangle}{\|Tx\| \|Ty\|} = \frac{\langle 2x, 2y \rangle}{\|2x\| \|2y\|} = \frac{\langle x, y \rangle}{\|x\| \|y\|}$$ where $\| \cdot\|$ is the norm induced by the inner product. Now that is the definition of the cosine of the angle between $Tx$ and $Ty$ on the left and the cosine of the angle between $x$ and $y$ on the right. It is not an isometry because lengths are not preserved (as has been pointed out).