A linear transformation $T : V \to V$ is said to be norm-preserving for a given inner product if for all vectors $v$ in $V$ the following is true: $\langle v,v \rangle = \langle T(v),T(v) \rangle$.
Prove that if $T$ is a norm-preserving transformation then for any vectors, $v_1$ and $v_2$ in ANY inner product space $V$, $\langle v_1, v_2 \rangle = \langle T (v_1), T (v_2) \rangle$.
I don’t get what I’m supposed to do, but I do know that the transformation is norm-preserving on the vector $v_1 + v_2$.
Notice that for any $v, w \in V$, $$\langle v + w, v + w \rangle = \langle v , v \rangle + 2 \langle v, w \rangle + \langle w, w \rangle$$
So that:
$$\langle v, w \rangle = \frac{\langle v + w, v + w \rangle - \langle v , v \rangle - \langle w , w \rangle}{2}$$
If $T$ is norm preserving, it preserves all the terms on the right hand side, and since it is linear we have:
$$\frac{\langle T(v + w), T(v + w) \rangle - \langle T(v) , T(v) \rangle - \langle T(w) , T(w) \rangle}{2} = \langle T(v), T(w) \rangle = \langle v, w \rangle $$