Show the $A$ is a orthogonal matrix when the $A$ satisfying $\left\langle {u,v} \right\rangle = \left\langle {Au,Av} \right\rangle$, $\forall u, v \in \mathbb{R}^3$ (The $A$ is a $3 \times3$ matrix.)
My book proved it by showing $A^tA = I$ But I solved the above question by different ways comparing my text book solution. The problem is I can't sure my solution is exact or not. Let me suggest my solution.
condition) $\langle e_i, e_j\rangle = \langle Ae_i, Ae_j\rangle $.
The $Ae_i$ is a $i$ th column of the $A$, $i$ th column and $j$ th column orthogonal each other if the $i \neq j$.
But if the $i=j$, $1= \langle e_i, e_i\rangle= \langle Ae_i, Ae_i\rangle$ (I.e. $\Vert Ae_i \Vert =1$ )
So all the norm of the $i$ th column of the $A$ is $1$
Consequently, Column of the $A$ are orthonormal. It is orthogonal matrix.
Is my proof right?