The question given below is from Intro to electrodynamics by Griffiths and I wanted to understand what he meant by the question. I am able to prove that part that is saying that is show that. But I am not able to understand what he means by the question. Does he mean the dot product of two vectors is same even after transformation of the vectors. I would be grateful to any comments or answers .
Yes, he means that the dot product of two vectors is the same before and after their transformation. In other words, if $A$ is the transformation and $\langle\cdot,\cdot\rangle$ is the dot product, he wants a proof of the fact that$$(\forall v,w\in\Bbb R^2):\langle Av,Aw\rangle=\langle v,w\rangle.$$