Can a person's idea show or provide their perspective?

A person's idea is in their head unless they tell you about it.

it can, although I suspect that an idea reveals a perspective rather than showing it.  It isn't obvious but the idea itself, but the idea comes from a perspective and which perspective that would have to be is inherent to the idea, and can often be figured out.

It isn't a direct action of showing or providing, but an indirect action of making it available to be seen if someone wants to look for it.

Of course, some ideas are so simplistic that the perspective is obvious and the revelation is equal in actual effect to display or show.  For example, racist comments are pretty obvious as to what perspective is held by the speaker, so if you make a lot of racist comments, I think we can safely call that showing that you are a racist.

Not only ideas, but any comments, observations, and the like cannot avoid reflecting one's perspective. The reason is all of our senses are filtered through our experience(s). Try though we might to be objective, we can only communicate what we perceive and that explains why our perceptions filter everything that comes out of our mouths. Thus, the short answer to this question is Yes.