Projective representation

Okay, imagine you have a toy train that you can move around on a track. Now, let's say you have a friend who has never seen your train before, and they want to know what it looks like. How can you show them?

One way might be to take a picture of your train and show them the photo. But what if your friend wants to see the train from a different angle? Or what if your train has moving parts that can't be captured in a still photo?

In this situation, you could use a projector to shine a light on your train and create a shadow on the wall. This shadow is called a "projective representation" of your train.

See, the shadow on the wall may not look exactly like your train, but it still gives your friend a good idea of what it looks like. And if you move your train around on the track, the shadow will move too - just like a real train would.

In mathematics, we use the idea of projective representation to help us study objects that can be hard to visualize or manipulate directly. Instead of working with the objects themselves, we use shadows or projections that give us information about the objects' properties and behavior.

So, projective representation is like using a projector to create a shadow or image that helps us understand something we can't see or touch directly. It's like a toy train's shadow on the wall - not exactly the real thing, but still a useful representation.