Projective space is a fundamental concept in geometry that extends the notion of Euclidean space by adding “points at infinity.” It has a rich structure and is crucial in various areas of mathematics and its applications.

give a simple example how a simplectic group is represented in projective space?

I am particularly interested in “how are homogeneous coordinates allow for a unified representation of points and vectors, and make transformations such as translation, rotation, scaling, and projection straightforward to compute using matrix multiplication”