Consider the function mapping the plane to itself that takes a vector to its projection into the line y = x. These two each show that the map is linear, the first one in a way that is coordinate-bound (that is, it fixes a basis and then computes) and the second in a way that is more conceptual.
(a) Produce a matrix that describes the function's action.
(b) Show that we can obtain this map by first rotating everything in the plane π/4 radians clockwise, then projecting into the x-axis, and then rotating π/4 radians counterclockwise.