Exercise 15

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.

• Produce a matrix that describes the function’s action.

• Show that we can obtain this map by first rotating everything in the plane $\pi/4$ radians clockwise, then projecting into the $x$-axis, and then rotating $\pi/4$ radians counterclockwise.