projection operator

A projection operator captures the idea of dropping a vector onto a subspace along a chosen direction. The defining property is algebraic and beautifully simple: P^2 = P. Applying the projection twice gives the same result as applying it once, because a vector already lying in the target subspace has nothing left to project.

Every such P splits the whole space into two complementary pieces. Its image im P is what P projects onto, and its kernel ker P is the direction it projects along. The key structural fact is that V is the direct sum of these two, V = im P (+) ker P: every vector decomposes uniquely as something in the image plus something in the kernel, namely v = P(v) + (v - P(v)). The first lies in im P; the second lies in ker P because P(v - P(v)) = P(v) - P^2(v) = 0.

Conversely, any direct-sum decomposition V = U (+) W gives a projection: send each vector to its U-component. So projections and direct-sum splittings are two views of one phenomenon. Note that the splitting is determined by both the image and the kernel — many different projections share the same image but project along different kernels, so a projection is not determined by where it lands alone.

An important refinement: if the space has an inner product and you require the projection direction to be orthogonal to the target subspace, you get an orthogonal projection, the operator behind least squares. That extra condition makes P self-adjoint, P^* = P, on top of P^2 = P. Without it, P is just an oblique projection.