A linear map on an inner-product (Euclidean) space that sends each vector to the unique closest point in a specified linear subspace by moving along directions orthogonal to that subspace.

Principle

Given an inner product, the orthogonal projection onto a subspace is the idempotent, self-adjoint linear operator whose range is the subspace and whose kernel is its orthogonal complement; it selects the nearest point in the subspace in norm.

Demonstration

In R^3 with the standard dot product, projecting a point p=(x,y,z) onto the plane z=0 yields (x,y,0); projecting onto the line spanned by unit vector u gives (u·p)u. The matrix P for projection onto a subspace spanned by orthonormal columns Q is P=QQ^T, which satisfies P^2=P and P^T=P.

Misapplication

Treating an orthogonal projection as if it were defined without an inner product or assuming uniqueness of nearest point on a non-closed subspace; or applying the same formulas in a space with a different metric leads to wrong nearest-point claims.

Consequence

When applied correctly, the map produces the closest-point approximation to data, decomposes vectors into orthogonal components, yields a symmetric idempotent operator, and preserves minimization properties (least-squares solutions).

Reversal

The inversion is not generally defined: information orthogonal to the subspace is lost, so one cannot recover the original vector from its projection; conversely, the complementary projection onto the orthogonal complement recovers the discarded component.

Boundary

Requires an inner-product structure (or Hilbert space) and a closed linear subspace for uniqueness; does not generalize as an orthogonal projection in arbitrary metric or purely affine settings without specifying orthogonality.

Semantic Tension

Often confused with oblique projection (projection along a non-orthogonal complement) and with nearest-point maps in non-Euclidean geometries; unlike oblique projections, orthogonal projections are self-adjoint and minimize distance.

Synthesis

An orthogonal projection is the canonical nearest-point linear operator in an inner-product space: a self-adjoint, idempotent map that decomposes any vector orthogonally into its projected part and its orthogonal remainder.