In a finite-dimensional inner product space (Hilbert space) V, for a subspace W and any vector v in V, there exists a unique vector w in W, called the orthogonal projection of v onto W, such that the error vector v−w is orthogonal to every vector in W. This projection is characterized as the unique minimum-norm approximation of v by an element of W (a generalization of the Pythagorean theorem), can be expressed explicitly via an orthogonal basis of W as a sum of coefficients (v,αk)/‖αk‖² times αk, and defines a linear operator P_W on V. The theory belongs to linear algebra/functional analysis and underlies direct sum decomposition of vector spaces—particularly orthogonal decomposition, where projections onto mutually orthogonal subspaces add—forming the mathematical foundation for linear estimation and adaptive filtering.
In a finite-dimensional inner product space (Hilbert space) V, for a subspace W and any vector v in V, there exists a unique vector w in W, called the orthogonal projection of v onto W, such that the…