Website owner:  James Miller

      Projection of a vector on a subspace

Projection of a vector on a subspace. Let L be a subspace of a vector space V. Then for any x in V there exists a unique representation x = y + z where y L and z L. The vector y is called the projection of vector x on subspace L. Vector y possesses the property that, as compared to other vectors of L, it is at least distance from X. Vector z is a member of the orthogonal complement M of subspace L. Thus we can say that every vector x of a vector space can be represented as a sum of vectors from a subspace L and its orthogonal complement M.

Example. Let V be three dimensional space, subspace L be any line through the origin, and x be any vector in V. Then M, the orthogonal complement of L, is a plane perpendicular to L passing through the origin, y is the projection of x on L, and z is the projection of x on M.