If W is a k-dimensional subspace of a vector space V with inner product 〈, 〉, then it is possible to project vectors from V to W. The most familiar projection is when W is the x-axis in the plane. In this case, P(x, y) = (x, 0) is the projection. This projection is an orthogonal projection. If the subspace W has an orthonormal basis {w_1, ..., w_k} then proj_W(v) = sum_(i = 1)^k 〈v, w_i〉 w_i is the orthogonal projection onto W. Any vector v element V can be written uniquely as v = v_W + v_W^⊥, where v_W element W and v_W^⊥ is in the orthogonal subspace W^⊥.

idempotent | inner product | orthogonal set | projection | projection matrix | symmetric matrix | vector space