Suppose that v1 v2 i vm is a basis for of subspace W of Rn S

Suppose that {v_1, v_2, ...i, v_m} is a basis for of subspace W of R^n. Show that the orthogonal complement of W consists of those vectors that are orthogonal to all of these basis vectors.

Solution

Since { v₁ , v₂ , …, v_m } is a basis for a subspace W of R, for any arbitrary vector u in W, we have u.v_i = 0 for 1 i m as u W , while the v_i s W ( by definition itself, every vector in W is orthogonal to all the vectors in W). Thus, the orthogonal complement of W, i.e. W consist of those vectors that are orthogonal to all the basis vectors of W.