Let S v1 v2 vn be a linearly independent subset of an inner

Let S = (v_1, v_2 v_n) be a linearly independent subset of an inner product space V, and w element V where w is orthogonal to each vector in S. Prove, using only the definition of linear independence, orthogonal vectors and the inner product space axioms that S union {w} is also linearly independent.

Solution

Let w be orthogonal to each v_i in S. Then w.v_i = 0 for each v_i . Then w is in the orthogonal complement of S. This means that w is not in span {v₁ , v₂ , ..., v_n} . Thus w can not be expressed as a linear combination of v₁ , v₂ , ..., v_n . Therefore, w, v₁ , v₂ , ..., v_n are linearly independent . Thus S U [w} is linearly independent.