Let S v 1 v 2 v n be a linearly independent subset of an i

Let S = {v 1 , v 2,..., v n} be a linearly independent subset of an inner product space V, and w 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 {w} is also linearly independent.

Solution

Given that S = {v 1 , v 2,..., v n} is a linearly independent subset of V, vector space.

w is another vector in V which is orthogonal to each vector in S.

Since w is orthogonal to each vi, w and vi are linearly independent as mutually perpendicular for each i.

Hence it follows that w cannot be represented as a linear combination v1, v2...vn.

So SU{w} is again linearly independent.

Note: If w is not orthogonal to even one vi, then SU{w} becomes linearly dependent.