Suppose V is a subspace of W and let beta upsilon1 upsilon

Suppose V is a subspace of W and let beta = {upsilon_1, ..., upsilon_n} are linearly independent vectors in V. If w is a vector in W but w is not in V, show that {upsilon_1, ...., upsilon_n, w} are linearly independent.

Solution

Every vector v in V is a linear combination of v₁,v₂,…,v_n . If w W , but w V, then w cannot be expressed as a linear combination of v₁,v₂,…,v_n. Hence { v₁,v₂,…,v_n ,w} are linearly independent.