Suppose v1 v2 vn are vectors and one of them is the zero vec

Suppose v_1, v_2, v_n are vectors and one of them is the zero vector. To be specific, suppose V_1 = 0. Show that v_1 v_2,..., v_n are linearly dependent. (Thus the zero vector is never part of a linearly independent set.)

Solution

The definition of linear dependence is : A set of n vectors { u1 , u2,,,,u_n} in a vector space S is said to be linearly dependent if there exist scalars a₁,a₂… …,a_n ,not all zero ,such a₁ u₁+a₂ u₂ + …+an un= 0 .

Let S = { 0 = u₁, u₂, …, u_n) be a set consisting of the zero vector. Then for some scalar p 0, pu₁+o u₂ + .+ o u_n = o . Hence, where a₁, a₂, …a_m are scalars, for the system a₁u₁ + a₂u₂ +… + a_mu_m = o, we have a non-zero solution a₁ = p, 0 = a₂ = …a_m. Therefore the set S is linearly dependent.