Suppose S v1 v2 v3 is a subset of a vector space V and u i

Suppose S = {v_1, v_2, ....., v_3} is a subset of a vector space V and u is a vector with u isin Span(S). Show that {v_1, v_2, ....., v_ n, u}is linearly dependent.

Solution

u belongs to span(S) hence there exist:a1,...,an so tha

u=a1v1+...+anvn

a1v1+...+anvn+(-1)*u=0

Hence, {v1,...,vn,u} is a linearly dependent set