Suppose S v1 v2 v n is a linearly independent subset of a

Suppose S = {v_1, v_2, ....., v_ n} is a linearly independent subset of a vector space V  and u is a vector in V with u not in Span(S)- Show that{v_1, v_2, ....., v_ n, u} linearly independent.

Solution

Let, {v1,...,vn,u} not be a lnearly independent set

So there exist:a1,..,an,b so that

a1v1+...+anvn+bu=0

Case 1:b=0

then since S is linearly independent so a1=...=an=0

HEnce, {v1,...,vn,u} is linearly independent set

Case 2: b not equal to 0

So, -bu=a1v1+.....+anvn

u=-a1/b v1-...-an/b vn

But u does not lie in span of S

SO a contradiction

HEnce, b not equal to 0 is not possible

Hence proved