True or False?
If S={v1, v2, v3}, T={v1, v2, v3, v1+v2}, and S is a basis, then T is not a basis.
Any help is appreciated. Thanx.
a subset B of V is called a basis for V if B is linear independent over F and B spans V.
suppose is basis, then there is no such that av1+bv2+cv3=0 unless a,b,c =0
is T a basis?
let
av1+bv2+cv3+d(v1+v2)=0
since V1 and V2 are in V hence d(v1+v2)= dv1+dv2
then we have (a+d)v1+(b+d)v2+cv3=0
clearly if a+d=0, then a need not to be 0 it could be -d
It is also true that all bases for a given vector space contain the same number of vectors. If S is a basis then T cannot be because it does not contain the same number of vectors as S.
You could also assert, similar to what Hamidr did, that v1+ v2- (v1+ v2)= 0 so the vectors in T are not independent.