Thread: Prove/Disprove set T is a basis.

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.

Originally Posted by azazelahh

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.