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Math Help - Prove/Disprove set T is a basis.

  1. #1
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    Question 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.
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  2. #2
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    Quote Originally Posted by azazelahh View Post
    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
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  3. #3
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    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.
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