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Math Help - linear combination problem dealing with uniqueness

  1. #1
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    linear combination problem dealing with uniqueness

    Let V be a vector space and S \subset V with the property that whenever v_1,v_2,...,v_n \in S and a_1{v_1}+a_2{v_2}+...+a_n{v_n} = 0, then a_1 = a_2 = ...=a_n = 0. Prove that every vector in the span of S can be uniquely written as a linear combination of vectors in S.
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  2. #2
    MHF Contributor kalagota's Avatar
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    Quote Originally Posted by lllll View Post
    Let V be a vector space and S \subset V with the property that whenever v_1,v_2,...,v_n \in S and a_1{v_1}+a_2{v_2}+...+a_n{v_n} = 0, then a_1 = a_2 = ...=a_n = 0. Prove that every vector in the span of S can be uniquely written as a linear combination of vectors in S.
    So, let s \in span \, S such that s = a_1{v_1}+a_2{v_2}+...+a_n{v_n} and s = b_1{v_1}+b_2{v_2}+...+b_n{v_n}. Then a_1{v_1}+a_2{v_2}+...+a_n{v_n} = b_1{v_1}+b_2{v_2}+...+b_n{v_n}. This means that (a_1-b_1){v_1}+(a_2-b_2){v_2}+...+(a_n-b_n){v_n} = 0. From the hypothesis, a_i - b_i = 0, \, \forall i=1,2,...,n. Therefore, a_i = b_i, \, \forall i=1,2,...,n
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