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Math Help - linearly independent of additive group

  1. #1
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    linearly independent of additive group

    Find the linearly independent subsets of the rational additive group over \mathbb{Z}

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  2. #2
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    Quote Originally Posted by roporte View Post

    Find the linearly independent subsets of the rational additive group over \mathbb{Z}

    Thanks!
    A \subset \mathbb{Q} is linearly independent over integers if and only if A=\{r\}, for some 0 \neq r \in \mathbb{Q}. such sets are obviously linearly independent. now suppose B \subset \mathbb{Q} is linearly independent over integers and

    |B| \geq 2. obviously 0 \notin B. choose any two elements of B, say r_1=\frac{a}{b}, \ r_2=\frac{c}{d}. then bcr_1 - adr_2 =0, which proves that \{r_1,r_2 \} and therefore B is not linearly independent over integers. Q.E.D.
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