Def. of lin independence for subsets I use following definition:


  let be $v:\Bbb{N} \to T$ with $T\subseteq V $ and $V $ a $K-$vector space: $$ (v_i)_{i \in \Bbb{N}} \text{ is lin. ind. if } \\ \forall \alpha \in K^\Bbb{N}: \sum_{i \in \Bbb{N}} \alpha_i \, v_i=0 \wedge \exists m \in \Bbb{N}:\forall n&gt;m: \alpha_n=0 \to \forall i \in \Bbb{N}: \alpha_i=0$$


Now I need a definition not for family of vectors but for subsets of $V $ . I thinked $T \subseteq V$ is lin ind if $\forall x\in T: x \notin &lt;T\setminus \{x\}&gt; $ but I have following problem: 


  the family of vectors  $((1,2), (1,2)) $ ist not lin ind, but the set $\{(1,2)\} $ ist lin ind...


Are there therefore two definitions of lin ind, one for families and one for sets?!
By definition singleton  sets are always linearly independent.

Let $\{\alpha \neq 0\}$ be a singleton  set. $a_1 \alpha = 0 \implies a_1 = 0 $.

Now $a_1(1,2) + a_2 (1,2) = (0,0) \implies (a_1 +a_2, 2a_1+2a_2) = (0,0) \implies a_1 = -a_2 $. Thus the family of vectors  $((1,2), (1,2)) $ is not linearly independent.
By definition singleton  sets are always linearly independent.

Let $\{\alpha \neq 0\}$ be a singleton  set. $a_1 \alpha = 0 \implies a_1 = 0 $.

Now $a_1(1,2) + a_2 (1,2) = (0,0) \implies (a_1 +a_2, 2a_1+2a_2) = (0,0) \implies a_1 = -a_2 $. Thus the family of vectors  $((1,2), (1,2)) $ is not linearly independent.