Linear independence of a set of vectors and vectors alone I have a linear algebra book with definition of linear independence like:
"Let $x_1, x_2, ..., x_k$ be vectors in $L$. If $k\geq2$, then the vectors $x_1, x_2, ..., x_k$ are said to be linearly dependent..."
And there are some problems (they aren't written in English, so I translate them):  Find out if a group of vectors is linear dependent or not.
I don't understand what is meant by "a group of vectors". I've tried to read about groups, but I don't think that a mathematical group and a group of vectors in my book are similar groups. In addition, all definitions which I found on Wikipedia (in different languages) are definitions of linear independence of a set, not just a "group" of vectors. 
What does that mean? Can we talk about linear independence of something which is not a set? 
 A: The word "group" is unusual in an English context, and I am quiet certain it has nothing to do with the algebraic object group. 
Beware though that there are two slightly different ways to define linear independence, namely for families and for sets. 


*

*A family $(v_i)_{i \in I}$ in $V$ is called linearly independent if 
$\sum_{i \in I} a_i v_i = 0$ with $a_i \in K$, and $a_i=0$ for all but finitely many $a_i$ implies that $a_i=0$ for all $I$.   

*A set $S$ in $V$ is called linearly independent if 
$\sum_{v \in S} a_v v = 0$ with $a_v \in K$,  and $a_v=0$ for all but finitely many $v$, implies that $a_v=0$ for all $v \in S$.
The difference is that in the former some element of the vector space could appear more than once. For example you could have the family $(v_1, v_2, v_3)$ where $v_1= v_2 = (1,1)$ and $v_3= (-1, 2)$ in $\mathbb{R}^2$ and this is not independent as $1v_1 + (-1)v_2 + 0 v_3 = (0,0)$.
But the set $\{v_1, v_2, v_3\}$ is independent as it is equal to $\{v_1, v_3\}$ as there are no repetitions in sets.
Thus, there is some need to be careful regarding what precisely is meant. It could be that your word "group" actually is the analogue of "family" in that language.     
