linearly independent elements of W I want to ask about the number of linearly independant elements of 
$W=(u_{1},u_{2},...,u_{m},u_{1}-u_{2},u_{2}-u_{3},...,u_{m-1}-u_{m},u_{m}-u_{1})$.
I say that it is $m$ and i need someone to confirm or not that.
Thanks 

The answer depends on the number of linearly independent vectors among $u_1, \ldots, u_m$.
Notice that $$\operatorname{span}\{u_1, u_2, \ldots, u_m, u_1 - u_2, u_2 -
 u_3, \ldots, u_{m-1} - u_m, u_m - u_1\} = \operatorname{span}\{u_1, u_2, \ldots, u_m\}$$
because all $u_i - u_{i+1}$ are linear combinations of $u_1, u_2, \ldots, u_m$.
Thus, the number of linearly independent vectors in $W$ is equal to the number of linearly independent vectors in $\{u_1, u_2, \ldots, u_m\}$.
In particular, assuming $u_1, u_2, \ldots, u_m$ are linearly independent, the number or linearly independent vectors in $W$ is $m$.
