linear independence of unknown value Guys I am given a set of vectors $(v_1,\dots,v_{10}), \mathbb{R}^{10}, n=10$, which are linearly independent.  It doesn't give me the values of the vectors but tells me to find if these are independent or not. 
a. $v_1+v_2, v_2+v_3, v_3+v_4, v_4+v_1$
b. $v_1+v_2, 2(v_2+v_3), 3(v_3+v_4),..., 10(v_{10}+v_1)$
Not sure how to solve this.  I know the definition of independency but I am having trouble applying it.  
 A: Since $v_1+v_2 = (v_2+v_3)-(v_3+v_4)+(v_4+v_1)$ we see that the vector $v_1+v_2$ is a linear combination of the other $3$ vectors thus $v_1+v_2$ is in span$\{v_2+v_3,v_3+v_4,v_4+v_1\}$ and so the set in part a cannot be linear independent. You can adapt this argument to show that, in part b, $v_1+v_2$ is a linear combination of $v_2+v_3, v_3+v_4, .., v_{10}+v_1$ which shows that this set is not linearly independent either.
A: For a: So consider:
$$
a_1(v_1 + v_2) + a_2(v_2 + v_3) + a_3(v_3+v_4) + a_4(v_4 + v_1) = 0 .
$$
Can you prove from this that all the $a_i$'s are zero? You get that this is
$$
(a_1 + a_4)v_1 + (a_1 + a_2)v_2 + (a_2 + a_3)v_3 + (a_3 + a_4)v_4 = 0.
$$
So since the $v_i$'s are linearly independent, you get
$$
\begin{align}
a_1 + a_4 &= 0 \\
a_1 + a_2 &= 0 \\
a_2 + a_3 &= 0 \\
a_3 + a_4 &= 0.
\end{align}
$$
Does this imply that all the $a_i$'s are zero? This set of equations will have a solution if and only if the matrix
$$
\pmatrix{1 & 0 & 0 &1 \\ 1 & 1 & 0 & 0 \\0 & 1 & 1 & 0 \\ 0 & 0 & 1 & 1\\}
$$
is invertible. Is it?
For b: Try the same thing. Start with
$$
a_1(v_1 + v_2) + a_22(v_2+ v_3) + \dots + a_{10}10(v_{10} + v_1) = 0.
$$
Again just expand the left hand side and try to do as in part a. 
