I don't really understand this proof given from the textbook:
Suppose some vectors $v_1,\ldots,v_m$ are linearly dependent and this $v_i$ is a redundant vector. $$v_i = c_1*v_1 +\cdots+c_{i-1}*v_{i-1}$$ Hence, $$-v_i + c_1*v_1+\cdots+c_{i-1}*v_{i-1} = 0$$
How exactly does this prove that $c_1*v_1+\cdots+c_m*v_m = 0$?
$c_i=-1$, as you can see just by subtracting $v_i$ from both sides... Just set $c_j=0$ for $j\gt i$...
Note: just one $c_k$ need be nonzero...
I.e. a nontrivial linear combination of the $v_1,\dots,v_m$ has been found which equals zero. It's nontrivial since $c_i=-1\not =0$...