prove that vectors $\displaystyle v_1$,..,$\displaystyle v_n$ on a vectorinc space V over feild F
are linearly dependant if and only if there is an index 1<=i<=n
so $\displaystyle v_i$ is a lenear combination of the previus vectors by its index
$\displaystyle v_1$,..,$\displaystyle v_{i-1}$

i got a prove but i cant fully understand it:
suppose v_i is a lenear combination of its previous

we transfer v_i on the other side

then they say that
so it lenear dependant

then they pick an index
all the index are 0 except the first one which is not
$\displaystyle i_0=max(i|a_i\neq0)$ for which a_i differs 0

but its true only for i_0>=2

so we get the expression

so there is a lenear dependance and we proved it.

the lecturer was in a hury
can you fill the gaps
make sense out of it