prove that vectors v_1,.., 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 v_i is a lenear combination of the previus vectors by its index
v_1,.., v_{i-1}
??

i got a prove but i cant fully understand it:
suppose v_i is a lenear combination of its previous
v_i=a_1v_1+..+a_i-1v_i-1

we transfer v_i on the other side
0=-1v_1+a_1v_1+..a_i-1v_i-1

then they say that
0c_i+..+0v_1
so it lenear dependant

(why??)
then they pick an index
all the index are 0 except the first one which is not
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

<br />
v_{i0}=(\frac{-a_1}{}a_{i0})v_1+..+()v_i<br />
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
??