prove that vectors ,.., on a vectorinc space V over feild F

are linearly dependant if and only if there is an index 1<=i<=n

so is a lenear combination of the previus vectors by its index

,..,

??

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

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

??