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

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

$\displaystyle i_0=max(i|a_i\neq0)$ for which a_i differs 0

but its true only for i_0>=2

so we get the expression

$\displaystyle

v_{i0}=(\frac{-a_1}{}a_{i0})v_1+..+()v_i

$

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

??