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
??