- #1
transgalactic
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prove that vectors [tex]v_1[/tex],..,[tex]v_n[/tex] on a vectorinc space V over field F
are linearly dependant if and only if there is an index 1<=i<=n
so [tex]v_i[/tex] is a lenear combination of the previus vectors by its index
[tex]v_1[/tex],..,[tex]v_{i-1}[/tex]
??
i got a prove but i can't 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
[tex]i_0=max(i|a_i\neq0)[/tex] for which a_i differs 0
but its true only for i_0>=2
so we get the expression
[tex]
v_{i0}=(\frac{-a_1}{}a_{i0})v_1+..+()v_i
[/tex]
so there is a lenear dependence and we proved it.
the lecturer was in a hury
can you fill the gaps
make sense out of it
??
are linearly dependant if and only if there is an index 1<=i<=n
so [tex]v_i[/tex] is a lenear combination of the previus vectors by its index
[tex]v_1[/tex],..,[tex]v_{i-1}[/tex]
??
i got a prove but i can't 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
[tex]i_0=max(i|a_i\neq0)[/tex] for which a_i differs 0
but its true only for i_0>=2
so we get the expression
[tex]
v_{i0}=(\frac{-a_1}{}a_{i0})v_1+..+()v_i
[/tex]
so there is a lenear dependence and we proved it.
the lecturer was in a hury
can you fill the gaps
make sense out of it
??