let S = v1 ,v2,v3, v4 , where v1 = [121]^T v2 =[-1-1 1 ] ^T v3 = [ -1 1 7 ]^T v4 = [-2 -4 -4 ] ^T find every subset of S that is a basis for R^3 the answers are [v1, v2 , v3 ] , [v1,v2,v4] and [v1,v3,v4] I some what get it but im stuck so if u could explain it to me step by step i would be very grateful. thank u :) Solution basis is the collection of the subspaces,that uniquely define the vector space. for example,for v1,v2,v3 to be a basis, there should exist unique a ,b and c such that a*v1+b*v2+c*v3=v4 now the three resulting equations are: a-b-c=-2 2*a-b+c=-4 a+b+7*c=-4 solving for a,b,c we get a=0 b=3 c=-1 so as a,b,c exist uniquely,[v1,v2,v3] consist a basis. similar analysis can be done for other sets. .

1. let S = v1 ,v2,v3, v4 , where
v1 = [121]^T
v2 =[-1-1 1 ] ^T
v3 = [ -1 1 7 ]^T
v4 = [-2 -4 -4 ] ^T
find every subset of S that is a basis for R^3
the answers are
[v1, v2 , v3 ] , [v1,v2,v4] and [v1,v3,v4]
I some what get it but im stuck so if u could explain it to me step by step i would be very
grateful. thank u :)
Solution
basis is the collection of the subspaces,that uniquely define the vector space.
for example,for v1,v2,v3 to be a basis,
there should exist unique a ,b and c such that a*v1+b*v2+c*v3=v4
now the three resulting equations are:
a-b-c=-2
2*a-b+c=-4
a+b+7*c=-4

2. solving for a,b,c
we get
a=0
b=3
c=-1
so as a,b,c exist uniquely,[v1,v2,v3] consist a basis.
similar analysis can be done for other sets.