Vectors are linearly independent if there doesn't exist a non-trivial linear combination which returns the zero vector. So, we must see if we can find three coefficients [tex] (a,b,c)\neq(0,0,0) [/tex] such that
[tex] av_1+bv_2+cv_3 = (0,0,0,0) [/tex]. We have
[tex] av_1 = (2a, -a, 2a, 3a) [/tex]
[tex] bv_2 = (b, 2b, 5b, -b) [/tex]
[tex] cv_3 = (7c, -c, 5c, 8c) [/tex]
So, [tex] av_1+bv_2+cv_3 = (2a+b+7c, -a+2b-c, 2a+5b+5c, 3a-b+8c) [/tex]
This linear combination returns the zero vector if
[tex] 2a+b+7c = 0 [/tex]
[tex] -a+2b-c = 0 [/tex]
[tex] 2a+5b+5c = 0 [/tex]
[tex] 3a-b+8c = 0 [/tex]
This system admits the only trivial solution a=b=c=0, so the vectors are linearly independent.
