The vectors v1 v2 v3 v4 v5 are linearly independent Determin

The vectors v₁, v₂, v₃, v₄, v₅ are linearly independent. Determine if the following vectors are also linearly independent. Show all work.

a) the vectors v₁+ v₂, v₂ + v₃, v₃ + v₄, v₄+ v₅, v₅- v₁

c) the vectors v₁ - v₂, v₂ - v₃, v₃ - v₄

if v₁, v₂, v₃, v₄, v₅ are linearly independent. then consider a, b , c, d , e as constants which satisfy,

av1+bv2+cv3+dv4+ev5 =0

then a = b = c = d = e = 0

a) So, let now the constants be k l m n o ,

k(v1+v2) + l (v2+v3) + m (v₃ + v_{4) + n(}v₄+ v_{5) + o (} v₅- v₁₎

(k- o)v1 + k+l(v2) + (l+m) v3 + (m+n) v4 + (n+ o) v5

so for these to be linearly independent, k - o = 0 ; k +l = 0 ; l + m = 0 ; m+n = 0 ; n + o = 0

b)apply the same steps for b) and c )