for some c1,c2..., \(\displaystyle{c}{k}∈{R}\) where at least one of c1,c2...ck is non zero.

Now, for any there scalars c1,c2 and c3 the linear combination of vectors u-v,v-w, and w-u can be written as

c1(u-v)+c2(v-w)+c3(w-u)

Taking c1=c2=c3=1 we have c1(u-v)+c2(v-w)+c3(w-u)=(u-v)+(v-w)+(w-u)=0

Therefore, there is a combination of scalars \(\displaystyle{c}{1}={c}{2}={c}{3}={1}\frac{=}{{0}}\) so that c1(u-v)+c2(v-w)+c3(w-u)=0.

Hence, by the definition of linearly dependent set, the vectors u-v, v-w, and w-u form a linearly dependent set.