For any vectors u, v and w, show that the vectors u-v, v-w and w-u form a linearly dependent set.

For any vectors u, v and w, show that the vectors u-v, v-w and w-u form a linearly dependent set.

Question
Vectors and spaces
asked 2020-10-21
For any vectors u, v and w, show that the vectors u-v, v-w and w-u form a linearly dependent set.

Answers (1)

2020-10-22
Given that the vectors u,v, and w. We are need to show that the vectors u-v, v-w, and w-u form a linearly dependent set. The set of vectors {x1,x2...xk} is linearly dependent if c1x1+c2x2+...+ckxk=0
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.
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