Expert Answer to "For any vectors u, v and w, show..."

College
Algebra
Linear algebra
Vectors and spaces

necessaryh

Answered question

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.

Answer & Explanation

funblogC

Skilled2020-10-22Added 91 answers

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 $x_{1}, x_{2} \dots x_{k}$ is linearly dependent if $c_{1} x_{1} + c_{2} x_{2} + \dots + c_{k} x_{k} = 0$
for some $c_{1}, c_{2} \dots,$ $c k \in R$ where at least one of $c_{1}, c_{2} \dots c_{k}$ is non zero.
Now, for any there scalars $c_{1}, c_{2} and c_{3}$ the linear combination of vectors u-v,v-w, and w-u can be written as
$c_{1} (u - v) + c_{2} (v - w) + c_{3} (w - u)$
Taking $c_{1} = c_{2} = c_{3} = 1$ we have $c_{1} (u - v) + c_{2} (v - w) + c_{3} (w - u) = (u - v) + (v - w) + (w - u) = 0$
Therefore, there is a combination of scalars $c 1 = c 2 = c 3 = 1 \frac{=}{0}$ so that $c_{1} (u - v) + c_{2} (v - w) + c_{3} (w - u) = 0.$
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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