 Let u, v, and w be any three vectors from Nannie Mack 2021-08-15 Answered
Let u, v, and w be any three vectors from a vector space V. Determine whether the set of vectors $$\displaystyle{\left\lbrace{v}-{u},{w}-{v},{u}-{w}\right\rbrace}$$ is linearly independent or linearly dependent.

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Step 1
Given data:
The given space vector is: $$\displaystyle{V}={\left\lbrace{v}-{u},{w}-{v},{u}-{w}\right\rbrace}$$.
The expression to check whether the space vector is linearly dependent or independent is
$$\displaystyle{c}_{{{1}}}{\left({v}-{u}\right)}+{c}_{{{2}}}{\left({w}-{v}\right)}+{c}_{{{3}}}{\left({u}-{w}\right)}={\left({0},{0},{0}\right)}$$
Further, solve the above expression.
$$\displaystyle{c}_{{{1}}}{v}-{c}_{{{1}}}{u}+{c}_{{{2}}}{w}-{c}_{{{2}}}{v}+{c}_{{{3}}}{u}-{c}_{{{3}}}{w}={\left({0},{0},{0}\right)}$$
$$\displaystyle{\left({c}_{{{1}}}-{c}_{{{2}}}\right)}{v}-{\left({c}_{{{1}}}-{c}_{{{3}}}\right)}{u}+{\left({c}_{{{2}}}-{c}_{{{3}}}\right)}{w}={\left({0},{0},{0}\right)}$$
$$\displaystyle{c}_{{{1}}}-{c}_{{{2}}}={0}$$
$$\displaystyle{c}_{{{1}}}-{c}_{{{3}}}={0}$$
Step 2
The above equation gives the value of the scalar multiples.
$$\displaystyle{c}_{{{1}}}={c}_{{{2}}}={c}_{{{3}}}$$
As the $$\displaystyle{c}_{{{1}}},{c}_{{{2}}}$$, and $$\displaystyle{c}_{{{3}}}$$ are the scalar multiples of the given space vector, and they are equal, so the space vector is linearly dependent.
Thus, the given space vector is linearly dependent.